// Numbas version: exam_results_page_options {"timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "showstudentname": true, "showQuestionGroupNames": false, "metadata": {"description": "

An exam using an experimental theme to use KaTeX to render maths instead of MathJax

", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Find gradient of scalar field", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "variables": {"p4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p4", "description": ""}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "e1", "description": ""}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p1", "description": ""}, "p12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "p12", "description": ""}, "p9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "p9", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "b1", "description": ""}, "p8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "p8", "description": ""}, "p7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p8=0 and p9=0,1,random(0,1))", "name": "p7", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "d1", "description": ""}, "p5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p5", "description": ""}, "p6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p6", "description": ""}, "p2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p2", "description": ""}, "p3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..4)", "name": "p3", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "c1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "a1", "description": ""}, "p11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "p11", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "t", "description": ""}, "p10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p11=0 and p12=0,1,random(0,1))", "name": "p10", "description": ""}}, "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": [{"customMarkingAlgorithm": "", "showCorrectAnswer": 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": [], "sortAnswers": false, "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, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "type": "jme", "marks": 1, "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": ["x", "y", "z"], "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}, {"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, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "type": "jme", "marks": 1, "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": ["x", "y", "z"], "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}, {"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, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "checkVariableNames": true, "checkingType": "absdiff", "vsetRange": [0, 1], "type": "jme", "marks": 1, "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": ["x", "y", "z"], "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Find $\\boldsymbol{\\nabla}f$ for the following function $f(x,y,z)$.

", "tags": [], "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Gradient of $f(x,y,z)$.

\n

Should include a warning to insert * between multiplied terms

This question is simply an exercise in partial differentiation, using the fact that

\n

\$\\boldsymbol{\\nabla}f=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}.\$

\n

The partial derivatives of $f$ are as follows:

\n

\\begin{align}
\\frac{\\partial f}{\\partial x} &= \\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})} \\\0.5em] \\frac{\\partial f}{\\partial y} &= \\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})} \\\\[0.5em] \\frac{\\partial f}{\\partial z} &= \\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})} \\end{align} \n Hence \n \\[ \\boldsymbol{\\nabla}f = \\pmatrix{ \\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}, & \\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}, & \\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})} }.\

The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$

\n \n \n \n

\$\\begin{eqnarray*}\n \n \\partial f \\over \\partial x &=&0\\\\\n \n \\\\\n \n \\partial f \\over \\partial y &=&0\n \n \\end{eqnarray*}\n \n \$

\n \n \n \n

In this case you get two equations to solve for $x$ and $y$

\n \n \n ", "variableReplacements": [], "marks": 0}], "prompt": "

$x$ – coordinate, $a=$ [[0]]

\n

$y$ – coordinate, $b=$ [[1]]

\n

Input value of $f(x,y)$ at $(a,b)$:

\n

$f(a,b)=$ [[2]]

\n

If you want some help, click on Show steps.

", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "

In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b) \\in D$ of the continuous function $f: D \\rightarrow \\mathbb{R}$:

\n

\$f(x,y) = \\simplify[std]{{a} + {b}*e^(-(x-{c})^2-(y-{d})^2)}\$

\n

where

\n

\$D = \\{(x,y): \\simplify[std]{(x-{c})^2+(y-{d})^2}\\} \\le \\var{r}\$

\n

that is, $D$ is a disk of radius $\\simplify[std]{sqrt({r})}$ and centre $(\\var{c},\\var{d})$.

\n

Input both cooordinates as fractions or integers and not decimals.

", "tags": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find the coordinates of the stationary point for $f: D \\rightarrow \\mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$

\n

\\begin{align}
\\partial f \\over \\partial x &= 0 \\\1em] \\partial f \\over \\partial y &= 0 \\end{align} \n In this case you get two equations to solve for x and y \n \\begin{align} \\simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)} &= 0 \\\\[1em] \\simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)} &= 0 \\end{align} \n We can cancel off the term \\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} in both equations as \\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \\neq 0,\\;\\forall x,\\;y. \n On solving these, we get \n \\[ x = \\var{c}, \\quad \\;y=\\var{d} \

\n

So the stationary point is $(\\var{c},\\var{d}) \\in D$.

