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(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.
\n(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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", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "unitTests": [], "scripts": {}, "answerSimplification": "std", "expectedVariableNames": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 6, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Consider the line integral:
\\[I=\\int_{\\Gamma} \\left( \\left(\\simplify[std]{x+y} \\right)\\;dx+\\left(\\simplify[std]{y-x}\\right)\\;dy\\right)\\]
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}\\]
\nSince $\\displaystyle{\\simplify[std]{y=({b}/{a})*x}}$ 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*} \\]
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*} \\]
$\\boldsymbol{(A\\cdot B)\\cdot C}$
", "$\\boldsymbol{(A\\cdot B)C}$
", "$\\boldsymbol{(A\\cdot B)\\times C}$
", "$\\boldsymbol{(A\\times B)\\times C}$
", "$\\boldsymbol{(A\\times B)\\cdot C}$
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", "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.
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\n \t\tAdded tags.
\n \t\t16/07/2012:
\n \t\tAdded tags.
\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 \n1. $\\boldsymbol{(A\\cdot B)\\cdot C}$ 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 \n2. $\\boldsymbol{(A\\cdot B)C}$ is a vector and is a multiple of $\\boldsymbol{C}$ as $\\boldsymbol{A \\cdot B}$ is a scalar.
\n \n \n \n3. $\\boldsymbol{(A\\cdot B)\\times C}$ is undefined as $\\boldsymbol{A\\cdot B}$ is a scalar and the cross product is only defined between vectors.
\n \n \n \n4. $\\boldsymbol{(A\\times B)\\times C}$ 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 \n5. $\\boldsymbol{(A\\times B)\\cdot C}$ 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]]
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"markPerCell": false, "allowFractions": false, "correctAnswer": "v", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": "0.6", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Calculate the following.
\n$\\lVert \\boldsymbol{v} \\rVert=$ [[0]]
\n$\\lVert \\boldsymbol{w} \\rVert = $ [[1]]
\n$\\lVert \\boldsymbol{v}+\\boldsymbol{w} \\rVert = $ [[2]]
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\nFind 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]] $\\big)$
\nYou 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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\n$\\boldsymbol{u_v} = \\big($ [[0]], [[1]], [[2]] $\\big)$
\n$-\\boldsymbol{u_w} = \\big($ [[3]], [[4]], [[5]] $\\big)$
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\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}
Enter your answers to the following questions exactly, using the function sqrt(x)
if necessary.
Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.
"}, "advice": "\\[\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \\]
\nIn general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.
\nHence:
\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}
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} \\]
\nFor 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}
\\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}
\\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}
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} \\]
\nand 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.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[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.
\nInput 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:
\nAdded tags.
\nMinor change to prompt.
\n19/07/2012:
\nAdded description.
\nDid not add Show steps.
\nChecked calculation.
\n23/07/2012:
\nAdded tags.
\n22/12/2012:(WHF)
\nCorrected 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.
\nChecked 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*}\\]
"}, {"name": "Use Green's theorem to convert line integral to double integral, ", "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": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "b"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "f"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "d"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{(b+d)*f*g}", "maxValue": "{(b+d)*f*g}", "marks": 8}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "$I=\\;\\;$[[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)\\]
\nwhere $\\Gamma$, 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:
\nAdded tags. Could include Show steps on Green's theorem.
\n19/07/2012:
\nAdded Show steps on Green's Theorem.
\nAdded description.
\nChecked calculation.
\n23/07/2012:
\nAdded tags.
\nIn the question and Steps added brackets so that Green's Theorem is valid.
\nQuestion appears to be working correctly.
\n23/12/2012: (WHF)
\nNo Shoe steps on Green's theorem.
\nChecked 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: $\\displaystyle \\oint_{\\Gamma} \\left(ax^2-by \\right)\\;dx+\\left(cy^2+px\\right)\\;dy$.
