// Numbas version: exam_results_page_options {"variable_groups": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Calculate line integral over a straight line", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..9)", "description": "", "name": "b1"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a=b1,b1+1,b1)", "description": "", "name": "b"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s"}}, "ungrouped_variables": ["a", "s", "b", "b1"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

(a) What is the equation of the line joining $(0,0)$ to $(\\var{a},\\var{b})$?

\n

$y=\\;\\;$[[0]] Input all numbers in your answer as integers or fractions, not as decimals.

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(b) Evaluate the line integral for $I$:

\n

$I=\\;\\;$[[1]] Input your answer as an integer or a fraction, not as a decimal

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Input all numbers in your answer as integers or fractions, not as decimals.

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Input all numbers in your answer as integers or fractions, not as decimals.

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Consider the line integral:
\$I=\\int_{\\Gamma} \\left( \\left(\\simplify[std]{x+y} \\right)\\;dx+\\left(\\simplify[std]{y-x}\\right)\\;dy\\right)\$

\n

where $\\Gamma$ is the path given by the straight line from $(0,0)$ to $(\\var{a},\\var{b})$

", "tags": ["Calculus", "calculus", "checked2015", "integration", "line integral", "path", "straight line", "Straight Line", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find $\\displaystyle \\int_{\\Gamma} \\left(x+y \\right)\\;dx+\\left(y-x\\right)\\;dy,\\;\\Gamma$ is the line from $(0,0)$ to $(a,b)$.

\n

"}, "type": "question", "advice": "

For a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$, the line equation is $\\displaystyle{y-y_1 = m(x-x_1)}$, where the gradient $\\displaystyle{m=\\frac{y_2-y_1}{x_2-x_1}}$.  Hence the equation of our line is:

\n

\$\\simplify[std]{y=({b}/{a})*x}\$

\n

Since $\\displaystyle{\\simplify[std]{y=({b}/{a})*x}}$ we have that $\\displaystyle{\\simplify[std]{dy=({b}/{a})*dx}}$.  Using these relations we can write the integrand of the line integral in terms of just $x$ or just $y$.  We will use $x$.  The integrand then becomes:
\$\\begin{eqnarray*} (x+y)\\;dx+(y-x)\\;dy&=&\\simplify[std]{(x+({b}/{a})*x)dx+(({b}/{a})*x-x)*({b}/{a})*dx}\\\\ &=&\\simplify{{a^2+b^2}/{a^2}}x\\;dx \\end{eqnarray*} \$

\n

In $x$, our line integral exists over the range $0 \\leq x\\leq\\var{a}$, and so we write our line integral as:
\$\\begin{eqnarray*} I&=&\\int_\\Gamma\\simplify{{a^2+b^2}/{a^2}}x\\;dx =\\int_0^{\\var{a}}\\simplify{{a^2+b^2}/{a^2}}x\\;dx \\\\ &=&\\simplify{{a^2+b^2}/{a^2}}\\left[\\frac{x^2}{2}\\right]_0^{\\var{a}}\\;dx =\\simplify{{a^2+b^2}/{a^2}}\\times \\frac{\\var{a^2}}{2}=\\simplify[std]{{a^2+b^2}/2} \\end{eqnarray*} \$

"}, {"name": "Dot and cross product combinations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["

$\\boldsymbol{(A\\cdot B)\\cdot C}$

", "

$\\boldsymbol{(A\\cdot B)C}$

", "

$\\boldsymbol{(A\\cdot B)\\times C}$

", "

$\\boldsymbol{(A\\times B)\\times C}$

", "

$\\boldsymbol{(A\\times B)\\cdot C}$

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Scalar

", "

Vector

", "

Undefined

"], "warningType": "none"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Given the vectors $\\boldsymbol{A},\\;\\;\\boldsymbol{B}$ and $\\boldsymbol{C}$ in $3$ dimensional space, state whether the following quantities are scalars, vectors or undefined.

", "tags": ["checked2015", "cross product", "dot product", "inner product", "mas1602", "MAS2104", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

15/07/2012:

\n \t\t

\n \t\t

16/07/2012:

\n \t\t

\n \t\t

\n \t\t

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Determine if various combinations of vectors are defined or not.

"}, "advice": "\n \n \n

1. $\\boldsymbol{(A\\cdot B)\\cdot C}$ is undefined as $\\boldsymbol{A\\cdot B}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{C}$.

\n \n \n \n

2. $\\boldsymbol{(A\\cdot B)C}$ is a vector and is a multiple of $\\boldsymbol{C}$ as $\\boldsymbol{A \\cdot B}$ is a scalar.

\n \n \n \n

3. $\\boldsymbol{(A\\cdot B)\\times C}$ is undefined as $\\boldsymbol{A\\cdot B}$ is a scalar and the cross product is only defined between vectors.

\n \n \n \n

4. $\\boldsymbol{(A\\times B)\\times C}$ is a vector as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the cross product between vectors produces a vector.

\n \n \n \n

5. $\\boldsymbol{(A\\times B)\\cdot C}$ is a scalar as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the inner or dot product is between vectors and produces a scalar.

\n \n \n "}, {"name": "Elementary operations on vectors, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(a,b,g)", "name": "v1", "description": ""}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s5", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..9)", "name": "c", "description": ""}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "v1+v2", "name": "v", "description": ""}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(c,d,f)", "name": "v2", "description": ""}, "ssquaresa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a)^2+(b)^2+(g)^2", "name": "ssquaresa", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)", "name": "b", "description": ""}, "ssquaresb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c)^2+(d)^2+(f)^2", "name": "ssquaresb", "description": ""}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "M+N", "name": "q", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([a,b],[c,d])", "name": "m", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "f", "description": ""}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-random(3..9)", "name": "b4", "description": ""}, "ssquares": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a+c)^2+(b+d)^2+(f+g)^2", "name": "ssquares", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([a,b],[c,d])", "name": "n", "description": ""}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "name": "a4", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s3", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "name": "g", "description": ""}}, "ungrouped_variables": ["b4", "q", "s3", "s2", "s1", "s5", "s4", "ssquares", "v1", "v2", "a4", "a", "c", "b", "d", "g", "f", "m", "n", "ssquaresb", "ssquaresa", "v"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find $\\boldsymbol{v}+\\boldsymbol{w} =$ [[0]]

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Calculate the following.

\n

$\\lVert \\boldsymbol{v} \\rVert=$ [[0]]

\n

$\\lVert \\boldsymbol{w} \\rVert =$ [[1]]

\n

$\\lVert \\boldsymbol{v}+\\boldsymbol{w} \\rVert =$ [[2]]

Let $\\boldsymbol{z}=\\boldsymbol{v}+\\boldsymbol{w}$.

\n

Find the unit vector $\\boldsymbol{u_z}$ in the direction of $\\boldsymbol{z}$. Write $\\boldsymbol{u_z}$ as a row vector.

\n

$\\boldsymbol{u_z}= \\big($ [[0]], [[1]], [[2]] $\\big)$

\n

You must enter your answers exactly, using the function sqrt(x) if necessary.

Find

\n

$\\var{a4}\\boldsymbol{v} =$ [[0]]

\n

$\\var{b4}\\boldsymbol{w} =$ [[1]]

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Find the unit vector $\\boldsymbol{u_v}$ parallel to $\\boldsymbol{v}$, and the unit vector $-\\boldsymbol{u_w}$ anti-parallel to $\\boldsymbol{w}$. Write both vectors as row vectors.

\n

$\\boldsymbol{u_v} = \\big($ [[0]], [[1]], [[2]] $\\big)$

\n

$-\\boldsymbol{u_w} = \\big($ [[3]], [[4]], [[5]] $\\big)$

You are given the vectors

\n

\\begin{align}
\\boldsymbol{v} & =\\simplify[std]{vector({a},{b},{g})}, &
\\boldsymbol{w} &= \\simplify[std]{vector({c},{d},{f})}\\qquad \\in{\\mathbb R}^3.
\\end{align}

\n

Enter your answers to the following questions exactly, using the function sqrt(x) if necessary.

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Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.

#### a)

\n

\$\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \$

\n

#### b)

\n

In general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.

\n

Hence:

\n

\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\sqrt{\\var{a^2}+\\var{b^2}+\\var{g^2}} = \\simplify[all]{ sqrt({a^2+b^2+g^2})} \\\\
\\lVert \\boldsymbol{w} \\rVert &= \\sqrt{\\var{c^2}+\\var{d^2}+\\var{f^2}} = \\simplify[all]{ sqrt({c^2+d^2+f^2})} \\\\
\\lVert \\boldsymbol{v+w} \\rVert &= \\sqrt{\\var{(a+c)^2}+\\var{(b+d)^2}+\\var{(g+f)^2}} = \\simplify[all]{ sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}
\\end{align}

\n

#### c)

\n

Given a vector $\\boldsymbol{x}= \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, the unit vector parallel to $\\boldsymbol{x}$ is given by:

\n

\$\\boldsymbol{u_x} = \\frac{1}{\\lVert \\boldsymbol{x} \\rVert} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix} = \\begin{pmatrix} \\frac{x_1}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_2}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_3}{\\lVert \\boldsymbol{x} \\rVert} \\end{pmatrix} \$

\n

For this example we have $\\lVert \\boldsymbol{v+w} \\rVert =\\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$, hence:

\n

\\begin{align}
&&\\boldsymbol{z} = \\boldsymbol{v} + \\boldsymbol{w} &= \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\
\\implies && \\boldsymbol{u_z} &= \\frac{1}{\\sqrt{\\var{ssquares}}} \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\1em] && &= \\begin{pmatrix} \\simplify[std]{{a+c}/sqrt({ssquares})} \\\\ \\simplify[std]{{b+d}/sqrt({ssquares})} \\\\ \\simplify[std]{{g+f}/sqrt({ssquares})} \\end{pmatrix} \\end{align} \n #### d) \n \\begin{align} \\var{a4}\\boldsymbol{v} &=\\var{a4}\\var{vector(a,b,g)}\\\\[1em] &= \\simplify{vector({a4}*{a}, {a4}*{b}, {a4}*{g})}\\\\[1em] \\end{align} \n \\begin{align} \\var{-b4}\\boldsymbol{v} &=\\var{-b4}\\var{vector(c,d,f)}\\\\[1em] &= \\simplify{vector({-b4}*{c}, {-b4}*{d}, {-b4}*{f})}\\\\[1em] \\end{align} \n \n Using the information above, the unit vector parallel to \\boldsymbol{v} is: \n \\[ \\boldsymbol{u_v} = \\begin{pmatrix} \\simplify[std]{{a}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{b}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{g}/sqrt({ssquaresA})} \\end{pmatrix} \

\n

and the unit vector anti-parallel to $\\boldsymbol{w}$ is:

\n

\$-\\boldsymbol{u_w} = \\begin{pmatrix} \\simplify[std]{{-c}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-d}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-f}/sqrt({ssquaresB})} \\end{pmatrix} \$

"}, {"name": "Evaluate double integrals with numerical limits, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "d", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "b", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "name": "g", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "parts": [{"scripts": {}, "gaps": [{"answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "

Input all numbers in your answer as integers or fractions, not as decimals.

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\$I=\\int^\\var{a}_{y=1} \\int^\\var{b}_{x=0} \\left(\\var{c}+\\simplify[std]{{4*d}xy} \\right) dx\\, dy \$

\n

$I=\\;\\;$[[0]]

", "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-h^(f+1)*((-1)^g-1)/(g*(f+1))}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "fractionnumbers", "type": "jme", "showCorrectAnswer": true, "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

\$I=\\int^\\pi_{x=0} \\int^\\var{h}_{y=0} \\simplify[std]{y^{f}sin({g}x)} dy \\, dx \$

\n

$I=\\;\\;$[[0]]

", "marks": 0}], "statement": "

Evaluate the following double integrals.

\n

Input your answer as an integer or a fraction, not as a decimal.

", "tags": ["Calculus", "calculus", "checked2015", "double integral", "MAS1603", "MAS2104", "tested1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

30/06/2012:

\n

\n

Minor change to prompt.

\n

19/07/2012:

\n

\n

Did not add Show steps.

\n

Checked calculation.

\n

23/07/2012:

\n

\n

22/12/2012:(WHF)

\n

Corrected mistake in last part, the upper limit in the integral was set as the value of a which was the upper limit in the first part, but it should have been the value of h.

\n

Checked calculations, OK. Added tested1 tag.

\n

\n

\n

\n

\n

Question appears to be working correctly.

\n

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Double integrals (2) with numerical limits

"}, "showQuestionGroupNames": false, "advice": "

(a) We proceed to evaluate the double-integral:

\n

\$\\begin{eqnarray*} I&=&\\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}xy} \\right) dx dy \\\\ &=& \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]^\\var{b}_0 dy \\\\ &=&\\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right) dy \\\\ &=& \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1 dy \\\\ &=&\\simplify[std]{{c*b*a}+{d*b^2*a^2}-{c*b}-{d*b^2}} \\\\ &=&\\simplify[std]{{(c*b*a)+(d*b^2*a^2)-(c*b)-(d*b^2)}}\\end{eqnarray*}\$

\n

(b) \$\\begin{eqnarray*} I&=&\\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} dy dx \\\\ &=& \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]^\\var{h}_0 dx \\\\ &=& \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)} dx \\\\ &=& \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0 \\\\ &=& -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\ &=& \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}}\\end{eqnarray*}\$

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$I=\\;\\;$[[0]]

", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Green’s theorem states that for a region R with boundary $\\Gamma$

\n

\$\\oint_{\\Gamma} \\left( u\\;dx+v\\;dy \\right)= \\int \\int_R\\left(\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}\\right)\\;dx\\;dy.\$ Use Green’s theorem to find the value of: \$I=\\oint_{\\Gamma} \\left( \\left(\\simplify[std]{{a}x^2-{b}y} \\right)\\;dx+\\left(\\simplify[std]{{c}y^2+{d}x}\\right)\\;dy\\right)\$

\n

where $\\Gamma$, mapped counter-clockwise, is the closed path, starting at $(0,0)$, around the boundary of a rectangle with vertices $(0,0),\\;(\\var{f},0),\\;(\\var{f},\\var{g}),\\;(0,\\var{g})$.

", "tags": ["Calculus", "calculus", "checked2015", "closed path", "differentiation", "Green's theorem", "green's theorem", "Greens theorem", "greens theorem", "integral over a closed path", "integral over a rectangle", "integral over a region", "line integral", "MAS2104", "partial differentiation", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

30/06/2012:

\n

Added tags. Could include Show steps on Green's theorem.

\n

19/07/2012:

\n

Added Show steps on Green's Theorem.

\n

\n

Checked calculation.

\n

23/07/2012:

\n

\n

In the question and Steps added brackets so that Green's Theorem is valid.

\n

Question appears to be working correctly.

\n

23/12/2012: (WHF)

\n

No Shoe steps on Green's theorem.

\n

Checked calculations, OK. Added tested1 tag. Few minor typos - full stops added.

\n

", "licence": "Creative Commons Attribution 4.0 International", "description": "

(Green’s theorem). $\\Gamma$ a rectangle, find: $\\displaystyle \\oint_{\\Gamma} \\left(ax^2-by \\right)\\;dx+\\left(cy^2+px\\right)\\;dy$.

