// Numbas version: exam_results_page_options {"name": "Week 10 Homework", "metadata": {"description": "Year 12 Mathematics Advanced ", "licence": "None specified"}, "duration": 7200, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Differentiation: Product rule", "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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..5)", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "b"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "name": "m"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}}, "ungrouped_variables": ["a", "b", "s2", "s1", "m", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

$\\simplify[std]{f(x) = sin({b}x + {a}) * e ^ ({n} * x)}$

\n\t\t\t

$\\displaystyle \\frac{df}{dx}=\\;$[[0]]

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Click on Show steps for more information, you will not lose any marks by doing  so.

\n\t\t\t", "steps": [{"type": "information", "prompt": "

The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "

Differentiate the following function $f(x)$ using the product rule.

", "tags": ["calculus", "Calculus", "checked2015", "derivative of a product", "differentiating a product", "differentiating exponential functions", "differentiating trigonometric functions", "differentiation", "mas1601", "MAS1601", "product rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

31/07/2012:

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Checked calculation.

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Added tags.

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Allowed no penalty on looking at Show steps.

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

Differentiate $ \\sin(ax+b) e ^ {nx}$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\t

The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]

\n\t \n\t \n\t \n\t

For this example:

\n\t \n\t \n\t \n\t

\\[\\simplify[std]{u = sin({a} + {b} * x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b} * cos({a} + {b} * x)}\\]

\n\t \n\t \n\t \n\t

\\[\\simplify[std]{v = e ^ ({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]

\n\t \n\t \n\t \n\t

Hence on substituting into the product rule above we get:

\n\t \n\t \n\t \n\t

\\[\\begin{eqnarray*}\\frac{df}{dx} &=& \\simplify[std]{{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)}\\\\\n\t \n\t &=&\\simplify[std]{({b}cos({a}+{b}x)+{n}sin({a}+{b}x))e^({n}x)}\n\t \n\t \\end{eqnarray*}\\]

\n\t \n\t \n\t"}, {"name": "Integration: Indefinite 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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..b-1)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*b+r", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "vsetrange": [1, 2], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Input all numbers as integers or fractions and not decimals.

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$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$

\n\t\t\t

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

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Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

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Click on Show steps to get more information. You will not lose any marks by doing so.

\n\t\t\t", "steps": [{"type": "information", "prompt": "

The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t

Integrate the following function $f(x)$.

\n\t


Input the constant of integration as $C$.

\n\t", "tags": ["Calculus", "calculus", "checked2015", "constant of integration", "indefinite integration", "integrating fractional powers", "integrating powers", "integration", "mas1601", "MAS1601", "standard integrals", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

2/08/2012:

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Added tags.

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Added description.

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Checked calculation. OK.

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Added decimal point to forbidden strings along with message to user re input of numbers.

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Message about Show steps included. Also another message about including the constant of integration.

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Changed checking range from 0 to 1 to 1 to 2.

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Improved display.

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

Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\t

Using
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]

\n\t \n\t \n\t"}, {"name": "Integration: Indefinite 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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..b-1)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*b+r", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "vsetrange": [1, 2], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Input all numbers as integers or fractions and not decimals.

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$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$

\n\t\t\t

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

\n\t\t\t

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

\n\t\t\t

Click on Show steps to get more information. You will not lose any marks by doing so.

\n\t\t\t", "steps": [{"type": "information", "prompt": "

The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t

Integrate the following function $f(x)$.

\n\t


Input the constant of integration as $C$.

\n\t", "tags": ["Calculus", "calculus", "checked2015", "constant of integration", "indefinite integration", "integrating fractional powers", "integrating powers", "integration", "mas1601", "MAS1601", "standard integrals", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

2/08/2012:

\n\t\t

Added tags.

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Added description.

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Checked calculation. OK.

\n\t\t

Added decimal point to forbidden strings along with message to user re input of numbers.

\n\t\t

Message about Show steps included. Also another message about including the constant of integration.

\n\t\t

Changed checking range from 0 to 1 to 1 to 2.

\n\t\t

Improved display.

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

Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\t

Using
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]

\n\t \n\t \n\t"}, {"name": "Pythagorean Identity recognition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["Pythagoras", "pythagoras", "Trigonometry", "trigonometry"], "metadata": {"description": "

Using $\\cos^2\\theta+\\sin^2\\theta=1$ to evaluate expressions.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The following questions require some familiarity with trigonometric identities. 