"}, "advice": "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*}\\]
\nHence: \\[\\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:
\\[\\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*}\\]
\nNow 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*} \\]
"}, {"name": "Vector equation of a 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/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([c,d,f])", "description": "", "name": "w"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([a,b,g])", "description": "", "name": "v"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+lam*c-mu*al", "description": "", "name": "a1"}, "mu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..5)", "description": "", "name": "mu"}, "be": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "be"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,g)+lam*vector(c,d,f)", "description": "Point of intersection of the two lines
", "name": "p"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "ga", "description": "", "name": "f1"}, "g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g+lam*f-mu*ga", "description": "", "name": "g1"}, "lam": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..5)", "description": "", "name": "lam"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b+lam*d-mu*be", "description": "", "name": "b1"}, "al": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "al"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "be", "description": "", "name": "d1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "al", "description": "", "name": "c1"}, "ga": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "ga"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "a1", "al", "b", "b1", "be", "c", "c1", "d", "d1", "f", "f1", "g", "g1", "ga", "lam", "mu", "s1", "s2", "s3", "s4", "v", "w", "p"], "preamble": {"css": "", "js": ""}, "variable_groups": [], "functions": {}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Find the vector equation of Line 1, which passes through the points $\\boldsymbol{x_0}$ and $\\boldsymbol{x_1}$.
\nInput 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} \\]
\nsuch 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]]
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(a,b,g)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(c,d,f)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "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} \\]
\nsuch 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]]
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(a1,b1,g1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "vector(c1,d1,f1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "0.75", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "You are told that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.
\nFind $\\boldsymbol{P}$.
\n$\\boldsymbol{P} = $ [[0]]
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "correctAnswer": "p", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "3", "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "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}
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": "For $\\lambda=0$ we have \\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}\\]
\nand we want this to be equal to $\\boldsymbol{x_0}$. So we need $a_1 = \\var{a}$, $a_2 = \\var{b}$, and $a_3 = \\var{g}$.
\nFor $\\lambda=1$ 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)}\\]
\nwhich 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)}\\]
\nProceeding as in part a), we find that
\n\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]
\nWrite 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}
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}\\]
"}, {"name": "Dot product - find angles between two pairs of 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/"}], "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "$\\boldsymbol{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$ and $\\boldsymbol{b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$
\n$\\cos({\\theta})=$ [[0]]. (Enter your answer to 2d.p.)
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans1-0.01", "maxValue": "ans1+0.01", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "$\\boldsymbol{c}=\\pmatrix{\\var{c[0]},\\var{c[1]},\\var{c[2]},\\var{c[3]}}$ 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.)
", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans2-0.01", "maxValue": "ans2+0.01", "unitTests": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variables": {"lend": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(d)", "name": "lend", "description": ""}, "ans2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(dot(c,d)/(lenc*lend),2)", "name": "ans2", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),4))", "name": "d", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "name": "a", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),4))", "name": "c", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "name": "b", "description": ""}, "ans1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(dot(a,b)/(lena*lenb),2)", "name": "ans1", "description": ""}, "lena": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(a)", "name": "lena", "description": ""}, "lenc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(c)", "name": "lenc", "description": ""}, "lenb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(b)", "name": "lenb", "description": ""}}, "ungrouped_variables": ["a", "lenb", "c", "b", "lenc", "d", "lend", "ans1", "ans2", "lena"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Find the cosine of the angle $\\theta$ between the following pairs of vectors.
", "tags": ["checked2015"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find the cosine of the angle between two pairs of 3D and 4D vectors.
\nThe 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.
"}, "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.
\nThe 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.\\]
\nIt is also given by
\n\\[\\boldsymbol{a\\cdot b}=ab\\cos(\\theta)\\]
\nwhere $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ 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 $\\boldsymbol{b}$.
\nEquating the two expressions gives
\n\\[a_1b_1+a_2b_2+a_3b_3=ab\\cos(\\theta)\\]
\nand so
\n\\[\\cos(\\theta)=\\frac{a_1b_1+a_2b_2+a_3b_3}{ab}.\\]
\nIn 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.,}\\]
\nand 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.}\\]
"}, {"name": "Identify correct parametric representation of a curve", "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/"}], "parts": [{"displayType": "radiogroup", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [1, 0, 0, 0], "choices": ["{formcorrectchoice}, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$
", "{formincorrectchoice1}, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$
", "{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?
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"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)}
\nYou 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.
\nThe graph of the curve was not displayed on my machine.
"}, "advice": "a)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t^3,t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nAlso, $\\frac{\\mathrm{d}y}{\\mathrm{d}x}=\\frac{1}{3}x^{-\\frac{2}{3}}$, which diverges at the origin, implying that the tangent to the curve is vertical there.
\n\n
b)
\nThe 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 $y=1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,\\frac{5}{t}}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y=5$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,\\arctan(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nSince $x=t$, the range of $x$ coincides with the range of $t$, and so $-5\\leqslant t\\leqslant 2$.