First we identify the functions $u$ and $v$ and their required derivatives:

\n

\$\\begin{eqnarray*} u &=& \\simplify[std]{{a}x^2-{b}y} \\Rightarrow \\frac{\\partial u}{\\partial y} = \\simplify[std]{-{b}} \\\\ v&=&\\simplify[std]{{c}y^2+{d}x} \\Rightarrow \\frac{\\partial v}{\\partial x} = \\simplify[std]{{d}}\\end{eqnarray*}\$

\n

Hence: \$\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}=\\simplify[std]{{d}-{-b}}=\\var{d+b}.\$

\n

So, using Green’s Theorem, the integral $I$ becomes:

\n

\$\\begin{eqnarray*} I&=&\\int \\int_R\\left(\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}\\right)\\;dx\\;dy\\\\ &=&\\int \\int_R \\var{d+b}\\;dx\\;dy = \\var{d+b}\\int \\int_R dx\\;dy\\\\ &=&\\var{d+b}\\times \\textrm{Area of }R. \\end{eqnarray*}\$

\n

Now the region $R$ is a rectangle of size $\\var{f}$ by $\\var{g}$.  Hence:

\n

\$\\begin{eqnarray*} I&=&\\var{b+d}\\times \\var{f} \\times \\var{g}\\\\ &=& \\var{(b+d)*f*g}. \\end{eqnarray*} \$

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Point of intersection of the two lines

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Find the vector equation of Line 1, which passes through the points $\\boldsymbol{x_0}$ and $\\boldsymbol{x_1}$.

\n

Input the vector equation in the form:

\n

\$\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \$

\n

such that $\\boldsymbol{r} = \\boldsymbol{x_0}$ when $\\lambda=0$ and $\\boldsymbol{r}=\\boldsymbol{x_1}$ when $\\lambda=1$ by filling in the appropriate fields below:

\n

$\\boldsymbol{r} =$ [[0]] $+ \\lambda$ [[1]]

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Now find the vector equation of Line 2, which passes through the points $\\boldsymbol{y_0}$ and $\\boldsymbol{y_1}$ in the form

\n

\$\\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \$

\n

such that $\\boldsymbol{r}=\\boldsymbol{y_0}$ when $\\mu=0$ and $\\boldsymbol{r}=\\boldsymbol{y_1}$ when $\\mu=1$ by filling in the appropriate fields below:

\n

$\\boldsymbol{r} =$ [[0]] $+ \\mu$ [[1]]

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You are told that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.

\n

Find $\\boldsymbol{P}$.

\n

$\\boldsymbol{P} =$ [[0]]

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You are given the vectors

\n

\\begin{align}
\\boldsymbol{x_0} &= \\var{vector(a,b,g)} , & \\boldsymbol{x_1} & = \\var{vector(a+c,b+d,g+f)}, \\\1em] \\boldsymbol{y_0} &= \\var{vector(a1,b1,g1)}, & \\boldsymbol{y_1} &=\\var{vector(a1+c1,b1+d1,g1+f1)} \\end{align} \n in \\mathbb{R^3}. ", "tags": ["checked2015", "equation of a line", "equation of a line through a vector in the direction of another vector", "Finding a common point for two lines in three dimensional space", "intersection of two lines in three dimensional space", "lines in three dimensional space", "three dimensional geometry", "vector equation of a line", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Given a pair of 3D position vectors, find the vector equation of the line through both. Find two such lines and their point of intersection. "}, "advice": " #### a) \n For \\lambda=0 we have \\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}\

\n

and we want this to be equal to $\\boldsymbol{x_0}$.  So we need $a_1 = \\var{a}$, $a_2 = \\var{b}$, and $a_3 = \\var{g}$.

\n

For $\\lambda=1$ we need $\\boldsymbol{r}=\\boldsymbol{x_1}$, and so

\n

\$\\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} = \\var{vector(a+c,b+d,g+f)}\$

\n

which tells us that $b_1=\\var{c}$, $b_2=\\var{d}$, and $b_3=\\var{f}$.  Thus the equation for Line 1 is

\n

\$\\boldsymbol{r} = \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)}\$

\n

#### b)

\n

Proceeding as in part a), we find that

\n

\$\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\$

\n

#### c)

\n

Write out a set of simultaneous equations for each component of $\\boldsymbol{P}$:

\n

\\begin{align}
\\simplify[]{{a} + lambda*{c}} &= \\simplify[]{{a1} + mu*{c1}} \\\\
\\simplify[]{{b} + lambda*{d}} &= \\simplify[]{{b1} + mu*{d1}} \\\\
\\simplify[]{{g} + lambda*{f}} &= \\simplify[]{{g1} + mu*{f1}}
\\end{align}

\n

By solving these equations, we find that the point $\\boldsymbol{P}$ common to both lines is given by $\\lambda=\\var{lam},\\mu=\\var{mu}$, and

\n

\$\\boldsymbol{P} = \\var{p}\$

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$\\boldsymbol{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$ and $\\boldsymbol{b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$

\n

$\\cos({\\theta})=$ [[0]].  (Enter your answer to 2d.p.)

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$\\boldsymbol{c}=\\pmatrix{\\var{c[0]},\\var{c[1]},\\var{c[2]},\\var{c[3]}}$ and $\\boldsymbol{d}=\\pmatrix{\\var{d[0]},\\var{d[1]},\\var{d[2]},\\var{d[3]}}$

\n

$\\cos({\\theta})=$ [[0]].  (Enter your answer to 2d.p.)

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Find the cosine of the angle $\\theta$ between the following pairs of vectors.

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Find the cosine of the angle between two pairs of 3D and 4D vectors.

\n

The calculations and answers are correct, however the Advice should display the interim calculations of the lengths of vectors and their products to say 6dps. At present the student may be mislead into using 2dps at each stage - the instruction at the start of Advice is somewhat confusing.

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

\n

The dot product of two vectors $\\boldsymbol{a}=\\pmatrix{a_1,a_2,a_3}$ and $\\boldsymbol{b}=\\pmatrix{b_1,b_2,b_3}$ is given by

\n

\$\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3.\$

\n

It is also given by

\n

\$\\boldsymbol{a\\cdot b}=ab\\cos(\\theta)\$

\n

where $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$ are the lengths of the vectors $\\boldsymbol{a}$ and $\\boldsymbol{b}$.

\n

Equating the two expressions gives

\n

\$a_1b_1+a_2b_2+a_3b_3=ab\\cos(\\theta)\$

\n

and so

\n

\$\\cos(\\theta)=\\frac{a_1b_1+a_2b_2+a_3b_3}{ab}.\$

\n

In part a) therefore, we have

\n

\$\\cos(\\theta)=\\frac{\\simplify[std]{{a[0]*b[0]}+{a[1]*b[1]}+{a[2]*b[2]}}}{\\var{precround(lena,2)}\\times\\var{precround(lenb,2)}}=\\frac{\\var{dot(a,b)}}{\\var{precround(lena*lenb,2)}}=\\var{ans1} \\; \\text{to 2d.p.,}\$

\n

and in part b) we have

\n

\$\\cos(\\theta)=\\frac{\\simplify[std]{{c[0]*d[0]}+{c[1]*d[1]}+{c[2]*d[2]}+{c[3]*d[3]}}}{\\var{precround(lenc,2)}\\times\\var{precround(lend,2)}}=\\frac{\\var{dot(c,d)}}{\\var{precround(lenc*lend,2)}}=\\var{ans2} \\; \\text{to 2d.p.}\$

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{formincorrectchoice2}, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$

", "

{formincorrectchoice3}, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$

"], "prompt": "

What is the correct parametric representation of the curve?

", "unitTests": [], "variableReplacements": [], "distractors": ["", "", "", ""], "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 1, "marks": 0}, {"displayType": "radiogroup", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [1, 0], "choices": ["

{dircorrectchoice}

", "

{dirincorrectchoice}

"], "prompt": "

In which direction is the curve traversed?