\n

", "advice": "

We will use the following Pythagorean identity  \\[\\cos^2\\theta+\\sin^2\\theta=1.\\] 

\n

For part a)

\n

Given $\\simplify{(cos^2(x)+sin^2(x))^{m1}+{m2}}$ we can replace $\\cos^2(x)+\\sin^2(x)$ with $1$. So our expression is $\\simplify{1^{m1}+{m2}}$. Therefore our expression simplifies to $\\var{m2+1}$.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{m1}+{m2}(2-sin^2(y)-cos^2(y))}$$=$$\\simplify{{m1}+{m2}(2-(sin^2(y)+cos^2(y)))}$
$=$$\\simplify[!collectNumbers]{{m1}+{m2}(2-1)}$
$=$$\\simplify[!collectNumbers]{{m1}+{m2}}$
$=$$\\var{m1+m2}$
\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{m1^2}cos^4(z)+{2*m1^2}cos^2(z)sin^2(z)+{m1^2}sin^4(z)}$$=$$\\simplify{{m1^2}(cos^4(z)+2cos^2(z)sin^2(z)+sin^4(z))}$
$=$$\\simplify{{m1^2}(cos^2(z)+sin^2(z))^2}$
$=$$\\var{m1^2}\\times 1^2$
$=$$\\var{m1^2}$
\n

\n

For part b)

\n

Rearranging the Pythagorean identity $\\cos^2\\theta+\\sin^2\\theta=1$ for $\\sin\\theta$ gives the equation \\[\\sin\\theta=\\pm\\sqrt{1-\\cos^2\\theta}\\]

\n

Recall that $\\sin\\theta$ is the $y$ value of a point on the unit circle, whether $\\sin\\theta$ is taken as the postive square root or as the negative square root depends on the whether the point on the circle is on the top semicircle (positive $y$ value) or the bottom semicircle (negative $y$ value).

\n

Since we are told $\\theta$ is in the first or second quadrant, the $y$ value must be postive, that is $\\sin\\theta=\\sqrt{1-\\cos^2\\theta}$. Therefore our expresson simplifies as follows

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}$$=$$\\simplify{{n}sin(theta)-{n}sin(theta)}$
$=$$0$
\n

\n

Since we are told $\\theta$ is in the third or fourth quadrant, the $y$ value must be negative, that is $\\sin\\theta=-\\sqrt{1-\\cos^2\\theta}$ or equivalently $-\\sin\\theta=\\sqrt{1-\\cos^2\\theta}$. Therefore our expresson simplifies as follows

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}$$=$$\\simplify{{n}sin(theta)-{n}sin(theta)}$
$=$$0$
\n

\n

Rearranging the Pythagorean identity $\\cos^2\\theta+\\sin^2\\theta=1$ for $\\cos\\theta$ gives the equation \\[\\cos\\theta=\\pm\\sqrt{1-\\sin^2\\theta}\\]

\n

Recall that $\\cos\\theta$ is the $x$ value of a point on the unit circle, whether $\\cos\\theta$ is taken as the postive square root or as the negative square root depends on the whether the point on the circle is on the right semicircle (positive $x$ value) or the left semicircle (negative $x$ value).

\n

Since we are told $\\theta$ is in the first or fourth quadrant, the $x$ value must be postive, that is $\\cos\\theta=\\sqrt{1-\\sin^2\\theta}$. Therefore our expresson simplifies as follows

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}$$=$$\\simplify{{n}cos(theta)-{n}cos(theta)}$
$=$$0$
\n

\n

Since we are told $\\theta$ is in the second or third quadrant, the $x$ value must be negative, that is $\\cos\\theta=-\\sqrt{1-\\sin^2\\theta}$ or equivalently $-\\cos\\theta=\\sqrt{1-\\sin^2\\theta}$. Therefore our expresson simplifies as follows

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}$$=$$\\simplify{{n}cos(theta)-{n}cos(theta)}$
$=$$0$
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The expression 

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\\[\\simplify{(cos^2(x)+sin^2(x))^{m1}+{m2}}\\] \\[\\simplify{{m1}+{m2}(2-sin^2(y)-cos^2(y))}\\] \\[\\simplify{{m1^2}cos^4(z)+{2*m1^2}cos^2(z)sin^2(z)+{m1^2}sin^4(z)}\\]

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can be simplified to [[0]].

\n

Note: For this question, your answer should be a number.

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The expression

\n

\\[\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}\\] \\[\\simplify{{n}sin(theta)+{n}sqrt(1-cos^2(theta))}\\]\\[\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}\\] \\[\\simplify{{n}cos(theta)+{n}sqrt(1-sin^2(theta))}\\]

\n

can be simplified to [[0]] for $\\theta$ in the first or second quadrant. third or fourth quadrant.  first or fourth quadrant. second or third quadrant.

\n

Note: For this question, your answer should be a number.

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Try and complete this question before you move on

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