\n\n
b)
\nThe 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 $y\\approx 1.11$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,\\tan(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nThe lines $x=-\\frac{\\pi}{2}$ and $x=\\frac{\\pi}{2}$ are asymptotes for $y=\\tan(x)$.
\n\n
b)
\nThe 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 $y\\rightarrow \\infty$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,\\cot(\\frac{t}{2}+5)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nSince $x=t$, the range of $t$ coincides with the range of $x$, and so $-2\\leqslant x\\leqslant 1$.
\n\n
b)
\nThe 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 $y\\approx -1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{3t,-7t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y=-7$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{-t^3,t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y=1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t^3,\\sin(-5t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nThe 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)
\nThe 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 $y\\approx -0.65$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{2\\sin(t),-2\\cos(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nIn 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)
\nThe 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 $y=0$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{3t,t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y=5$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{-t^2,t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y=1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,t^\\frac{3}{2}}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y\\approx 2.83$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t,-t^2}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nSince $x=t$, the range of $x$ coincides with the range of $t$.
\nIn addition, substituting values of $t$ for $x$ and $y$ in the given range determines points along the curve.
\n \nb)
\nThe 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 $y=-1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t^2,\\sin(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nThe 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.
\nAlso 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 $x=\\frac{\\pi^2}{4}$.
\nAlso note that $\\frac{\\mathrm{d}y}{\\mathrm{d}x}$ diverges at the origin, so the curve is vertical there.
\n \nb)
\nThe 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 $y\\approx 0.14$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{3\\cos(t),-3\\sin(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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 \nb)
\nThe 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 $y=-3$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{1,2t}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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 \nb)
\nThe 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 $y=2$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t^2,t^4}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nAlso, 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 \nb)
\nThe 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 $y=0$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{5\\sin(t),3\\cos(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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 $3$.
\nSince $t$ takes values from $-\\pi$ to $\\pi$, the curve is defined over the entire $\\pmatrix{x,y}$-plane.
\n \nb)
\nThe 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 $\\pmatrix{0,-3}$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{t^2,t^3}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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.
\nAlso, 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)
\nThe 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 $y=1$.
\na)
\nThe correct answer in this case is $t\\rightarrow\\pmatrix{3\\cosh(t),3\\sinh(t)}$, $\\var{range[0]}\\leqslant t\\leqslant\\var{range[1]}$.
\nTo 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)
\nThe 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 $y\\approx 3.53$.
\nTheir 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]]$)$.
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"}, "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
\na)
\nLengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$, which are scalar quantities.
\n\n
b)
\nDistance 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)
\nSum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{a_1+b_1,a_2+b_2,a_3+b_3}$ 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)
\nDot product: $\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3$, which is a scalar quantity.
\n\n
e)
\nCross 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:
\na)
\nLengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{\\var{a[0]^2}+\\var{a[1]^2}+\\var{a[2]^2}}=\\var{lena}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{\\var{b[0]^2}+\\var{b[1]^2}+\\var{b[2]^2}}=\\var{lenb}$.
\n\n
b)
\nDistance 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)
\nSum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}+{b[0]}},\\simplify[std]{{a[1]}+{b[1]}},\\simplify[std]{{a[2]}+{b[2]}}}=\\pmatrix{\\var{sumab[0]},\\var{sumab[1]},\\var{sumab[2]}}$ 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)
\nDot 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)
\nCross product: $\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[1]*b[2]}-{a[2]*b[1]}},\\simplify[std]{{a[2]*b[0]}-{a[0]*b[2]}},\\simplify[std]{{a[0]*b[1]}-{a[1]*b[0]}}}=\\pmatrix{\\var{crossab[0]},\\var{crossab[1]},\\var{crossab[2]}}$.
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\n$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$.
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\n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.
\nThe 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)
\nTo 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)
\nDifferentiate 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}}.\\]
\nThe 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)
\nThe 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]]$)$.
\nThe 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)
\nTo 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)
\nDifferentiate 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)}}.\\]
\nThe 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)
\nThe 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}\\]
\nintersect 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.
\nThe two curves $\\mathcal{C}_1$ and $\\mathcal{C}_2$ intersect 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}\\]
\nFrom equation (1)
\n\\[\\tau=\\frac{\\var{a1}}{\\var{d1}}t=\\simplify{{a1}/{d1}t},\\tag{4}\\]
\nwhich 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}.\\]
\nThen either $t=0$ or $t=\\simplify{{b1*d1^2}/{e1*a1^2}}$.