", "unitTests": [], "variableReplacements": [], "distractors": ["", ""], "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 1, "marks": 0}], "variables": {"g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, random(1..9)*sign(random(-1,1)),\n false\n )", "name": "g1", "description": ""}, "formincorrectchoice3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, '$t\\\\rightarrow\\\\pmatrix{t,t^3}$',\n index=1, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},\\\\simplify{t^{d1}}}$',\n index=2, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},\\\\simplify{t^{d1}}}$',\n index=3, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},\\\\arctan(t)}$',\n index=4, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},\\\\tan(t)}$',\n index=5, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},1}$',\n index=6, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t^3},3}$',\n index=7, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},\\\\sin(3)}$',\n index=8, '$t\\\\rightarrow\\\\pmatrix{t,t^2}$',\n index=9, '$t\\\\rightarrow\\\\pmatrix{t,3t}$',\n index=10, '$t\\\\rightarrow\\\\pmatrix{t,\\\\sin(t)}$',\n index=11, '$t\\\\rightarrow\\\\pmatrix{t^2,t^\\\\frac{5}{3}}$',\n index=12, '$t\\\\rightarrow\\\\pmatrix{t,\\\\cos(t)}$',\n index=13, '$t\\\\rightarrow\\\\pmatrix{t^2,\\\\sqrt{t}}$',\n index=14, '$t\\\\rightarrow\\\\pmatrix{\\\\cos(t),t^\\\\frac{1}{2}}$',\n index=15, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},1}$',\n index=16, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},t}$',\n index=17, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},t^2}$',\n index=18, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t},t^4}$',\n index=19, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{f1}*t*cos(3t)},t^4}$',\n false\n )", "name": "formincorrectchoice3", "description": ""}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0 or index=5 or index=7 or index=11 or index=13 or index=14 or index=15 or index=16 or index=17, random(1..9)*sign(random(-1,1)),\n index=6 or index=9 or index=10 or index=12, random(1..9),\n false\n )", "name": "e1", "description": ""}, "dirincorrectchoice": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, 'From $\\\\pmatrix{1,1}$ to $\\\\pmatrix{-1,-1}$',\n index=1, 'From $\\\\pmatrix{1,5}$ to $\\\\pmatrix{0,\\\\infty}$',\n index=2, 'From $\\\\pmatrix{2,1.11}$ to $\\\\pmatrix{-5,-1.37}$',\n index=3, 'From $\\\\pmatrix{\\\\frac{\\\\pi}{2},\\\\infty}$ to $\\\\pmatrix{-\\\\frac{\\\\pi}{2},-\\\\infty}$',\n index=4, 'From $\\\\pmatrix{1,-1}$ to $\\\\pmatrix{-2,0.86}$',\n index=5, 'From $\\\\pmatrix{3,-7}$ to $\\\\pmatrix{0,0}$',\n index=6, 'From $\\\\pmatrix{-1,1}$ to $\\\\pmatrix{1,-1}$',\n index=7, 'From $\\\\pmatrix{27,-0.65}$ to $\\\\pmatrix{0,0}$',\n index=8, 'From $\\\\pmatrix{2,0}$ to $\\\\pmatrix{-2,0}$',\n index=9, 'From $\\\\pmatrix{15,5}$ to $\\\\pmatrix{3,1}$',\n index=10, 'From $\\\\pmatrix{-1,1}$ to $\\\\pmatrix{-1,-1}$',\n index=11, 'From $\\\\pmatrix{2,2.83}$ to $\\\\pmatrix{1,1}$',\n index=12, 'From $\\\\pmatrix{1,-1}$ to $\\\\pmatrix{-1,-1}$',\n index=13, 'From $\\\\pmatrix{9,0.14}$ to $\\\\pmatrix{0,0}$',\n index=14, 'From $\\\\pmatrix{0,-3}$ to $\\\\pmatrix{0,3}$',\n index=15, 'From $\\\\pmatrix{1,2}$ to $\\\\pmatrix{1,0}$',\n index=16, 'From $\\\\pmatrix{0,0}$ to $\\\\pmatrix{1,1}$ and then back to $\\\\pmatrix{0,0}$',\n index=17, 'Counter-clockwise',\n index=18, 'From $\\\\pmatrix{1,1}$ to $\\\\pmatrix{1,-1}$',\n index=19, 'From $\\\\pmatrix{4.63,3.53}$ to $\\\\pmatrix{4.63,-3.53}$',\n false\n )", "name": "dirincorrectchoice", "description": ""}, "dircorrectchoice": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, 'From $\\\\pmatrix{-1,-1}$ to $\\\\pmatrix{1,1}$',\n index=1, 'From $\\\\pmatrix{0,\\\\infty}$ to $\\\\pmatrix{1,5}$',\n index=2, 'From $\\\\pmatrix{-5,-1.37}$ to $\\\\pmatrix{2,1.11}$',\n index=3, 'From $\\\\pmatrix{-\\\\frac{\\\\pi}{2},-\\\\infty}$ to $\\\\pmatrix{\\\\frac{\\\\pi}{2},\\\\infty}$',\n index=4, 'From $\\\\pmatrix{-2,0.86}$ to $\\\\pmatrix{1,-1}$',\n index=5, 'From $\\\\pmatrix{0,0}$ to $\\\\pmatrix{3,-7}$',\n index=6, 'From $\\\\pmatrix{1,-1}$ to $\\\\pmatrix{-1,1}$',\n index=7, 'From $\\\\pmatrix{0,0}$ to $\\\\pmatrix{27,-0.65}$',\n index=8, 'From $\\\\pmatrix{-2,0}$ to $\\\\pmatrix{2,0}$',\n index=9, 'From $\\\\pmatrix{3,1}$ to $\\\\pmatrix{15,5}$',\n index=10, 'From $\\\\pmatrix{-1,-1}$ to $\\\\pmatrix{-1,1}$',\n index=11, 'From $\\\\pmatrix{1,1}$ to $\\\\pmatrix{2,2.83}$',\n index=12, 'From $\\\\pmatrix{-1,-1}$ to $\\\\pmatrix{1,-1}$',\n index=13, 'From $\\\\pmatrix{0,0}$ to $\\\\pmatrix{9,0.14}$',\n index=14, 'From $\\\\pmatrix{0,3}$ to $\\\\pmatrix{0,-3}$',\n index=15, 'From $\\\\pmatrix{1,0}$ to $\\\\pmatrix{1,2}$',\n index=16, 'From $\\\\pmatrix{1,1}$ to $\\\\pmatrix{0,0}$ and then back to $\\\\pmatrix{1,1}$',\n index=17, 'Clockwise',\n index=18, 'From $\\\\pmatrix{1,-1}$ to $\\\\pmatrix{1,1}$',\n index=19, 'From $\\\\pmatrix{4.63,-3.53}$ to $\\\\pmatrix{4.63,3.53}$',\n false\n )", "name": "dircorrectchoice", "description": ""}, "formcorrectchoice": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, '$t\\\\rightarrow\\\\pmatrix{t^3,t}$',\n index=1, '$t\\\\rightarrow\\\\pmatrix{t,\\\\frac{5}{t}}$',\n index=2, '$t\\\\rightarrow\\\\pmatrix{t,\\\\arctan(t)}$',\n index=3, '$t\\\\rightarrow\\\\pmatrix{t,\\\\tan(t)}$',\n index=4, '$t\\\\rightarrow\\\\pmatrix{t,\\\\cot(\\\\frac{t}{2}+5)}$',\n index=5, '$t\\\\rightarrow\\\\pmatrix{3t,-7t}$',\n index=6, '$t\\\\rightarrow\\\\pmatrix{-t^3,t}$',\n index=7, '$t\\\\rightarrow\\\\pmatrix{t^3,\\\\sin(-5t)}$',\n index=8, '$t\\\\rightarrow\\\\pmatrix{2\\\\sin(t),-2\\\\cos(t)}$',\n index=9, '$t\\\\rightarrow\\\\pmatrix{3t,t}$',\n index=10, '$t\\\\rightarrow\\\\pmatrix{-t^2,t}$',\n index=11, '$t\\\\rightarrow\\\\pmatrix{t,t^\\\\frac{3}{2}}$',\n index=12, '$t\\\\rightarrow\\\\pmatrix{t,-t^2}$',\n index=13, '$t\\\\rightarrow\\\\pmatrix{t^2,\\\\sin(t)}$',\n index=14, '$t\\\\rightarrow\\\\pmatrix{3\\\\cos(t),-3\\\\sin(t)}$',\n index=15, '$t\\\\rightarrow\\\\pmatrix{1,2t}$',\n index=16, '$t\\\\rightarrow\\\\pmatrix{t^2,t^4}$',\n index=17, '$t\\\\rightarrow\\\\pmatrix{5\\\\sin(t),3\\\\cos(t)}$',\n index=18, '$t\\\\rightarrow\\\\pmatrix{t^2,t^3}$',\n index=19, '$t\\\\rightarrow\\\\pmatrix{3\\\\cosh(t),3\\\\sinh(t)}$',\n false\n )", "name": "formcorrectchoice", "description": ""}, "formincorrectchoice1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}},\\\\simplify{t^{d1}}}$',\n index=1, '$t\\\\rightarrow\\\\pmatrix{\\\\frac{5}{t},\\\\simplify{{c1}*t}}$',\n index=2, '$t\\\\rightarrow\\\\pmatrix{-\\\\tan(t),\\\\simplify{{c1}*t^2}}$',\n index=3, '$t\\\\rightarrow\\\\pmatrix{\\\\sin(t),\\\\simplify{{c1}*t}}$',\n index=4, '$t\\\\rightarrow\\\\pmatrix{\\\\sin(t),\\\\cot(t)}$',\n index=5, '$t\\\\rightarrow\\\\pmatrix{3+\\\\simplify{t^{c1}},t}$',\n index=6, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{5-t^{c1}},t^5}$',\n index=7, '$t\\\\rightarrow\\\\pmatrix{t^2,\\\\sin(5t)}$',\n index=8, '$t\\\\rightarrow\\\\pmatrix{4\\\\cos(t),4\\\\sin(t)}$',\n index=9, '$t\\\\rightarrow\\\\pmatrix{t,t}$',\n index=10, '$t\\\\rightarrow\\\\pmatrix{t,t^2}$',\n index=11, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}},\\\\simplify{t^{d1}+t}}$',\n index=12, '$t\\\\rightarrow\\\\pmatrix{t,t^2}$',\n index=13, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}+t},\\\\simplify{t^{d1}}}$',\n index=14, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}+t},t^2}$',\n index=15, '$t\\\\rightarrow\\\\pmatrix{3+\\\\simplify{t^{c1}},t}$',\n index=16, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}},t+\\\\sin(t)}$',\n index=17, '$t\\\\rightarrow\\\\pmatrix{\\\\cos(5t),3\\\\sin(t)}$',\n index=18, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{c1}},t}$',\n index=19, '$t\\\\rightarrow\\\\pmatrix{3\\\\cos(t),3\\\\sin(t)}$',\n false\n )", "name": "formincorrectchoice1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, -1,\n index=1, 0,\n index=2, -5,\n index=3, '$-\\\\frac{\\\\pi}{2}$',\n index=4, -2,\n index=5, 0,\n index=6, -1,\n index=7, 0,\n index=8, '$-\\\\frac{\\\\pi}{2}$',\n index=9, 1,\n index=10, -1,\n index=11, 1,\n index=12, -1,\n index=13, 0,\n index=14, '$-\\\\frac{\\\\pi}{2}$',\n index=15, 0,\n index=16, -1,\n index=17, '$-\\\\pi$',\n index=18, -1,\n index=19, -1,\n false\n )", "name": "a1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0 or index=11, random(1,2,4,5),\n index=1 or index=2 or index=3, random(1..5),\n index=4, random(2..5),\n index=13 or index=16, -random(1..9),\n false\n )", "name": "d1", "description": ""}, "formincorrectchoice2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^{f1}+{g1}},\\\\sin(t)}$',\n index=1, '$t\\\\rightarrow\\\\pmatrix{t^2,t}$',\n index=2, '$t\\\\rightarrow\\\\pmatrix{t^3,t^2}$',\n index=3, '$t\\\\rightarrow\\\\pmatrix{\\\\tan(t),t}$',\n index=4, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{t^{d1-1}},\\\\simplify{t^{d1}}}$',\n index=5, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t},t}$',\n index=6, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^3},t}$',\n index=7, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*cos(t)},t}$',\n index=8, '$t\\\\rightarrow\\\\pmatrix{\\\\mathrm{e}^t,\\\\sin(2t)}$',\n index=9, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^3},t}$',\n index=10, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^3},t^4}$',\n index=11, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^{f1}},\\\\cos(t)}$',\n index=12, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t},t^4}$',\n index=13, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^{f1}},\\\\cos(t)}$',\n index=14, '$t\\\\rightarrow\\\\pmatrix{\\\\simplify{{e1}*t^{f1}},\\\\cos(t)\\\\sin(t)}$',\n index=15, '$t\\\\rightarrow\\\\pmatrix{\\\\mathrm{e}^{\\\\simplify{{e1}*t}},t}$',\n index=16, '$t\\\\rightarrow\\\\pmatrix{\\\\mathrm{e}^{\\\\simplify{{e1}*t}},\\\\simplify{t^{d1}}}$',\n index=17, '$t\\\\rightarrow\\\\pmatrix{\\\\mathrm{e}^{\\\\simplify{{e1}*t}},\\\\sin(3t)}$',\n index=18, '$t\\\\rightarrow\\\\pmatrix{t^3,t^2}$',\n index=19, '$t\\\\rightarrow\\\\pmatrix{\\\\sqrt{t},\\\\mathrm{e}^t}$',\n false\n )", "name": "formincorrectchoice2", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, 1,\n index=1, 1,\n index=2, 2,\n index=3, '$\\\\frac{\\\\pi}{2}$',\n index=4, 1,\n index=5, 1,\n index=6, 1,\n index=7, 3,\n index=8, '$\\\\frac{\\\\pi}{2}$',\n index=9, 5,\n index=10, 1,\n index=11, 2,\n index=12, 1,\n index=13, 3,\n index=14, '$\\\\frac{\\\\pi}{2}$',\n index=15, 1,\n index=16, 1,\n index=17, '$\\\\pi$',\n index=18, 1,\n index=19, 1,\n false\n )", "name": "b1", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0, random(1,2,4),\n index=1 or index=5 or index=6 or index=13 or index=14 or index=15 or index=16 or index=18, random(2..9),\n index=2 or index=3, random(1..9),\n index=4, random(1..3),\n index=11, -random(1,2,4),\n false\n )", "name": "c1", "description": ""}, "range": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[a1,b1]", "name": "range", "description": ""}, "index": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..19)", "name": "index", "description": ""}, "image": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'parametric-'+index+'.png'", "name": "image", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n index=0 or index=11, random(1..9),\n index=1 or index=2 or index=3 or index=4 or index=19, 3*random(1..3),\n index=5 or index=6 or index=7 or index=13 or index=14 or index=15 or index=16 or index=17 or index=18, random(1..4),\n false\n )", "name": "f1", "description": ""}}, "ungrouped_variables": ["index", "formcorrectchoice", "g1", "f1", "image", "range", "a1", "dirincorrectchoice", "formincorrectchoice3", "formincorrectchoice1", "formincorrectchoice2", "b1", "c1", "e1", "dircorrectchoice", "d1"], "functions": {"show": {"type": "html", "language": "javascript", "definition": "return $('');", "parameters": [["n", "number"]]}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": " {show(index)} \n You are given the following parametric representation of a curve. \n {image('resources/images/'+image)} ", "tags": ["checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Determine the correct parametric representation of a given curve. Curve is randomly chosen from a set of 20. \n The graph of the curve was not displayed on my machine. "}, "advice": " \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t^3,t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t^3$and$y=t$, therefore$x=y^3$or$y=x^\\frac{1}{3}$. Substituting the values of$t=-1$and$t=1$for$x$and$y$reveals the end-points of the curve. \n Also,$\\frac{\\mathrm{d}y}{\\mathrm{d}x}=\\frac{1}{3}x^{-\\frac{2}{3}}$, which diverges at the origin, implying that the tangent to the curve is vertical there. \n \n b) \n The curve is traversed from$\\pmatrix{-1,-1}$to$\\pmatrix{1,1}$, because$t=-1$implies$x=-1$and$y=-1$, and$t=1$implies$x=1$and$y=1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,\\frac{5}{t}}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=\\frac{5}{t}$, therefore$y=\\frac{5}{x}$or$xy=5$. This is the equation of a rectangular hyperbola in the right half-plane, where the asymptotes coincide with the$x$- and$y$-axes. You can check this by taking the limit of$y=\\frac{5}{x}$as$x\\rightarrow 0$. This limit diverges. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{0,\\infty}$to$\\pmatrix{1,5}$, because$t=0$implies$x=0$and$y\\rightarrow\\infty$, and$t=1$implies$x=1$and$y=5$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,\\arctan(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=\\arctan(t)$, therefore$y=\\arctan(x)$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n Since$x=t$, the range of$x$coincides with the range of$t$, and so$-5\\leqslant t\\leqslant 2$. \n \n b) \n The curve is traversed from$\\pmatrix{-5,-1.37}$to$\\pmatrix{2,1.11}$, because$t=-5$implies$x=-5$and$y\\approx -1.37$, and$t=2$implies$x=2$and$y\\approx 1.11$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,\\tan(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=\\tan(t)$, therefore$y=\\tan(x)$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n The lines$x=-\\frac{\\pi}{2}$and$x=\\frac{\\pi}{2}$are asymptotes for$y=\\tan(x)$. \n \n b) \n The curve is traversed from$\\pmatrix{-\\frac{\\pi}{2},-\\infty}$to$\\pmatrix{\\frac{\\pi}{2},\\infty}$, because$t=-\\frac{\\pi}{2}$implies$x=-\\frac{\\pi}{2}$and$y\\rightarrow -\\infty$, and$t=\\frac{\\pi}{2}$implies$x=\\frac{\\pi}{2}$and$y\\rightarrow \\infty$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,\\cot(\\frac{t}{2}+5)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=\\cot(\\frac{t}{2}+5)$, therefore$y=\\cot(\\frac{x}{2}+5)$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n Since$x=t$, the range of$t$coincides with the range of$x$, and so$-2\\leqslant x\\leqslant 1$. \n \n b) \n The curve is traversed from$\\pmatrix{-2,0.86}$to$\\pmatrix{1,-1}$, because$t=-2$implies$x=-2$and$y\\approx 0.86$, and$t=1$implies$x=1$and$y\\approx -1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{3t,-7t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=3t$and$y=7t$, therefore$y=-\\frac{7}{3}x$, and this is the equation of a straight line with gradient$-7/3$, passing through the origin. In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{0,0}$to$\\pmatrix{3,-7}$, because$t=0$implies$x=0$and$y=0$, and$t=1$implies$x=3$and$y=-7$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{-t^3,t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=-t^3$and$y=t$, therefore$x=-y^3$or$y=-x^\\frac{1}{3}$. Also note that$\\frac{\\mathrm{d}y}{\\mathrm{d}x}=-\\frac{1}{3}x^{-\\frac{2}{3}}$, which diverges at the origin, so the tangent to the curve is vertical there. In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{1,-1}$to$\\pmatrix{-1,1}$, because$t=-1$implies$x=1$and$y=-1$, and$t=1$implies$x=-1$and$y=1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t^3,\\sin(-5t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t^3$and$y=\\sin(-5t)$, therefore$y=\\sin\\left(-5x^\\frac{1}{3}\\right)$. The curve is a non-uniformly stretched sinusoid, due to the term$x^\\frac{1}{3}$, and the stretching is stronger at larger$x$, because$x=t^3$grows faster as$t$increases. \n The values of$x$range from$0$to$27$because$t$ranges from$0$to$3$and$x=t^3$. In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{0,0}$to$\\pmatrix{27,-0.65}$, because$t=0$implies$x=0$and$y=0$, and$t=3$implies$x=27$and$y\\approx -0.65$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{2\\sin(t),-2\\cos(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=2\\sin(t)$and$y=-2\\cos(t)$, therefore$x^2+y^2=4$, and this is the equation for a circle of radius$2$, centred at the origin. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n In addition, since$-\\frac{\\pi}{2}\\leqslant t\\leqslant\\frac{\\pi}{2}$, the curve is only defined in the lower half of the$\\pmatrix{x,y}$-plane. \n \n b) \n The curve is traversed from$\\pmatrix{-2,0}$to$\\pmatrix{2,0}$, because$t=-\\frac{\\pi}{2}$implies$x=-2$and$y=0$, and$t=\\frac{\\pi}{2}$implies$x=2$and$y=0$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{3t,t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=3t$and$y=t$, therefore$y=\\frac{1}{3}x$, which is the equation of a straight line, with gradient$1/3$, passing through the origin. Since the range of$t$given does not allow$x$and$y$to be equal to zero, the segment of the line shown does not include the origin. In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{3,1}$to$\\pmatrix{15,1}$, because$t=1$implies$x=3$and$y=1$, and$t=5$implies$x=15$and$y=5$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{-t^2,t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=-t^2$and$y=t$, therefore$x=-y^2$, which is only defined in the left half-plane, and the derivative at the origin is divergent, so the tangent to the curve is vertical there. In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{-1,-1}$to$\\pmatrix{-1,1}$, because$t=-1$implies$x=-1$and$y=-1$, and$t=1$implies$x=-1$and$y=1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,t^\\frac{3}{2}}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=t^\\frac{3}{2}$, therefore$y=x^\\frac{3}{2}$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{1,1}$to$\\pmatrix{2,2.83}$, because$t=1$implies$x=1$and$y=1$, and$t=2$implies$x=2$and$y\\approx 2.83$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t,-t^2}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t$and$y=-t^2$, therefore$y=-x^2$, which is only defined in the lower half-plane, and the derivative at the origin is zero, corresponding to a maximum. \n Since$x=t$, the range of$x$coincides with the range of$t$. \n In addition, substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{-1,-1}$to$\\pmatrix{1,-1}$, because$t=-1$implies$x=-1$and$y=-1$, and$t=1$implies$x=1$and$y=-1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t^2,\\sin(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t^2$and$y=\\sin(t)$, therefore$y=\\sin(\\sqrt{x})$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n The curve is part of a sinusoid that has been stretched in the$x$-direction due to the term$\\sqrt{x}$, with the stretching stronger at larger$x$, because$x=t^2$grows faster as$t$increases. \n Also note that$\\frac{\\mathrm{d}y}{\\mathrm{d}x}=\\frac{\\cos(\\sqrt{x})}{2\\sqrt{x}}$, so the curve has a maximum where$\\sqrt{x}=\\frac{1}{2}(2n+1)\\pi$, or$x=\\frac{1}{4}(2n+1)^2\\pi^2$, for$n=0,1,2,\\ldots$. Given the range of$t$, the only valid value of$x$is when$n=0$, so the curve has a maximum at$x=\\frac{\\pi^2}{4}$. \n Also note that$\\frac{\\mathrm{d}y}{\\mathrm{d}x}$diverges at the origin, so the curve is vertical there. \n \n b) \n The curve is traversed from$\\pmatrix{0,0}$to$\\pmatrix{9,0.14}$, because$t=0$implies$x=0$and$y=0$, and$t=3$implies$x=9$and$y\\approx 0.14$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{3\\cos(t),-3\\sin(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=3\\cos(t)$and$y=-3\\sin(t)$, therefore$x^2+y^2=9$, which is the equation of a circle, centred at the origin, having radius$3$. From the range of$t$we can determine that the range of valid$x$and$y$values corresponds to the right half-plane. \n \n b) \n The curve is traversed from$\\pmatrix{0,3}$to$\\pmatrix{0,-3}$, because$t=-\\frac{\\pi}{2}$implies$x=0$and$y=3$, and$t=\\frac{\\pi}{2}$implies$x=0$and$y=-3$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{1,2t}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=1$and$y=2t$therefore, for all values of$t$,$x=1$and$y$increases with$t$. The graph is therefore a segment of a straight line. \n \n b) \n The curve is traversed from$\\pmatrix{1,0}$to$\\pmatrix{1,2}$, because$t=0$implies$x=1$and$y=0$, and$t=1$implies$x=1$and$y=2$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t^2,t^4}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t^2$and$y=t^4$, therefore$y=x^2$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n Also, note that$\\frac{\\mathrm{d}y}{\\mathrm{d}x}=2x$, which is zero at the origin, so the tangent to the curve is horizontal there. \n \n b) \n The curve is traversed from$\\pmatrix{1,1}$to$\\pmatrix{0,0}$, then back to$\\pmatrix{1,1}$, because$t=-1$implies$x=1$and$y=1$, and$t=1$also implies$x=1$and$y=1$, but$t=0$implies$x=0$and$y=0$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{5\\sin(t),3\\cos(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=5\\sin(t)$and$y=3\\cos(t)$, therefore$\\left(\\frac{x}{5}\\right)^2+\\left(\\frac{y}{3}\\right)^2=1$, which is the equation of an ellipse, with semi-major axis equal to$5$, and semi-minor axis equal to$3$. \n Since$t$takes values from$-\\pi$to$\\pi$, the curve is defined over the entire$\\pmatrix{x,y}$-plane. \n \n b) \n The curve is traversed in a clockwise direction. Substituting$t=-\\pi$for$x$and$y$gives$x=0$and$y=-3$. Then choosing$t=-\\frac{\\pi}{2}$, say, gives$x=-5$and$y=0$, which is$90^\\circ$clockwise from$\\pmatrix{0,-3}$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{t^2,t^3}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=t^2$and$y=t^3$, therefore$y=x^\\frac{3}{2}$. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n Also, note that$\\frac{\\mathrm{d}y}{\\mathrm{d}x}=\\frac{3}{2}x^\\frac{1}{2}$, which is zero at the origin, so the tangent to the curve is horizontal there. \n \n b) \n The curve is traversed from$\\pmatrix{1,-1}$to$\\pmatrix{1,1}$, because$t=-1$implies$x=1$and$y=-1$, and$t=1$implies$x=1$and$y=1$. \n \n \n a) \n The correct answer in this case is$t\\rightarrow\\pmatrix{3\\cosh(t),3\\sinh(t)}$,$\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$. \n To see this note that$x=3\\cosh(t)$and$y=3\\sinh(t)$, therefore$x^2-y^2=9$, which is the equation of a rectangular hyperbola in the right half-plane. Substituting values of$t$for$x$and$y$in the given range determines points along the curve. \n \n b) \n The curve is traversed from$\\pmatrix{4.63,-3.53}$to$\\pmatrix{4.63,3.53}$, because$t=-1$implies$x\\approx 4.63$and$y\\approx -3.53$, and$t=1$implies$x\\approx 4.63$and$y\\approx 3.53$. \n "}, {"name": "Lengths of and distance between vectors, dot and cross products, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "name": "a"}, "crossab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(a,b)", "description": "", "name": "crossab"}, "lena": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(a),2)", "description": "", "name": "lena"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "name": "b"}, "dotab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "dot(a,b)", "description": "", "name": "dotab"}, "sumab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+b", "description": "", "name": "sumab"}, "dist": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(a-b),2)", "description": "", "name": "dist"}, "lenb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(b),2)", "description": "", "name": "lenb"}, "diffab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a-b", "description": "", "name": "diffab"}}, "ungrouped_variables": ["a", "lenb", "lena", "b", "dist", "dotab", "diffab", "sumab", "crossab"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lena+0.01", "minValue": "lena-0.01", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lenb+0.01", "minValue": "lenb-0.01", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Their lengths:$a=\\lvert\\boldsymbol{a}\\rvert=$[[0]],$b=\\lvert\\boldsymbol{b}\\rvert=$[[1]]. (Enter your answers to 2d.p.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dist+0.01", "minValue": "dist-0.01", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " The distance,$d=$[[0]], between$\\boldsymbol{a}$and$\\boldsymbol{b}$, assuming their common initial point is at the origin. (Enter your answer to 2d.p.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[0]", "minValue": "sumab[0]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[1]", "minValue": "sumab[1]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[2]", "minValue": "sumab[2]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[0]", "minValue": "diffab[0]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[1]", "minValue": "diffab[1]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[2]", "minValue": "diffab[2]", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Their sum,$\\boldsymbol{a}+\\boldsymbol{b}=($[[0]]$,$[[1]]$,$[[2]]$)$, and difference,$\\boldsymbol{a}-\\boldsymbol{b}=($[[3]]$,$[[4]]$,$[[5]]$)$. ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dotab", "minValue": "dotab", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Their dot product$\\boldsymbol{a\\cdot b}=$[[0]]. ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "crossab[0]", "minValue": "crossab[0]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "crossab[1]", "minValue": "crossab[1]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "crossab[2]", "minValue": "crossab[2]", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Their cross product$\\boldsymbol{a}\\times\\boldsymbol{b}=($[[0]]$,$[[1]]$,$[[2]]$)$. ", "showCorrectAnswer": true, "marks": 0}], "statement": " Given the vectors$\\boldsymbol{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$and$\\boldsymbol{b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$find: ", "tags": ["checked2015", "MAS1602", "MAS2104"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " For the general 3-component vectors$\\boldsymbol{a}=\\pmatrix{a_1,a_2,a_3}$and$\\boldsymbol{b}=\\pmatrix{b_1,b_2,b_3}$, we have \n a) \n Lengths:$a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$and$b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$, which are scalar quantities. \n \n b) \n Distance between the terminal points:$d=\\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$, which is a scalar quantity. \n \n c) \n Sum$\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{a_1+b_1,a_2+b_2,a_3+b_3}$and difference$\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{a_1-b_1,a_2-b_2,a_3-b_3}$, which are vector quantities. \n \n d) \n Dot product:$\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3$, which is a scalar quantity. \n \n e) \n Cross product:$\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1}$, which is a vector quantity. \n \n In this question, therefore, we have: \n a) \n Lengths:$a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{\\var{a[0]^2}+\\var{a[1]^2}+\\var{a[2]^2}}=\\var{lena}$and$b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{\\var{b[0]^2}+\\var{b[1]^2}+\\var{b[2]^2}}=\\var{lenb}$. \n \n b) \n Distance between the terminal points:$d=\\sqrt{(\\simplify[std]{{a[0]}-{b[0]}})^2+(\\simplify[std]{{a[1]}-{b[1]}})^2+(\\simplify[std]{{a[2]}-{b[2]}})^2}=\\var{dist}$. \n \n c) \n Sum$\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}+{b[0]}},\\simplify[std]{{a[1]}+{b[1]}},\\simplify[std]{{a[2]}+{b[2]}}}=\\pmatrix{\\var{sumab[0]},\\var{sumab[1]},\\var{sumab[2]}}$and difference$\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}-{b[0]}},\\simplify[std]{{a[1]}-{b[1]}},\\simplify[std]{{a[2]}-{b[2]}}}=\\pmatrix{\\var{diffab[0]},\\var{diffab[1]},\\var{diffab[2]}}$. \n \n d) \n Dot product:$\\boldsymbol{a\\cdot b}=(\\var{a[0]}\\times\\var{b[0]})+(\\var{a[1]}\\times\\var{b[1]})+(\\var{a[2]}\\times\\var{b[2]})=\\var{dotab}$. \n \n e) \n Cross product:$\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[1]*b[2]}-{a[2]*b[1]}},\\simplify[std]{{a[2]*b[0]}-{a[0]*b[2]}},\\simplify[std]{{a[0]*b[1]}-{a[1]*b[0]}}}=\\pmatrix{\\var{crossab[0]},\\var{crossab[1]},\\var{crossab[2]}}$. "}, {"name": "Parameterisation of a curve - tangent and coordinates at given point, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"speed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(sqrt(dxdtd^2+b^2),2)", "description": "", "name": "speed"}, "x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c^a", "description": "", "name": "x"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..10)*sign(random(1,-1))", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "name": "c"}, "dxdtd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*d^(a-1)", "description": "", "name": "dxdtd"}, "dxdtc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*c^(a-1)", "description": "", "name": "dxdtc"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "name": "a"}, "y": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b*c", "description": "", "name": "y"}}, "ungrouped_variables": ["a", "c", "b", "d", "dxdtc", "dxdtd", "y", "x", "speed"], "rulesets": {}, "showQuestionGroupNames": false, "variable_groups": [], "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "x", "maxValue": "x", "marks": 1}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "y", "maxValue": "y", "marks": 1}], "type": "gapfill", "prompt": " Find the coordinates$\\pmatrix{x,y}$of the point corresponding to$t=\\var{c}$. \n$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$. ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a}*t^{a-1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "dxdtc", "maxValue": "dxdtc", "marks": 1}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "b", "maxValue": "b", "marks": 1}], "type": "gapfill", "prompt": " Find the components of the tangent vector$\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$. \n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$. \n The components of the same tangent vector, given$t=\\var{c}$. \n$\\boldsymbol{u}|_{t=\\var{c}}=($[[2]]$,$[[3]]$)$. ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "speed-0.01", "maxValue": "speed+0.01", "marks": 1}], "type": "gapfill", "prompt": " Interpreting$t$as time, and hence the tangent vector$\\boldsymbol{u}$as velocity, find the speed$u=|\\boldsymbol{u}|$at$t=\\var{d}$. \n$u=$[[0]]. (Enter your answer to 2d.p.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " You are given the following curve,$t\\rightarrow\\pmatrix{t^\\var{a},\\simplify{{b}t}}$, defined with respect to the parameter$t$. ", "tags": ["checked2015", "MAS1902", "MAS2104"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Parametric form of a curve, cartesian points, tangent vector, and speed. "}, "functions": {}, "advice": " a) \n To find the coordinates of the point corresponding to$t=\\var{c}$, substitute$t=\\var{c}$into the expression for the curve, i.e. \n \$\\pmatrix{x,y}=\\pmatrix{\\var{c}^\\var{a},\\var{b}\\times\\var{c}}=\\pmatrix{\\var{x},\\var{y}}.\$ \n \n b) \n Differentiate each component of the vector in part a) to find the tangent vector$\\boldsymbol{u}$, i.e. \n \$\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}t^\\var{a},\\frac{\\mathrm{d}}{\\mathrm{d}t}\\simplify{{b}t}}=\\pmatrix{\\var{a}t^\\var{a-1},\\var{b}}.\$ \n The tangent vector at$t=\\var{c}$is found by substituting$t=\\var{c}$into the tangent vector$\\boldsymbol{u}$, i.e. \n \$\\boldsymbol{u}\\vert_{t=\\var{c}}=\\pmatrix{\\var{a}\\times\\var{c}^\\var{a-1},\\var{b}}=\\pmatrix{\\var{dxdtc},\\var{b}}.\$ \n \n c) \n The velocity$u$is given by$u=\\lvert\\boldsymbol{u}\\rvert=\\sqrt{\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)^2+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}t}\\right)^2}$. We must calculate the speed at$t=\\var{d}$, however, therefore \n \$u\\vert_{t=\\var{d}}=\\sqrt{\\left(\\var{a}\\times\\var{d}^\\var{a-1}\\right)^2+\\var{b}^2}=\\sqrt{\\var{dxdtd^2}+\\var{b^2}}=\\var{speed} \\; \\text{to 2d.p.}\$ "}, {"name": "Parametric curve - find tangent and coordinates at given point, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"speed": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(sqrt(dxdtf1^2+dydtf1^2),3)", "description": "", "name": "speed"}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(a1*cos(b1*e1),3)", "description": "", "name": "x"}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "e1"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "f1"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "b1"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "a1"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "d1"}, "dydte1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(c1*d1*cos(d1*e1),3)", "description": "", "name": "dydte1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "c1"}, "dydtf1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "c1*d1*cos(d1*f1)", "description": "", "name": "dydtf1"}, "dxdtf1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-a1*b1*sin(b1*f1)", "description": "", "name": "dxdtf1"}, "dxdte1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(-a1*b1*sin(b1*e1),3)", "description": "", "name": "dxdte1"}, "y": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(c1*sin(d1*e1),3)", "description": "", "name": "y"}}, "ungrouped_variables": ["f1", "dxdte1", "dxdtf1", "dydte1", "dydtf1", "a1", "b1", "y", "x", "c1", "e1", "speed", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "x", "minValue": "x", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "y", "minValue": "y", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Find the coordinates$\\pmatrix{x,y}$of the point corresponding to$t=\\var{e1}$. \n$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$. (Enter your answers to 3d.p.