\nSubstitute 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 $\\mathcal{C}_2$.
\nThe 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 $\\boldsymbol{p}=\\pmatrix{0,0,0}$.
\nThe 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 $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t},\\frac{\\mathrm{d}z}{\\mathrm{d}t}}$.
\nThe 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 $t$.
\nThe 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 $\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$.
\nIn 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},\\]
\nwhere $\\lvert\\boldsymbol{x}\\rvert=\\sqrt{x_1^2+x_2^2+x_3^2}$ is the length of the vector $\\boldsymbol{x}$.
\nThe 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.}\\]
\nThen $\\theta=\\arccos(\\var{precround(dotuv/(lenu*lenv),4)})=\\var{theta}^\\circ$ to 2d.p.
\nIn 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.
\nUses $\\wedge$ 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.
\nA 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} \\]
\nThe magnitude of $\\boldsymbol{a}$ is given by
\n\\[ \\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2} \\]
\nA 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} \\]
\nIn 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} \\]
\nand
\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.} \\]
\nThe 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.
\nInterim 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.
\nThere 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 $\\lvert\\boldsymbol{u}\\rvert=\\sqrt{u_1^2+u_2^2+u_3^2}$.
\nIn 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.
\nThere 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.
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", "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 $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.
\nThe 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},\\]
\nwhere $u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$.
\nIn 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 $u^2=9\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)$.
\nThen
\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}\\]
\nso $f(t)=\\simplify{2/{b^2}}\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}$.
\nFinally, 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 $t$.
\nHence $s=\\simplify{2/{b^2}}\\left\\{\\left(\\var{a^2+t2*b^2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right\\}=\\var{s}$ to 2d.p.
\nAn alternative parametric representation, using $s$ as 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}\\]
\nNow rearrange this expression for $t(s)$, 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\\},\\]
\nand 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}}\\]
\nwith $0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2}+\\simplify{{b^2}*t_2}\\right)^\\frac{3}{2}-\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right)$.
\nFinally, 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}}\\]
\nwith $0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2+b^2*t2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right)$.
"}, {"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 $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.
\nThe 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},\\]
\nwhere $u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$.
\nIn 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 $u^2=\\var{(a*b)^2}$.
\nThen
\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.\\]
\nFinally, 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 $t$.
\nHence $s=\\simplify{{2*a*b}/{t2}}\\pi$.
\nAn 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.\\]
\nNow rearrange this expression for $t(s)$, so
\n\\[t(s)=\\frac{s}{\\var{a*b}}+t_1\\pi,\\]
\nand 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)},\\]
\nwith $0\\leqslant s\\leqslant\\var{a*b}(t_2-t_1)\\pi$.
\nFinally, 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)},\\]
\nwith $0\\leqslant s\\leqslant \\simplify{{2*a*b}/{t2}}\\pi$.
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\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.
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\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.
\nAccuracy 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}\\]
\nThen
\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,\\]
\nbut we know that $z=\\simplify{{d}*v^2}$, so
\n\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{z/{d}},\\]
\nand 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}.\\]
\nIn this case
\n\\[\\boldsymbol{n}=\\pmatrix{\\simplify{{-2d}/{a^2}}x,\\simplify{{-2d}/{c^2}}y,1}\\]
\nby 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": 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"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
\nSet p1/=0, p4=p7=0, so that the x-derivative in part a) is never zero. This avoids division by zero when calculating n1hat.
\nq[1]->q[0] in definition of n2.
\nSet p12=0, so that the x-derivative in part b) is never zero. This avoids division by zero when calculating n2hat.
\n12/11/2013
\nSet 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.
\na)
\nThe 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}\\]
\nThen
\n\\[\\boldsymbol{n}\\vert_{\\pmatrix{\\var{r[0]},\\var{r[1]},\\var{r[2]}}}=\\pmatrix{\\var{n1[0]},\\var{n1[1]},\\var{n1[2]}}\\]
\nis 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.
\nThe 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}.\\]
\nIn 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.}\\]
\nNow 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)
\nThe 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}.\\]
\nThen
\n\\[\\boldsymbol{n}\\vert_{\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}}=\\pmatrix{\\var{n2[0]},\\var{n2[1]},\\var{n2[2]}}\\]
\nis 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.