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-a1*b1}*sin({b1}*t)", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{c1*d1}*cos({d1}*t)", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dxdte1+0.001", "minValue": "dxdte1-0.001", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dydte1+0.001", "minValue": "dydte1-0.001", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Find the components of the tangent vector$\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$. \n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$. \n The components of the same tangent vector, given$t=\\var{e1}$. \n$\\boldsymbol{u}|_{t=\\var{e1}}=($[[2]]$,$[[3]]$)$. (Enter your answers to 3d.p.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "speed+0.001", "minValue": "speed-0.001", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": " Interpreting$t$as time, and hence the tangent vector$\\boldsymbol{u}$as velocity, find the speed$u=|\\boldsymbol{u}|$at$t=\\var{f1}$. \n$u=$[[0]]. (Enter your answer to 3d.p.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " You are given the following curve,$t\\rightarrow\\pmatrix{\\simplify{{a1}*cos({b1}t)},\\simplify{{c1}*sin({d1}t)}}$, defined with respect to the parameter$t$. ", "tags": ["checked2015", "MAS1902", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Parametric form of a curve, cartesian points, tangent vector, and speed. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " a) \n To find the coordinates of the point corresponding to$t=\\var{e1}$, substitute$t=\\var{e1}$into the expression for the curve, i.e. \n \$\\pmatrix{x,y}=\\pmatrix{\\simplify{{a1}*cos({b1*e1})},\\simplify{{c1}*cos({d1*e1})}}=\\pmatrix{\\var{x},\\var{y}}.\$ \n \n b) \n Differentiate each component of the vector in part a) to find the tangent vector$\\boldsymbol{u}$, i.e. \n \$\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{a1}*cos({b1}*t)}\\right),\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{c1}*sin({d1}*t)}\\right)}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1}*t)},\\simplify{{c1*d1}*cos({d1}*t)}}.\$ \n The tangent vector at$t=\\var{e1}$is found by substituting$t=\\var{e1}$into the tangent vector$\\boldsymbol{u}$, i.e. \n \$\\boldsymbol{u}\\vert_{t=\\var{e1}}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1*e1})},\\simplify{{c1*d1}*cos({d1*e1})}}=\\pmatrix{\\var{dxdte1},\\var{dydte1}}.\$ \n \n c) \n The velocity$u$is given by$u=\\lvert\\boldsymbol{u}\\rvert=\\sqrt{\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)^2+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}t}\\right)^2}$. We must calculate the speed at$t=\\var{f1}$, however, therefore \n \$u\\vert_{t=\\var{f1}}=\\sqrt{\\left(\\simplify{{-a1*b1}*sin({b1*f1})}\\right)^2+\\left(\\simplify{{c1*d1}*cos({d1*f1})}\\right)^2}=\\sqrt{\\var{dxdtf1}^2+\\var{dydtf1}^2}=\\var{speed} \\; \\text{to 3d.p.}\$ "}, {"name": "Find points of intersection, tangents, and angles between parametric curves", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"lenu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(u)", "name": "lenu", "description": ""}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(a1*t,b1*t,c1*t)", "name": "q", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(d1,0,0)", "name": "v", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(d1,2*e1*tau,3*f1*tau^2)", "name": "w", "description": ""}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "f1", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(e1*a1^2)*t/d1^2", "name": "b1", "description": ""}, "phi": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuw/(lenu*lenw))),2)", "name": "phi", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "t", "description": ""}, "dotuv": {"templateType": "anything", "group": "Ungrouped variables", "definition": "dot(u,v)", "name": "dotuv", "description": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "d1", "description": ""}, "dotuw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "dot(u,w)", "name": "dotuw", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(f1*d1*b1^2)/(a1*e1^2)", "name": "c1", "description": ""}, "e1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "e1", "description": ""}, "lenw": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(w)", "name": "lenw", "description": ""}, "theta": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuv/(lenu*lenv))),2)", "name": "theta", "description": ""}, "lenv": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(v)", "name": "lenv", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d1*random(-2..2 except 0)", "name": "a1", "description": ""}, "tau": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1*t/d1", "name": "tau", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(a1,b1,c1)", "name": "u", "description": ""}}, "ungrouped_variables": ["f1", "phi", "lenu", "dotuw", "tau", "e1", "dotuv", "a1", "u", "t", "w", "v", "lenw", "lenv", "d1", "q", "theta", "c1", "b1"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Enter the least value of$t$, and the corresponding value of$\\tau$, defining the first intersection point. Hence enter the values of the intersection point$\\boldsymbol{p}$for these values of$t$and$\\tau$. \n$t=$[[0]];$\\tau=$[[1]]. \n$\\boldsymbol{p}=($[[2]]$,$[[3]]$,$[[4]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "0", "maxValue": "0", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "0", "maxValue": "0", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "0", "maxValue": "0", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "0", "maxValue": "0", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "0", "maxValue": "0", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Enter the greatest value of$t$, and the corresponding value of$\\tau$, defining the second intersection point. Hence enter the values of the intersection point$\\boldsymbol{q}$for these values of$t$and$\\tau$. \n$t=$[[0]];$\\tau=$[[1]]. \n$\\boldsymbol{q}=($[[2]]$,$[[3]]$,$[[4]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "t", "maxValue": "t", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "tau", "maxValue": "tau", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "q[0]", "maxValue": "q[0]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "q[1]", "maxValue": "q[1]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "q[2]", "maxValue": "q[2]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Find the tangent vector$\\boldsymbol{u}$of the curve$\\mathcal{C}_1$. \n$\\boldsymbol{u}=($[[0]]$,$[[1]]$,$[[2]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "u[0]", "maxValue": "u[0]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "u[1]", "maxValue": "u[1]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "u[2]", "maxValue": "u[2]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Find the tangent vector$\\boldsymbol{v}$of the curve$\\mathcal{C}_2$at the point$\\boldsymbol{p}$. \n$\\boldsymbol{v}=($[[0]]$,$[[1]]$,$[[2]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "v[0]", "maxValue": "v[0]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "v[1]", "maxValue": "v[1]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "v[2]", "maxValue": "v[2]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Find the tangent vector$\\boldsymbol{w}$of the curve$\\mathcal{C}_2$at the point$\\boldsymbol{q}$. \n$\\boldsymbol{w}=($[[0]]$,$[[1]]$,$[[2]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "w[0]", "maxValue": "w[0]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "w[1]", "maxValue": "w[1]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "w[2]", "maxValue": "w[2]", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Calculate the angle$\\theta$(in degrees) between the tangent vectors of each curve, at the point$\\boldsymbol{p}$. \n$\\theta=$[[0]]$^\\circ$. (Enter your answer to 2d.p.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "theta-0.01", "maxValue": "theta+0.01", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Calculate the angle$\\phi$(in degrees) between the tangent vectors of each curve, at the point$\\boldsymbol{q}$. \n$\\phi=$[[0]]$^\\circ. (Enter your answer to 2d.p.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "phi-0.01", "maxValue": "phi+0.01", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": " The pair of curves \n \\\begin{align}\\mathcal{C}_1&:t\\rightarrow\\pmatrix{\\simplify{{a1}*t},\\simplify{{b1}*t},\\simplify{{c1}*t}},-\\infty\\leqslant t\\leqslant\\infty\\\\\\mathcal{C}_2&:\\tau\\rightarrow\\pmatrix{\\simplify{{d1}*tau},\\simplify{{e1}*tau^2},\\simplify{{f1}*tau^3}},-\\infty\\leqslant \\tau\\leqslant\\infty\\end{align}\ \n intersect at two distinct points\\boldsymbol{p}$and$\\boldsymbol{q}$. ", "tags": ["checked2015", "intersection of curves", "parametric curves", "tangent vectors"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Intersection points, tangent vectors, angles between pairs of curves, given in parametric form. "}, "advice": " Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity. \n The two curves$\\mathcal{C}_1$and$\\mathcal{C}_2intersect where \n \\\begin{align}\\simplify{{a1}*t}&=\\simplify{{d1}*tau}\\tag{1},\\\\\\simplify{{b1}t}&=\\simplify{{e1}*tau^2},\\tag{2}\\\\\\simplify{{c1}*t}&=\\simplify{{f1}*tau^3}.\\tag{3}\\end{align}\ \n From equation (1) \n \$\\tau=\\frac{\\var{a1}}{\\var{d1}}t=\\simplify{{a1}/{d1}t},\\tag{4}\$ \n which we substitute into equation (2) to determine that \n \$\\var{b1}t=\\var{e1}\\times\\left(\\simplify{{a1}/{d1}t}\\right)^2=\\simplify{{e1*a1^2}/{d1^2}t^2}.\$ \n Then eithert=0$or$t=\\simplify{{b1*d1^2}/{e1*a1^2}}$. \n Substitute these two expressions into equation (4), then either$\\tau=0$(when$t=0$), or$\\tau=\\simplify{{b1*d1}/{e1*a1}}$(when$t=\\var{t}$). \n (As a check, substitute these pairs of values into equation (3), to show that equality holds.) \n \n To determine the intersection points$\\boldsymbol{p}$and$\\boldsymbol{q}$, substitute the values of$t$and$\\tau$into either expression for the curves$\\mathcal{C}_1$and$\\mathcal{C}_2$. \n The point$\\boldsymbol{p}$is given by the least value of$t$, which is$t=0$(and correspondingly$\\tau=0$). The point$\\boldsymbol{p}$is therefore$\\boldsymbol{p}=\\pmatrix{0,0,0}$. \n The point$\\boldsymbol{q}$is given by the greatest value of$t$, which is$t=\\var{t}$(and correspondingly$\\tau=\\var{tau}$). The point$\\boldsymbol{q}$is therefore$\\boldsymbol{q}=\\pmatrix{\\var{a1}\\times\\var{t},\\var{b1}\\times\\var{t},\\var{c1}\\times\\var{t}}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$. \n \n In general, the tangent vector$\\boldsymbol{u}$, of a curve$t\\rightarrow\\pmatrix{x(t),y(t),z(t)}$, is given by$\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t},\\frac{\\mathrm{d}z}{\\mathrm{d}t}}$. \n The tangent vector of the curve$\\mathcal{C}_1$is therefore given by$\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$, which is constant, and independent of$t$. \n The tangent vector of$\\mathcal{C}_2$is given by$\\pmatrix{\\var{d1},\\var{2*e1}\\tau,\\var{3*f1}\\tau^2}$, so the tangent vector at the point$\\boldsymbol{p}$(where$\\tau=0$) is given by$\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$. \n In a similar way, the tangent vector of$\\mathcal{C}_2$at the point$\\boldsymbol{q}$(where$\\tau=\\var{tau}$) is given by$\\boldsymbol{w}=\\pmatrix{\\var{w[0]},\\var{w[1]},\\var{w[2]}}$. \n \n The angle$\\theta$between any two vectors$\\boldsymbol{a}$and$\\boldsymbol{b}$can be calculated using \n \$\\cos(\\theta)=\\frac{\\boldsymbol{a\\cdot b}}{\\lvert\\boldsymbol{a}\\rvert\\lvert\\boldsymbol{b}\\rvert},\$ \n where$\\lvert\\boldsymbol{x}\\rvert=\\sqrt{x_1^2+x_2^2+x_3^2}$is the length of the vector$\\boldsymbol{x}$. \n The angle$\\theta$between the tangent vectors at the point$\\boldsymbol{p}$is the angle between the vectors$\\boldsymbol{u}$and$\\boldsymbol{v}$, so \n \$\\cos(\\theta)=\\frac{(\\var{u[0]}\\times\\var{v[0]})+(\\var{u[1]}\\times\\var{v[1]})+(\\var{u[2]}\\times\\var{v[2]})}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}\\sqrt{(\\var{v[0]})^2+(\\var{v[1]})^2+(\\var{v[2]})^2}}=\\frac{\\var{dotuv}}{\\var{precround(lenu,4)}\\times\\var{precround(lenv,4)}}=\\var{precround(dotuv/(lenu*lenv),4)}\\;\\text{to 4d.p.}\$ \n Then$\\theta=\\arccos(\\var{precround(dotuv/(lenu*lenv),4)})=\\var{theta}^\\circ$to 2d.p. \n In an identical way, the angle$\\phi$between the tangent vectors at the point$\\boldsymbol{q}$is the angle between the vectors$\\boldsymbol{u}$and$\\boldsymbol{w}$, so$\\phi=\\var{phi}^\\circ$to 2d.p. "}, {"name": "Find unit vector orthogonal to two others, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(-9..9 except 0),3))", "description": "", "name": "v"}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(-9..9 except 0),3))", "description": "", "name": "u"}, "apos": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a[0]<0,-a,a)", "description": "", "name": "apos"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(u,v)", "description": "", "name": "a"}, "unitapos": {"group": "Ungrouped variables", "templateType": "anything", "definition": "apos/len(a)", "description": "", "name": "unitapos"}}, "ungrouped_variables": ["u", "v", "a", "apos", "unitapos"], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "$\\boldsymbol{\\hat{a}}=$[[0]] (Enter your answers to 3d.p.) ", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowResize": false, "type": "matrix", "numRows": 1, "precisionMessage": "You have not given your answer to the correct precision.", "tolerance": 0, "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "correctAnswer": "matrix([unitapos])", "precision": "3", "unitTests": [], "correctAnswerFractions": false, "precisionType": "dp", "strictPrecision": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "numColumns": "3", "markPerCell": true, "marks": "3", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "a[0]<>0", "maxRuns": 100}, "statement": " Find the unit vector$\\boldsymbol{\\hat{a}}$, with positive$x$-component, which is orthogonal to both$\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$and$\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$. ", "tags": ["checked2015", "cross product", "vector", "Vector"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Find a unit vector orthogonal to two others. \n Uses$\\wedge$for the cross product. The interim calculations should all be displayed to enough dps, not 3, to ensure accuracy to 3 dps. If the cross product has a negative x component then it is not explained that the negative of the cross product is taken for the unit vector. "}, "advice": " Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity. \n A vector$\\boldsymbol{a}$, which is orthogonal to both$\\boldsymbol{u}$and$\\boldsymbol{v}$, is given by \n \$\\boldsymbol{u}\\wedge\\boldsymbol{v}=\\pmatrix{u_2 v_3 - u_3 v_2, & u_3 v_1 - u_1 v_3, & u_1 v_2 - u_2 v_1} \$ \n The magnitude of$\\boldsymbol{a}$is given by \n \$\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2} \$ \n A unit vector$\\boldsymbol{\\hat{a}}$is obtained by dividing the components of the vector$\\boldsymbol{a}$by its magnitude, i.e. \n \$\\boldsymbol{\\hat{a}}=\\frac{\\boldsymbol{a}}{\\lvert\\boldsymbol{a}\\rvert} \$ \n In this question, \n \$\\boldsymbol{a} = \\pmatrix{\\simplify[basic]{{u[1]}*{v[2]}-{u[2]}*{v[1]}}, & \\simplify[basic]{{u[2]}*{v[0]}-{u[0]}*{v[2]}}, & \\simplify[basic]{{u[0]}*{v[1]} - {u[1]}*{v[0]}}} = \\var[rowvector]{a} \$ \n and \n \$\\lvert\\boldsymbol{a}\\rvert = \\sqrt{(\\var{a[0]})^2+(\\var{a[1]})^2+(\\var{a[2]})^2} = \\var{precround(len(a),3)} \\text{ to 3 decimal places.} \$ \n The unit vector with positive$x$-component is therefore$\\boldsymbol{\\hat{a}}=\\frac{1}{\\var{precround(len(a),3)}}\\var[rowvector]{apos} = \\pmatrix{\\var{precround(unitapos[0],3)}, & \\var{precround(unitapos[1],3)}, & \\var{precround(unitapos[2],3)}}$to 3d.p. "}, {"name": "Find unit vector parallel to given vector, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"lenu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(u)", "description": "", "name": "lenu"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "name": "u"}, "unitu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(precround(u[0]/lenu,3),precround(u[1]/lenu,3),precround(u[2]/lenu,3))", "description": "", "name": "unitu"}}, "ungrouped_variables": ["u", "lenu", "unitu"], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "$\\boldsymbol{\\hat{a}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 3d.p.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "unitu[0]-0.001", "maxValue": "unitu[0]+0.001", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "unitu[1]-0.001", "maxValue": "unitu[1]+0.001", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}, {"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "unitu[2]-0.001", "maxValue": "unitu[2]+0.001", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": " Find a unit vector$\\boldsymbol{\\hat{a}}$, which is parallel to$\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$. ", "tags": ["checked2015", "vector", "Vector"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Find the unit vector parallel to a given vector. \n Interim calculations in Advice should be presented in enough accuracy to ensure that the final calculations are to 3dps. "}, "advice": " Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity. \n There is only one unit vector parallel to a vector$\\boldsymbol{u}=\\pmatrix{u_1,u_2,u_3}$, namely the unit vector$\\boldsymbol{\\hat{u}}=\\boldsymbol{u}/\\lvert\\boldsymbol{u}\\rvert$, where$\\lvert\\boldsymbol{u}\\rvert=\\sqrt{u_1^2+u_2^2+u_3^2}$. \n In this question$\\lvert\\boldsymbol{u}\\rvert=\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}=\\var{precround(lenu,3)}$, and so$\\boldsymbol{\\hat{a}}=\\boldsymbol{\\hat{u}}=\\frac{1}{\\var{precround(lenu,3)}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}=\\pmatrix{\\var{unitu[0]},\\var{unitu[1]},\\var{unitu[2]}}$to 3d.p. \n There is also an anti-parallel unit vector$-\\boldsymbol{\\hat{u}}=\\pmatrix{\\var{-unitu[0]},\\var{-unitu[1]},\\var{-unitu[2]}}$. "}, {"name": "Parameterisation of a curve - find tangent and length", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"num3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "4*9*(a^2+t1*b^2)", "description": "", "name": "num3"}, "num4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "3*a", "description": "", "name": "num4"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((8/(27*4*b^2))*((9*a^2+((9*t2*4*b^2)/4))^(3/2)-(9*a^2+((9*t1*4*b^2)/4))^(3/2)),2)", "description": "", "name": "s"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "t1"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)+t1", "description": "", "name": "t2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "a"}, "num1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "9*4*b^2", "description": "", "name": "num1"}, "num2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "27*4*b^2", "description": "", "name": "num2"}}, "ungrouped_variables": ["a", "num4", "b", "num1", "num2", "num3", "t2", "t1", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": " Find the tangent vector$\\boldsymbol{u}$to the curve. \n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$. (Do not enter decimals in your answers.) ", "scripts": {}, "gaps": [{"answer": "{3*a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answers. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{3*b}*t^(1/2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "all", "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answers. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": " The arc-length along the curve can be written in the form$s(t)=f(t)-f(t_1)$. Find$f(t)$. \n$f(t)=$[[0]]. (Do not enter decimals in your answers.) ", "scripts": {}, "gaps": [{"answer": "2/{b^2}*({a^2}+({b^2})*t)^(3/2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "all", "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": false, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": " Find the total length of the curve,$S$, given$t_1=\\var{t1}$and$t_2=\\var{t2}$. \n$S=$[[0]]. (Enter your answer to 2d.p.) ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "s+0.01", "minValue": "s-0.01", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"displayType": "radiogroup", "choices": ["$s\\longmapsto\\pmatrix{\\simplify{{3a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\left(\\simplify{{b^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$", "$s\\longmapsto\\pmatrix{\\simplify{{a}/{3*b^2}}\\left\\{\\left[\\simplify{{a^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{27*b^2}}\\left\\{\\left[\\left(\\simplify{{a^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$", "$s\\longmapsto\\pmatrix{\\simplify{{3*a}}\\left\\{\\left[\\simplify{{a^2+t1*b^2}s/2}+\\left(\\var{b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{{2*b}}\\left\\{\\left[\\left(\\simplify{{a^2+t1*b^2}s/2}+\\left(\\var{b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$", "$s\\longmapsto\\pmatrix{\\simplify{{3}/{a*b^2}}\\left\\{\\left[\\simplify{{b^2}s/2}+\\left(\\var{a^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2+t1*b^2}\\right\\},\\simplify{2/{a^3*b^2}}\\left\\{\\left[\\left(\\simplify{{b^2}s/2}+\\left(\\var{a^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2+t1*b^2}\\right]\\right\\}^\\frac{3}{2}}$"], "showCorrectAnswer": true, "displayColumns": 0, "prompt": " Which of the following corresponds to an alternative parametric representation of the curve, again with$t_1=\\var{t1}$and$t_2=\\var{t2}$, using the arc-length$s$as the curve parameter? ", "distractors": ["", "", "", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [1, 0, 0, 0], "marks": 0}], "statement": " You are given the curve$t\\longmapsto\\pmatrix{\\var{3*a}t,\\var{2*b}t^\\frac{3}{2}}$, where$t_1\\leqslant t\\leqslant t_2$. ", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Calculation of the length and alternative form of the parameteric representation of a curve. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " The tangent vector to the curve$t\\longmapsto\\pmatrix{x,y}$is given by$\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$. \n The length$s$of the curve in the range$t_1\\leqslant t\\leqslant t_2$is given by \n \$s=\\int_{t_1}^{t_2}{u\\,\\mathrm{d}t},\$ \n where$u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$. \n In this question, therefore,$\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}(\\var{3*a}t),\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\var{2*b}t^\\frac{3}{2}\\right)}=\\pmatrix{\\var{3*a},\\simplify{{3*b}*t^(1/2)}}$, and$u^2=9\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right). \n Then \n \\\begin{align}s=\\int_{t_1}^{t_2}{u\\,\\mathrm{d}t}&=3\\int_{t_1}^{t_2}{\\sqrt{\\var{a^2}+\\simplify{{b^2}*t}}\\,\\mathrm{d}t}\\\\&=\\simplify{2/{b^2}}\\left[\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}\\right]_{t_1}^{t_2},\\end{align}\ \n sof(t)=\\simplify{2/{b^2}}\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}$. \n Finally, substitute$t_1=\\var{t1}$and$t_2=\\var{t2}$into the expression for$s$to find the length of the curve over the given range of$t$. \n Hence$s=\\simplify{2/{b^2}}\\left\\{\\left(\\var{a^2+t2*b^2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right\\}=\\var{s}$to 2d.p. \n An alternative parametric representation, using$sas the curve parameter is given by \n \\\begin{align}s(t)=\\int_{t_1}^t{u\\,\\mathrm{d}\\tau}&=3\\int_{t_1}^t{\\sqrt{\\var{a^2}+\\simplify{{b^2}*tau}}\\,\\mathrm{d}\\tau}\\\\&=\\simplify{2/{b^2}}\\left[\\left(\\var{a^2}+\\simplify{{b^2}*tau}\\right)^\\frac{3}{2}\\right]_{t_1}^t\\\\&=\\simplify{2/{b^2}}\\left\\{\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}-\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right\\}.\\end{align}\ \n Now rearrange this expression fort(s)$, so \n \$t(s)=\\simplify{1/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\$ \n and substitute into the original representation of the curve$t\\longmapsto\\pmatrix{\\var{3*a}t,\\var{2*b}t^\\frac{3}{2}}$with$t_1\\leqslant t\\leqslant t_2$. Hence \n \$s\\longmapsto\\pmatrix{\\simplify{{3*a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\}^\\frac{3}{2}}\$ \n with$0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2}+\\simplify{{b^2}*t_2}\\right)^\\frac{3}{2}-\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right)$. \n Finally, substitute$t_1=\\var{t1}$and$t_2=\\var{t2}$into the above expressions, to find the specific parametric representation corresponding to the given range of t: \n \$s\\longmapsto\\pmatrix{\\simplify{{3*a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\}^\\frac{3}{2}}\$ \n with$0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2+b^2*t2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right)$. "}, {"name": "Parametric representations of a curve - find tangent and length", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,4)", "description": "", "name": "t2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "b", "t2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{-a*b}*sin({b}*t)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "{-a*b}*cos({b}*t)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": " Find the tangent vector$\\boldsymbol{u}$to the curve. \n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$. (Do not enter decimals in your answers.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "({2*a*b}/{t2})*pi", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": " Find the length of the curve$s$, given$t_1=-\\simplify{1/{t2}}$and$t_2=\\simplify{1/{t2}}$. \n$s=$[[0]]. (Enter your answer as a fractional multiple of$\\pi$. Do not enter decimals.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a}*cos(s/{a}-{b}*pi/{t2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Enter your answer as a fractional multiple of$\\pi$. Do not enter decimals. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "{-a}*sin(s/{a}-{b}*pi/{t2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Enter your answer as a fractional multiple of$\\pi$. Do not enter decimals. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": " Find another parametric representation of the curve, again with$t_1=-\\simplify{1/{t2}}$and$t_2=\\simplify{1/{t2}}$, using$s$as the curve parameter, such that$0\\leqslant s\\leqslant \\simplify{{2*a*b}/{t2}}\\pi$. \n$s\\rightarrow($[[0]]$,$[[1]]$)$. (Enter your answers as fractional multiples of$\\pi$. Do not enter decimals.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " You are given the curve$t\\rightarrow\\pmatrix{\\var{a}\\cos(\\simplify{{b}t}),\\var{-a}\\sin(\\simplify{{b}t})}$, where$t_1\\pi\\leqslant t\\leqslant t_2\\pi$. ", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " The tangent vector to the curve$t\\rightarrow\\pmatrix{x,y}$is given by$\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$. \n The length$s$of the curve in the range$t_1\\pi\\leqslant t\\leqslant t_2\\pi$is given by \n \$s=\\int_{t_1\\pi}^{t_2\\pi}{u\\mathrm{d}t},\$ \n where$u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$. \n In this question, therefore,$\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left\\{\\var{a}\\cos(\\simplify{{b}*t})\\right\\},\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left\\{\\var{-a}\\sin(\\simplify{{b}*t})\\right\\}}=\\pmatrix{\\var{-a*b}\\sin(\\simplify{{b}*t}),\\var{-a*b}\\cos(\\simplify{{b}*t})}$, and so$u^2=\\var{(a*b)^2}$. \n Then \n \$s=\\int_{t_1\\pi}^{t_2\\pi}{u\\mathrm{d}t}=\\var{a*b}\\int_{t_1\\pi}^{t_2\\pi}{\\mathrm{d}t}=\\var{a*b}(t_2-t_1)\\pi.\$ \n Finally, substitute$t_1=-\\simplify{1/{t2}}$and$t_2=\\simplify{1/{t2}}$into this expression for$s$, to find the length of the curve over the given range of$t$. \n Hence$s=\\simplify{{2*a*b}/{t2}}\\pi$. \n An alternative parametric representation, using$s$as the curve parameter is given by \n \$s=\\int_{t_1\\pi}^{t}{u\\mathrm{d}\\tau}=\\var{a*b}\\int_{t_1\\pi}^{t}{\\mathrm{d}\\tau}=\\var{a*b}(t-t_1)\\pi.\$ \n Now rearrange this expression for$t(s)$, so \n \$t(s)=\\frac{s}{\\var{a*b}}+t_1\\pi,\$ \n and substitute into the original representation of the curve$t\\rightarrow\\pmatrix{\\var{a}\\cos(\\simplify{{b}t}),\\var{-a}\\sin(\\simplify{{b}t})}$with$t_1\\pi\\leqslant t\\leqslant t_2\\pi$. Hence \n \$s\\rightarrow\\pmatrix{\\var{a}\\cos\\left(\\simplify{s/{a}}+\\var{b}t_1\\pi\\right),\\var{-a}\\sin\\left(\\simplify{s/{a}}+\\var{b}t_1\\pi\\right)},\$ \n with$0\\leqslant s\\leqslant\\var{a*b}(t_2-t_1)\\pi$. \n Finally, substitute$t_1=-\\simplify{1/{t2}}$and$t_2=\\simplify{1/{t2}}$into the above expressions, to find the specific parametric representation corresponding to the given range of t: \n \$s\\rightarrow\\pmatrix{\\var{a}\\cos\\left(\\simplify{s/{a}}-\\simplify{{b*pi}/{t2}}\\right),\\var{-a}\\sin\\left(\\simplify{s/{a}}-\\simplify{{b*pi}/{t2}}\\right)},\$ \n with$0\\leqslant s\\leqslant \\simplify{{2*a*b}/{t2}}\\pi$. "}, {"name": "Cartesian parameterisation of surface, normal vector, and magnitude", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "a", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "t", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "c", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "b", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "t"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Express the surface in the form$z=g(x,y)$. \n$z=g(x,y)=$[[0]]. (Do not enter decimals in your answer.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "({d}/{a^2})*x^2+({d}/{c^2})*y^2", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": " Do not enter decimals in your answer. ", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Using the above Cartesian form$z=g(x,y)$of the surface you have just found, find a normal vector$\\boldsymbol{n}$with$z$-component equal to$1$. \n$\\boldsymbol{n}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Do not enter decimals in your answer.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "({-2d}/{a^2})*x", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": " Do not enter decimals in your answer. ", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "({-2d}/{c^2})*y", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": " Do not enter decimals in your answer. ", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "1", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": " Do not enter decimals in your answer. ", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": " Calculate the magnitude$|\\boldsymbol{n}|$of the normal vector$\\boldsymbol{n}$. \n$|\\boldsymbol{n}|=[[0]]. (Do not enter decimals in your answer.) ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "sqrt(({4*d^2}/{a^4})x^2+({4*d^2}/{c^4})y^2+1)", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": " Do not enter decimals in your answer. ", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": " You are given the following surface, defined in parametric form \n \$\\pmatrix{u,v}\\rightarrow\\pmatrix{\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\simplify{{d}*v^2}}, \\quad 0\\leqslant u\\leqslant 2\\pi, 0\\leqslant v\\leqslant 1\$ ", "tags": ["checked2015", "normals", "parametric form", "surfaces"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Cartesian form of the parametric representation of a surface, normal vector, and magnitude. \n Accuracy for part c) should be made more stringent as can be marked correct for an incorrect answer. Use a different sample range rather than 0 to 1 would help as would setting accuracy to something less than 0.001. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " The given surface has components \n \\\begin{align}x&=\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\\\y&=\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\\\z&=\\simplify{{d}*v^2}.\\end{align}\ \n Then \n \$\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=(\\simplify{v*({1-t}*cos({b}*u)+{t}*sin({b}*u))})^2+(\\simplify{v*({1-t}*sin({b}*u)+{t}*cos({b}*u))})^2=v^2,\$ \n but we know thatz=\\simplify{{d}*v^2}$, so \n \$\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{z/{d}},\$ \n and hence \n \$z=\\simplify{{d}/{a^2}x^2+{d}/{c^2}y^2}.\$ \n \n Given the above form for$z=g(x,y)$, a normal vector$\\boldsymbol{n}$, with positive$z$-component is \n \$\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}.\$ \n In this case \n \$\\boldsymbol{n}=\\pmatrix{\\simplify{{-2d}/{a^2}}x,\\simplify{{-2d}/{c^2}}y,1}\$ \n by straightforward partial differentiation. \n \n The magnitude$\\lvert\\boldsymbol{n}\\rvert$of the normal vector$\\boldsymbol{n}$is given by$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}$, and hence \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{\\simplify{{4*d^2}/{a^4}}x^2+\\simplify{{4*d^2}/{c^4}}y^2+1}.