\nThe 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}.\\]
\nIn 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.}\\]
\nNow 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": "\nExpress 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}\\]
\nThen
\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,\\]
\nbut we know that $z=\\simplify{{d}*v}$, so
\n\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{(z/{d})^2},\\]
\nand hence
\n\\[z=\\var{d}\\sqrt{\\simplify{x^2/{a^2}}+\\simplify{y^2/{c^2}}},\\]
\nwhere 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}.\\]
\nIn this case
\n\\[\\boldsymbol{n}=\\pmatrix{\\simplify{{-d^2}/{a^2}}\\frac{x}{z},\\simplify{{-d^2}/{c^2}}\\frac{y}{z},1}\\]
\nby 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 $f(x,y,z)$.
\nShould 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}}.\\]
\nHence
\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.
\nShould 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. $(\\boldsymbol{\\nabla}f)_z=0$.
\nThe $z$-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}},\\]
\nand 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\\]
\ndefines 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}.\\]
\na)
\nThe 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}\\]
\nand so, cyclically permuting the variables,
\n\\[\\frac{\\partial u_2}{\\partial y}=\\simplify{{a1}*({b1}*z-{c1}*x)}\\]
\nand
\n\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{a1}*({b1}*x-{c1}*y)}.\\]
\nFinally, 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)
\nAs in part a) the variables $x$, $y$, 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}\\]
\nand 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),\\]
\nand
\n\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right).\\]
\nFinally, 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)
\nFirst note that $f_1$ 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.
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", "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}}.\\]
\nThe 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}.\\]
\nBy 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}}}.\\]
\nAgain, by partial differentiation
\n\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.\\]
\nA 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.
\nSince $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ {irrequal} to the zero vector, the vector field {isirr}.
\nSince $\\boldsymbol{\\nabla\\cdot u}$ {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.
\na)
\nThe gradient of $f$ is 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}\\]
\nby straight forward partial differentiation.
\n\n
b)
\nThe gradient of $f$ 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)
\nThe 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)
\nThe 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}\\]
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\n$\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=($[[0]]$,$[[1]]$,$[[2]]$)$.
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\n$\\boldsymbol{\\hat{u}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 3d.p.)
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\n$\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=$ [[0]]. (Enter your answer to 3d.p., and be sure to use the full calculator display from any previous parts in calculating your answer.)
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", "tags": ["checked2015"], "rulesets": {}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Curl of a vector field.
\nShould warn that multiplied terms need * to denote multiplication.
"}, "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}}.\\]
\nHence, in this example, after straight forward partial differentiation
\n\\[\\boldsymbol{\\nabla}\\times\\boldsymbol{u}=\\pmatrix{\\simplify{{c1*p8}*x^{p7}*y^{p8-1}*z^{p9}-{b1*p6}*x^{p4}*y^{p5}*z^{p6-1}},\\simplify{{a1*p3}*x^{p1}*y^{p2}*z^{p3-1}-{c1*p7}*x^{p7-1}*y^{p8}*z^{p9}},\\simplify{{b1*p4}*x^{p4-1}*y^{p5}*z^{p6}-{a1*p2}*x^{p1}*y^{p2-1}*z^{p3}}}.\\]
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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)
\nThe unit outward normals can most easily be identified by sketching the region bounded by the given surfaces which, in this case, is a wedge.
\nThen the unit outward normals to the non-slanted surfaces are given by
\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}},\\]
\nand 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)
\nThe 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.\\]
\nThe 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.\\]
\nThe integrals in $y$ and $z$ are 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}\\]
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Outward normals to the surfaces enclosing a region; volume of that enclosed region.
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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.)
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\nThe 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.
\nThen the unit outward normals to the non-slanted surfaces are given by
\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},\\]
\nand so the unit outward normal is given by
\n\\[\\frac{1}{\\sqrt{x^2+y^2}}\\pmatrix{x,y,0}.\\]
\n\n
b)
\nThe 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,\\]
\nwhich in this case is
\n\\[V=\\int_R\\left[z\\right]_0^{\\var{b1}}\\mathrm{d}x\\mathrm{d}y,\\]
\nwhere $R$ is the projection of $V$ onto the $\\pmatrix{x,y}$-plane.
\nDue 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}\\]
\nthen the integral becomes
\n\\[V=\\var{b1}\\int_{\\var{theta1}}^{\\frac{\\pi}{2}}\\mathrm{d}\\theta\\int_0^{\\var{a1}}r\\mathrm{d}r,\\]
\nwhich, after some straight forward integration, gives
\n\\[V=\\simplify{{b1*a1^2}/4}\\pi=\\var{vol}\\;\\text{to 3d.p.}\\]
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