\$ "}, {"name": "Find a unit normal vector to a surface given in Cartesian form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"n2hat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sign(n2[0])*vector(precround(n2[0]/lenn2,4),precround(n2[1]/lenn2,4),precround(n2[2]/lenn2,4))", "description": "", "name": "n2hat"}, "p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p4"}, "p13": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "name": "p13"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "p1"}, "p12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p12"}, "p9": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p9"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p3"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "n2andmultiply": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n2[0]<0,\", and multiply by$-1$,\",\",\")", "description": "", 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"description": "", "name": "q"}, "lenn2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(n2)", "description": "", "name": "lenn2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a2"}, "p8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p8"}, "p7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p7"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a2)*sign(random(-1,1))", "description": "", "name": "b2"}, "p5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p5"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p2"}, "lenn1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(n1)", "description": "", "name": "lenn1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)))", "description": "", "name": "r"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(p1*a1*r[0]^(p1-1)*r[1]^p2*r[2]^p3+p4*b1*r[0]^(p4-1)*r[1]^p5*r[2]^p6+p7*c1*r[0]^(p7-1)*r[1]^p8*r[2]^p9,p2*a1*r[0]^p1*r[1]^(p2-1)*r[2]^p3+p5*b1*r[0]^p4*r[1]^(p5-1)*r[2]^p6+p8*c1*r[0]^p7*r[1]^(p8-1)*r[2]^p9,p3*a1*r[0]^p1*r[1]^p2*r[2]^(p3-1)+p6*b1*r[0]^p4*r[1]^p5*r[2]^(p6-1)+p9*c1*r[0]^p7*r[1]^p8*r[2]^(p9-1))", "description": "", "name": "n1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "p6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p6"}, "p10": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "p10"}}, "ungrouped_variables": ["n1hat", "n2hat", "n1andmultiply", "b1", "b2", "lenn2", "lenn1", "r", "p10", "a1", "p11", "p12", "p13", "c1", "p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "q", "a2", "n1", "n2", "n2andmultiply"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[0]+0.001", "minValue": "n1hat[0]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[1]+0.001", "minValue": "n1hat[1]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[2]+0.001", "minValue": "n1hat[2]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$\\simplify{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*x^{p7}*y^{p8}*z^{p9}}=0$at$\\pmatrix{\\var{r[0]},\\var{r[1]},\\var{r[2]}}$. \n$\\boldsymbol{\\hat{n}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 3d.p.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[0]+0.0001", "minValue": "n2hat[0]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[1]+0.0001", "minValue": "n2hat[1]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[2]+0.0001", "minValue": "n2hat[2]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$z=\\simplify{{a2}*x^{p10}*y^{p11}+{b2}*x^{p12}*y^{p13}}$at$\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$. \n$\\boldsymbol{\\hat{n}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 4d.p.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " Find a unit normal vector$\\boldsymbol{\\hat{n}}$, with positive$x$-component, to the following surfaces, at the given point. ", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": " 19/11/2013 \n Set p1/=0, p4=p7=0, so that the x-derivative in part a) is never zero. This avoids division by zero when calculating n1hat. \n q[1]->q[0] in definition of n2. \n Set p12=0, so that the x-derivative in part b) is never zero. This avoids division by zero when calculating n2hat. \n 12/11/2013 \n Set a2 not equal to b2, to avoid any degenerate surfaces in part b). ", "licence": "Creative Commons Attribution 4.0 International", "description": " Unit normal vector to a surface, given in Cartesian form. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity. \n a) \n The surface is given in the form$f(x,y,z)=0, and so a normal vector to this surface is given by \n \\\begin{align}\\boldsymbol{n}=\\boldsymbol{\\nabla}f&=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}\\\\&=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{p7*c1}*x^{p7-1}*y^{p8}*z^{p9}},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{p8*c1}*x^{p7}*y^{p8-1}*z^{p9}},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{p9*c1}*x^{p7}*y^{p8}*z^{p9-1}}}.\\end{align}\ \n Then \n \$\\boldsymbol{n}\\vert_{\\pmatrix{\\var{r[0]},\\var{r[1]},\\var{r[2]}}}=\\pmatrix{\\var{n1[0]},\\var{n1[1]},\\var{n1[2]}}\$ \n is one such normal to the surface at the given point, by straight forward substitution of the components of the point into the components of the normal. \n The unit normal vector to the surface at the given point is given by\\boldsymbol{\\hat{n}}=\\frac{\\boldsymbol{n}}{\\lvert\\boldsymbol{n}\\rvert}$, where \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}.\$ \n In this case \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{(\\var{n1[0]})^2+(\\var{n1[1]})^2+(\\var{n1[2]})^2}=\\var{precround(lenn1,3)}\\;\\text{to 3d.p.}\$ \n Now divide each component of$\\boldsymbol{n}$by this value{n1andmultiply} so that the normal vector to the surface at the given point, with positive$x$-component is \n \$\\boldsymbol{\\hat{n}}=\\pmatrix{\\var{n1hat[0]},\\var{n1hat[1]},\\var{n1hat[2]}}\\;\\text{to 3d.p.}\$ \n \n b) \n The surface is given in the form$z=g(x,y)$. This expression can either be rearranged into the form$f(x,y,z)=0$, and then adopt the same method of solution as in part a) or, alternatively, a normal vector to this surface is given by \n \$\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}=\\pmatrix{\\simplify{{-p10*a2}*x^{p10-1}*y^{p11}-{p12*b2}*x^{p12-1}*y^{p13}},\\simplify{{-p11*a2}*x^{p10}*y^{p11-1}-{p13*b2}*x^{p12}*y^{p13-1}},1}.\$ \n Then \n \$\\boldsymbol{n}\\vert_{\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}}=\\pmatrix{\\var{n2[0]},\\var{n2[1]},\\var{n2[2]}}\$ \n is one such normal to the surface at the given point, by straight forward substitution of the components of the point into the components of the normal. \n The unit normal vector to the surface at the given point is given by$\\boldsymbol{\\hat{n}}=\\frac{\\boldsymbol{n}}{\\lvert\\boldsymbol{n}\\rvert}$, where \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}.\$ \n In this case \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{(\\var{n2[0]})^2+(\\var{n2[1]})^2+(\\var{n2[2]})^2}=\\var{precround(lenn2,4)}\\;\\text{to 4d.p.}\$ \n Now divide each component of$\\boldsymbol{n}$by this value{n2andmultiply} so that the normal vector to the surface at the given point, with positive$x$-component is \n \$\\boldsymbol{\\hat{n}}=\\pmatrix{\\var{n2hat[0]},\\var{n2hat[1]},\\var{n2hat[2]}}\\;\\text{to 4d.p.}.\$ "}, {"name": "Find Cartesian form of a surface, and a normal vector", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "t"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "d", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{d}*sqrt(x^2/{a^2}+y^2/{c^2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n Express the surface in the form$z=g(x,y)$. \n$z=g(x,y)=$[[0]]. (Do not enter decimals in your answer.) \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-({d^2}/{a^2})*x", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "-({d^2}/{c^2})*y", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": " Using the above Cartesian form$z=g(x,y)$of the surface you have just found, a normal vector$\\boldsymbol{n}$with$z$-component equal to$1$, can be written in the form$\\boldsymbol{n}=\\pmatrix{\\frac{p(x)}{z},\\frac{q(y)}{z},k}$. Fill in the values for$p(x)$,$q(y)$, and$k$below. \n$\\boldsymbol{n}=($[[0]]$/z,$[[1]]$/z,$[[2]]$)$. (Do not enter decimals in your answer.) ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "sqrt(({d^4}/{a^4})*(x^2/z^2)+({d^4}/{c^4})*(y^2/z^2)+1)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": " Do not enter decimals in your answer. ", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": " Calculate the magnitude$|\\boldsymbol{n}|$of the normal vector$\\boldsymbol{n}$, using the expression for$z$to simplify your answer. \n$|\\boldsymbol{n}|=[[0]]. (Do not enter decimals in your answer.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " You are given the following surface, defined in parametric form \n \$\\pmatrix{u,v}\\rightarrow\\pmatrix{\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\simplify{{d}*v}}, \\quad 0\\leqslant u\\leqslant 2\\pi, 0\\leqslant v\\leqslant 1\$ ", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Cartesian form of the parametric representation of a surface, normal vector, and magnitude. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " The given surface has components \n \\\begin{align}x&=\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\\\y&=\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\\\z&=\\simplify{{d}*v}.\\end{align}\ \n Then \n \$\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=(\\simplify{v*({1-t}*cos({b}*u)+{t}*sin({b}*u))})^2+(\\simplify{v*({1-t}*sin({b}*u)+{t}*cos({b}*u))})^2=v^2,\$ \n but we know thatz=\\simplify{{d}*v}$, so \n \$\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{(z/{d})^2},\$ \n and hence \n \$z=\\var{d}\\sqrt{\\simplify{x^2/{a^2}}+\\simplify{y^2/{c^2}}},\$ \n where we have taken the positive square root because$v\\geqslant 0\\implies z\\geqslant 0$. \n \n Given the above form for$z=g(x,y)$, a normal vector$\\boldsymbol{n}$, with positive$z$-component is \n \$\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}.\$ \n In this case \n \$\\boldsymbol{n}=\\pmatrix{\\simplify{{-d^2}/{a^2}}\\frac{x}{z},\\simplify{{-d^2}/{c^2}}\\frac{y}{z},1}\$ \n by straightforward partial differentiation. \n \n The magnitude$\\lvert\\boldsymbol{n}\\rvert$of the normal vector$\\boldsymbol{n}$is given by$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}$, and hence \n \$\\lvert\\boldsymbol{n}\\rvert=\\sqrt{\\simplify{{d^4}/{a^4}}\\frac{x^2}{z^2}+\\simplify{{d^4}/{c^4}}\\frac{y^2}{z^2}+1}.\$ "}, {"name": "Find gradient of scalar field, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"p4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p4"}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "e1"}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p1"}, "p12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "description": "", "name": "p12"}, "p9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "description": "", "name": "p9"}, "p3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p3"}, "p8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "description": "", "name": "p8"}, "p7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p8=0 and p9=0,1,random(0,1))", "description": "", "name": "p7"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "d1"}, "p5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p5"}, "p2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p2"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "description": "", "name": "t"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "p11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "description": "", "name": "p11"}, "p6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "description": "", "name": "p6"}, "p10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p11=0 and p12=0,1,random(0,1))", "description": "", "name": "p10"}}, "ungrouped_variables": ["p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "a1", "p11", "p12", "t", "b1", "p10", "c1", "e1", "d1"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "$f(x,y,z)=\\simplify[std]{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*({1-t}*sin({d1}*x^{p7}*y^{p8}*z^{p9})+{t}*cos({e1}*x^{p10}*y^{p11}*z^{p12}))}$. \n$\\boldsymbol{\\nabla}f=($[[0]]$,$[[1]]$,$[[2]]$)$. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"answer": "{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}, {"answer": "{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "failureRate": 1, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": " Find$\\boldsymbol{\\nabla}f$for the following function$f(x,y,z)$. ", "tags": ["checked2015", "gradient", "nabla", "partial derivatives", "scalar field"], "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Gradient of$f(x,y,z)$. \n Should warn that multiplied terms need * to denote multiplication. "}, "advice": " This question is simply an exercise in partial differentiation, using the fact that \n \$\\boldsymbol{\\nabla}f=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}.\$ \n Hence \n \$\\boldsymbol{\\nabla}f=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}}.\$ "}, {"name": "Find surface of points in scalar field orthogonal to the z axis, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p4", "description": ""}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p1", "description": ""}, "p12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p12", "description": ""}, "p9": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p9", "description": ""}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,3,5,7)", "name": "p3", "description": ""}, "p8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p8", "description": ""}, "p7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p7", "description": ""}, "p5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p5", "description": ""}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p2", "description": ""}, "p11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p11", "description": ""}, "p6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5 except p3)", "name": "p6", "description": ""}, "p10": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..5)", "name": "p10", "description": ""}}, "ungrouped_variables": ["p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "p10", "p11", "p12"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "marks": 1, "checkingType": "absdiff", "scripts": {}, "answerSimplification": "all", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": ["x", "y", "z"], "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}], "prompt": "$f(x,y,z)=\\simplify{x^{p1}*y^{p2}*z^{p3}+x^{p4}*y^{p5}*z^{p6}+x^{p7}*y^{p8}*z^{p9}+x^{p10}*y^{p11}*z^{p12}}$. \n$g(x,y,z)=$[[0]]. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": " For the following scalar field$f$, find all the points for which$\\boldsymbol{\\nabla}f$is orthogonal to the$z$-axis. Enter your answer in the form of a surface$g(x,y,z)=0$. ", "tags": ["checked2015", "nabla", "scalar field", "surface"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Find all points for which the gradient of a scalar field is orthogonal to the$z$-axis. \n Should warn that multiplied terms need * to denote multiplication. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " A vector that is orthogonal to the$z$-axis has its$z$-component equal to zero. We need to find all the points, therefore, for which the$z$-component of$\\boldsymbol{\\nabla}f$is zero, i.e.$(\\boldsymbol{\\nabla}f)_z=0$. \n The$z$-component of$\\boldsymbol{\\nabla}f$is \n \$(\\boldsymbol{\\nabla}f)_z=\\frac{\\partial f}{\\partial z}=\\simplify{{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}},\$ \n and so the surface \n \$g(x,y,z)=\\simplify{{p3}*x^{p1}*y^{p2}*z^{p3-1}+{p6}*x^{p4}*y^{p5}*z^{p6-1}+{p9}*x^{p7}*y^{p8}*z^{p9-1}+{p12}*x^{p10}*y^{p11}*z^{p12-1}}=0\$ \n defines the set of points for which$\\boldsymbol{\\nabla}f$is orthogonal to the$z$-axis. "}, {"name": "Calculate divergence of vector fields", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a3"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c3"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "p1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,1))", "description": "", "name": "a2"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b3"}}, "ungrouped_variables": ["p1", "a1", "a3", "a2", "b1", "b3", "c3", "c1"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "$\\boldsymbol{u}=\\pmatrix{(\\simplify[std]{{a1}*x+{b1}*y+{c1}*z})(\\simplify[std]{{b1}*y-{c1}*z}),(\\simplify[std]{{a1}*y+{b1}*z+{c1}*x})(\\simplify[std]{{b1}*z-{c1}*x}),(\\simplify[std]{{a1}*z+{b1}*x+{c1}*y})(\\simplify[std]{{b1}*x-{c1}*y})}$. \n$\\boldsymbol{\\nabla\\cdot u}=$[[0]]. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a1*b1-a1*c1}*x+{a1*b1-a1*c1}*y+{a1*b1-a1*c1}*z", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "$\\boldsymbol{u}=\\pmatrix{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\left(y^\\var{p1}+z^\\var{p1}\\right)\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\\left(z^\\var{p1}+x^\\var{p1}\\right)\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)}$. \n$\\boldsymbol{\\nabla\\cdot u}=$[[0]]. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "0", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "$\\boldsymbol{u}=\\pmatrix{\\simplify{{a3}*x}+f_1(y,z),\\simplify{{b3}*y}+f_2(x,z),\\simplify{{c3}*z}+f_3(x,y)}$, for any general functions$f_1$,$f_2$, and$f_3$. \n$\\boldsymbol{\\nabla\\cdot u}=$[[0]]. ", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a3+b3+c3}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "all", "expectedVariableNames": ["x", "y", "z"], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": " For each of the following vector fields$\\boldsymbol{u}$, find the divergence$\\boldsymbol{\\nabla\\cdot u}$. ", "tags": ["checked2015"], "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Divergence of vector fields. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " The divergence of a vector field$\\boldsymbol{u}=\\pmatrix{u_1,u_2,u_3}$is given by \n \$\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_1}{\\partial x}+\\frac{\\partial u_2}{\\partial y}+\\frac{\\partial u_3}{\\partial z}.\$ \n a) \n The variables$x$,$y$, and$z$appear in a cyclical manner in each of the three components of$\\boldsymbol{u}$. Once you have calculated$\\frac{\\partial u_1}{\\partial x}, you can use cyclic permutations to determine the other two derivatives. Hence \n \\\begin{align}\\frac{\\partial u_1}{\\partial x}&=\\frac{\\partial}{\\partial x}\\left(\\simplify{({a1}*x+{b1}*y+{c1}*z)*({b1}*y-{c1}*z)}\\right)\\\\&=\\simplify{{a1}*({b1}*y-{c1}*z)},\\end{align}\ \n and so, cyclically permuting the variables, \n \$\\frac{\\partial u_2}{\\partial y}=\\simplify{{a1}*({b1}*z-{c1}*x)}\$ \n and \n \$\\frac{\\partial u_3}{\\partial z}=\\simplify{{a1}*({b1}*x-{c1}*y)}.\$ \n Finally, adding the components together gives the divergence \n \$\\boldsymbol{\\nabla\\cdot u}=\\simplify[std]{{a1}*({b1}*y-{c1}*z)+{a1}*({b1}*z-{c1}*x)+{a1}*({b1}*x-{c1}*y)}=\\simplify{{a1*b1-a1*c1}*x+{a1*b1-a1*c1}*y+{a1*b1-a1*c1}*z}.\$ \n \n b) \n As in part a) the variablesx$,$y$, and$z$appear cyclically in each component of$\\boldsymbol{u}, so we only need calculate one derivative explicitly, then use cyclic permutations to calculate the other two. Hence \n \\\begin{align}\\frac{\\partial u_1}{\\partial x}&=\\frac{\\partial}{\\partial x}\\left\\{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)\\right\\}\\\\&=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\end{align}\ \n and by symmetry \n \$\\frac{\\partial u_2}{\\partial y}=\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\$ \n and \n \$\\frac{\\partial u_3}{\\partial z}=\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right).\$ \n Finally, adding the derivatives together gives the divergence \n \$\\boldsymbol{\\nabla\\cdot u}=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)+\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right)+\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)=0.\$ \n \n c) \n First note thatf_1$does not depend on$z$,$f_2$does not depend on$y$, and$f_3$does not depend on$z. This makes the differentiation very straight forward, and hence \n \\\begin{align}\\boldsymbol{\\nabla\\cdot u}&=\\frac{\\partial}{\\partial x}\\left(\\simplify{{a3}*x}+f_1(y,z)\\right)+\\frac{\\partial}{\\partial y}\\left(\\simplify{{b3}*y}+f_2(x,z)\\right)+\\frac{\\partial}{\\partial z}\\left(\\simplify{{c3}*z}+f_3(x,y)\\right)\\\\&=\\simplify[all,!collectNumbers]{{a3}+{b3}+{c3}}\\\\&=\\var{a3+b3+c3}.\\end{align}\ "}, {"name": "Find curl and divergence of a vector field", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": " For the vector field\\boldsymbol{u}=\\pmatrix{\\simplify{{a1}*x^{p1}+{b1}*y^{p2}*z^{p3}},\\simplify{{c1}*y^{p4}+{d1}*x^{p5}*z^{p6}},\\simplify{{e1}*z^{p7}+{f1}*x^{p8}*y^{p9}}}$, calculate$\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$and$\\boldsymbol{\\nabla\\cdot u}$, and determine whether$\\boldsymbol{u}$is irrotational or solenoidal, or both. ", "functions": {}, "rulesets": {}, "tags": ["checked2015"], "preamble": {"css": "", "js": ""}, "variables": {"p9": {"description": "", "definition": "random(1..9 except p6)", "name": "p9", "group": "Ungrouped variables", "templateType": "anything"}, "p5": {"description": "", "definition": "random(1..9 except p2)", "name": "p5", "group": "Ungrouped variables", "templateType": "anything"}, "irrequal": {"description": "", "definition": "if(n=1,\"is equal\",\"is not equal\")", "name": "irrequal", "group": "Ungrouped variables", "templateType": "anything"}, "a1": {"description": "", "definition": "if(n=2,0,random(1..9))", "name": "a1", "group": "Ungrouped variables", "templateType": "anything"}, "e1": {"description": "", "definition": "if(n=2,0,random(1..9))", "name": "e1", "group": "Ungrouped variables", "templateType": "anything"}, "n": {"description": "", "definition": "random(0..2)", "name": "n", "group": "Ungrouped variables", "templateType": "anything"}, "notirrotational": {"description": "", "definition": "if(n=1,\"No\",\"Yes\")", "name": "notirrotational", "group": "Ungrouped variables", "templateType": "anything"}, "solenoidal": {"description": "", "definition": "if(n=2,\"Yes\",\"No\")", "name": "solenoidal", "group": "Ungrouped variables", "templateType": "anything"}, "p4": {"description": "", "definition": "random(1..9)", "name": "p4", "group": "Ungrouped variables", "templateType": "anything"}, "isirr": {"description": "", "definition": "if(n=1,\"is irrotational\",\"is not irrotational\")", "name": "isirr", "group": "Ungrouped variables", "templateType": "anything"}, "issol": {"description": "", "definition": "if(n=2,\"is solenoidal\",\"is not solenoidal\")", "name": "issol", "group": "Ungrouped variables", "templateType": "anything"}, "p6": {"description": "", "definition": "random(1..9)", "name": "p6", "group": "Ungrouped variables", "templateType": "anything"}, "irrotational": {"description": "", "definition": "if(n=1,\"Yes\",\"No\")", "name": "irrotational", "group": "Ungrouped variables", "templateType": "anything"}, "b1": {"description": "", "definition": "if(n=1,0,random(1..9))", "name": "b1", "group": "Ungrouped variables", "templateType": "anything"}, "c1": {"description": "", "definition": "if(n=2,0,random(1..9))", "name": "c1", "group": "Ungrouped variables", "templateType": "anything"}, "p8": {"description": "", "definition": "random(1..9 except p3)", "name": "p8", "group": "Ungrouped variables", "templateType": "anything"}, "p1": {"description": "", "definition": "random(1..9)", "name": "p1", "group": "Ungrouped variables", "templateType": "anything"}, "d1": {"description": "", "definition": "if(n=1,0,random(1..9))", "name": "d1", "group": "Ungrouped variables", "templateType": "anything"}, "solequal": {"description": "", "definition": "if(n=2,\"is equal\",\"is not equal\")", "name": "solequal", "group": "Ungrouped variables", "templateType": "anything"}, "p2": {"description": "", "definition": "random(1..9)", "name": "p2", "group": "Ungrouped variables", "templateType": "anything"}, "p7": {"description": "", "definition": "random(1..9)", "name": "p7", "group": "Ungrouped variables", "templateType": "anything"}, "p3": {"description": "", "definition": "random(1..9)", "name": "p3", "group": "Ungrouped variables", "templateType": "anything"}, "f1": {"description": "", "definition": "if(n=1,0,random(1..9))", "name": "f1", "group": "Ungrouped variables", "templateType": "anything"}, "notsolenoidal": {"description": "", "definition": "if(n=2,\"No\",\"Yes\")", "name": "notsolenoidal", "group": "Ungrouped variables", "templateType": "anything"}}, "metadata": {"description": " Curl and divergence of a vector field. Determine whether the vector field is irrotational or solenoidal. ", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "advice": " The curl of a vector field$\\boldsymbol{u}=\\pmatrix{u_x,u_y,u_z}$is given by \n \$\\boldsymbol{\\nabla}\\times\\boldsymbol{u}=\\pmatrix{\\frac{\\partial u_z}{\\partial y}-\\frac{\\partial u_y}{\\partial z},\\frac{\\partial u_x}{\\partial z}-\\frac{\\partial u_z}{\\partial x},\\frac{\\partial u_y}{\\partial x}-\\frac{\\partial u_x}{\\partial y}}.\$ \n The divergence of the same vector field is given by \n \$\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}.\$ \n #### a) \n By straightforward partial differentiation \n \$\\boldsymbol{\\nabla\\cdot u}=\\pmatrix{\\simplify{{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}},\\simplify{{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}},\\simplify{{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}}}.\$ \n #### b) \n Again, by partial differentiation \n \$\\boldsymbol{\\nabla\\cdot u}=\\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.\$ \n A vector field is irrotational if its curl is equal to the zero vector; a vector field is solenoidal if its divergence is equal to zero. \n #### c) \n Since$\\boldsymbol{\\nabla}\\times\\boldsymbol{u}${irrequal} to the zero vector, the vector field {isirr}. \n #### d) \n Since$\\boldsymbol{\\nabla\\cdot u}${solequal} to zero, the vector field {issol}. 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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": " You are given the scalar field$f=\\simplify{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}}$. ", "advice": " Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity. \n a) \n The gradient of$fis given by \n \\\begin{align}\\boldsymbol{\\nabla}f&=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}\\\\&=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}}},\\end{align}\ \n by straight forward partial differentiation. \n \n b) \n The gradient off$at the point$\\boldsymbol{q}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$is found by substituting$\\boldsymbol{q}$into$\\boldsymbol{\\nabla}f$, hence \n \$\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}.\$ \n \n c) \n The unit vector$\\boldsymbol{\\hat{u}}$in the direction of$\\boldsymbol{u}is given by \n \\\begin{align}\\boldsymbol{\\hat{u}}=\\frac{\\boldsymbol{u}}{\\lvert\\boldsymbol{u}\\rvert}&=\\frac{1}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\frac{1}{\\var{precround(lenu,3)}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}.\\end{align}\ \n \n d) \n The directional derivative\\frac{\\partial f}{\\partial\\boldsymbol{u}}$, of the scalar field$f$, in the direction of$\\boldsymbol{u}$, at the point$\\boldsymbol{q}is given by \n \\\begin{align}\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}&=\\boldsymbol{\\hat{u}\\cdot\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}\\boldsymbol{\\cdot}\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}\\\\&=\\var{uhatdotgradfq}\\;\\text{to 3d.p., using the full calculator display for the answers in the previous part.}\\end{align}\ ", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p4": {"name": "p4", "group": "Ungrouped variables", "definition": "random(0..2 except p1)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..2)*sign(random(-1,1)),3))", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", 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", "tags": ["checked2015"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Curl of a vector field. \n Should warn that multiplied terms need * to denote multiplication. 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outward normal at$x=\\var{d1}$:$($[[0]]$,$[[1]]$,$[[2]]$)$• \n • Unit outward normal at$y=\\var{e1}$:$($[[3]]$,$[[4]]$,$[[5]]$)$• \n • Unit outward normal at$y=\\var{f1}$:$($[[6]]$,$[[7]]$,$[[8]]$)$• \n • Unit outward normal at$z=\\var{g1}$:$($[[9]]$,$[[10]]$,$[[11]]$)$• \n • Unit outward normal at$\\simplify{{a1}*x+{b1}*z}=\\var{c1}$:$($[[12]]$,$[[13]]$,$[[14]]$)$(Enter your answer to 3d.p.) • \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "vol+0.001", "minValue": "vol-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": " By calculating a volume integral, find the volume$V$of the region enclosed by the above surfaces. \n$V=$[[0]]. (Enter your answer to 3d.p.) ", "showCorrectAnswer": true, "marks": 0}], "statement": " A region is enclosed by the surfaces$\\simplify{{a1}*x+{b1}*z}=\\var{c1}$,$x=\\var{d1}$,$y=\\var{e1}$,$y=\\var{f1}$, and$z=\\var{g1}$. ", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": " Outward normals to the surfaces enclosing a region; volume of that enclosed region. "}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " a) \n The unit outward normals can most easily be identified by sketching the region bounded by the given surfaces which, in this case, is a wedge. \n Then the unit outward normals to the non-slanted surfaces are given by \n \n • Unit outward normal at$x=\\var{d1}$:$\\pmatrix{-1,0,0}$• \n • Unit outward normal at$y=\\var{e1}$:$\\pmatrix{0,-1,0}$• \n • Unit outward normal at$y=\\var{f1}$:$\\pmatrix{0,1,0}$• \n • Unit outward normal at$z=\\var{g1}$:$\\pmatrix{0,0,-1}$• \n \n \n The outward normal to the final, slanted surface, is given by \n \$\\boldsymbol{\\nabla}(\\simplify{{a1}*x+{b1}*z-{c1}})=\\pmatrix{\\var{a1},0,\\var{b1}},\$ \n and so the unit outward normal is given by \n \$\\frac{1}{\\sqrt{(\\var{a1})^2+(\\var{b1})^2}}\\pmatrix{\\var{a1},0,\\var{b1}}=\\pmatrix{\\var{gradfhat[0]},\\var{gradfhat[1]},\\var{gradfhat[2]}}.\$ \n \n b) \n The volume of a region$V$bounded by some particular surfaces is given by \n \$V=\\int_V\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z.\$ \n The relevant integral for the wedge in this question is therefore \n \$V=\\int_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\mathrm{d}x\\int_{y=\\var{e1}}^{\\var{f1}}\\mathrm{d}y\\int_{z=\\var{g1}}^{\\simplify{{c1}/{b1}-{a1}/{b1}*x}}\\mathrm{d}z.\$ \n The integrals in$y$and$zare straight forward, and we are left with \n \\\begin{align}V&=\\var{f1-e1}\\int_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\left(\\simplify{{-a1}/{b1}*x+{c1}/{b1}-{g1}}\\right)\\mathrm{d}x\\\\&=\\var{f1-e1}\\left[\\simplify{{-a1}/{2*b1}*x^2+{c1-g1*b1}/{b1}*x}\\right]_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\\\&=\\var{f1-e1}\\left\\{\\left(\\simplify{{-(c1-b1*g1)^2}/{2*a1*b1}+{(c1-g1*b1)^2}/{a1*b1}}\\right)-\\left(\\simplify{{-a1*d1^2}/{2*b1}+{d1*(c1-g1*b1)}/{b1}}\\right)\\right\\}\\\\&=\\var{vol}\\;\\text{to 3d.p.}\\end{align}\ \n "}, {"name": "Find normals to surfaces enclosing a region, and the volume of the region", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b1"}, "theta2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi/2", "description": "", "name": "theta2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a1"}, "vol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(0.5*b1*(theta2-theta1)*a1^2,3)", "description": "", "name": "vol"}, "theta1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "theta1"}}, "ungrouped_variables": ["a1", "vol", "theta1", "b1", "theta2"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": " Outward normals to the surfaces enclosing a region; volume of that enclosed region. "}, "parts": [{"prompt": " Find the unit outward normal to each component of the region's boundary. \n \n • Unit outward normal atx=0$:$($[[0]]$,$[[1]]$,$[[2]]$)$• \n • Unit outward normal at$y=0$:$($[[3]]$,$[[4]]$,$[[5]]$)$• \n • Unit outward normal at$z=0$:$($[[6]]$,$[[7]]$,$[[8]]$)$• \n • Unit outward normal at$z=\\var{b1}$:$($[[9]]$,$[[10]]$,$[[11]]$)$• \n • Unit outward normal at curved face:$($[[12]]$,$[[13]]$,$[[14]]$)$(Enter your answer to 3d.p.) • \n \n ", "scripts": {}, "gaps": [{"answer": "-1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, 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"originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "x/sqrt(x^2+y^2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "all", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "y/sqrt(x^2+y^2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "all", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\n By calculating a volume integral, find the volume$V$of the region enclosed by the above surfaces. \n$V=$[[0]]. (Enter your answer to 3d.p.) \n ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "vol+0.001", "minValue": "vol-0.001", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": " A region is enclosed by the surfaces$x^2+y^2\\leqslant\\var{a1^2}$,$0\\leqslant z\\leqslant\\var{b1}$,$x\\geqslant 0$,$y\\geqslant 0$. ", "tags": ["checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": " a) \n The unit outward normals can most easily be identified by sketching the region bounded by the given surfaces which, in this case, is a quarter cylinder. \n Then the unit outward normals to the non-slanted surfaces are given by \n \n • Unit outward normal at$x=0$:$\\pmatrix{-1,0,0}$• \n • Unit outward normal at$y=0$:$\\pmatrix{0,-1,0}$• \n • Unit outward normal at$z=0$:$\\pmatrix{0,0,-1}$• \n • Unit outward normal at$z=\\var{b1}$:$\\pmatrix{0,0,1}$• \n \n \n The outward normal to the final, curved surface, is given by \n \$\\boldsymbol{\\nabla}(\\simplify{x^2+y^2-{a1^2}})=\\pmatrix{2x,2y,0},\$ \n and so the unit outward normal is given by \n \$\\frac{1}{\\sqrt{x^2+y^2}}\\pmatrix{x,y,0}.\$ \n \n b) \n The volume of a region$V$bounded by some particular surfaces is given by \n \$V=\\int_V\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z,\$ \n which in this case is \n \$V=\\int_R\\left[z\\right]_0^{\\var{b1}}\\mathrm{d}x\\mathrm{d}y,\$ \n where$R$is the projection of$V$onto the$\\pmatrix{x,y}$-plane. \n Due to the nature of the surface, however, it is convenient to simplify the integrals in$x$and$y\$, by using polar coordinates

\n

\\\begin{align}x&=r\\cos(\\theta),\\\\y&=r\\sin(\\theta),\\end{align}\

\n

then the integral becomes

\n

\$V=\\var{b1}\\int_{\\var{theta1}}^{\\frac{\\pi}{2}}\\mathrm{d}\\theta\\int_0^{\\var{a1}}r\\mathrm{d}r,\$

\n

which, after some straight forward integration, gives

\n

\$V=\\simplify{{b1*a1^2}/4}\\pi=\\var{vol}\\;\\text{to 3d.p.}\$

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