\n

On substituting these values into $f(x,y)$ we get:

\n

\$f(\\var{c},\\var{d})=\\simplify[std,!zeropower,!othernumbers]{{a}+{b}*e^0={a+b}} \$

"}, {"name": "Find partial derivatives of $f(x,y)$ and identify its stationary points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variablesTest": {"condition": "ch<>0\nand\ns5=-1", "maxRuns": 100}, "variables": {"ch": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a1*d1=b1*c1,0,1)", "name": "ch", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "name": "s5", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1*c1+a1*d1", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1*d1", "name": "c", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,6)", "name": "b1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "a1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,6)", "name": "d1", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "c2", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "name": "c1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a1*c1", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-(b1*c1-3*a1*d1)/2*m^2", "name": "d", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "m", "description": ""}}, "ungrouped_variables": ["a", "c", "ch", "d", "m", "s5", "a1", "b", "b1", "c2", "c1", "d1"], "functions": {}, "variable_groups": [], "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find the partial derivatives of $f$ with respect to $x$ and $y$.

\n

Note that if you want to enter a product of two unknowns, such as $xy$, then you input the expression in the form x*y.

\n

$\\displaystyle { \\partial f \\over \\partial x} =$ [[0]]

\n

$\\displaystyle {\\partial f \\over \\partial y} =$ [[1]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "vsetRangePoints": 5, "expectedVariableNames": ["x", "y"], "failureRate": 1, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "marks": 2, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "checkingType": "absdiff"}, {"answer": "((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "vsetRangePoints": 5, "expectedVariableNames": ["x", "y"], "failureRate": 1, "checkVariableNames": true, "unitTests": [], "vsetRange": [0, 1], "marks": 2, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "checkingType": "absdiff"}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"displayType": "checkbox", "prompt": "\n

#### Finding Stationary Points.

\n

Tick the two choices which give stationary points for $f(x,y)$.

\n

Note that the easiest way to do this question is to substitute the values for $x$ and for $y$ into the expressions for $\\displaystyle {\\partial f \\over \\partial x}$ and $\\displaystyle{\\partial f \\over \\partial y}$ and see if you get $0$ for both.

\n ", "distractors": ["", "", "", "", "", ""], "type": "m_n_2", "maxAnswers": 0, "shuffleChoices": true, "showFeedbackIcon": true, "minAnswers": 0, "minMarks": 0, "variableReplacements": [], "showCorrectAnswer": true, "choices": ["

$x=\\var{m},\\;\\;y=\\simplify[std]{-{a1*m}/{b1}}$

", "

$x=\\var{-m},\\;\\;y=\\simplify[std]{{a1*m}/{b1}}$

", "

$x=\\var{m+1},\\;\\;y=\\simplify[std]{-{c1*(m+1)}/{d1}}$

", "

$x=\\var{-m-1},\\;\\;y=\\simplify[std]{{c1*(m+1)}/{d1}}$

", "

$x=\\var{m-1},\\;\\;y=\\simplify[std]{-{a1+2*b1}/{b1}}$

", "

$x=\\var{-m+1},\\;\\;y=\\simplify[std]{{a1+2*b1}/{b1}}$

"], "extendBaseMarkingAlgorithm": true, "matrix": [2, 2, 0, 0, 0, 0], "customMarkingAlgorithm": "", "unitTests": [], "scripts": {}, "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 3, "marks": 0, "warningType": "none"}], "statement": "\n

\n

\$f(x,y)=\\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}\$

\n ", "tags": ["calculus", "functions", "multivariable", "partial derivatives", "stationary points"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

\n

Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

\n

\\begin{align}
\\partial f \\over \\partial x &= \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} \\\1em] \\partial f \\over \\partial y &= \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} \\end{align} \n #### b) \n (a,b) is a stationary point for the function f(x,y) if f_x=0 and f_y=0, where the partial derivatives are evaluated at x=a, y=b. \n So you have to make sure that both of these partial derivatives are 0 at the stationary point. \n For this example we have from the above equations that: \n \\begin{align} \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} &= 0, & \\mathbf{(1)}\\\\ \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} &= 0, & \\mathbf{(2)} \\end{align} \n The left hand side of equation (1) can be factorised as: \n \\[ \\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0} \

\n

and so we have:

\n

\$y=\\simplify[std]{{-a1}/{b1}*x}, \\text{ or } y= \\simplify[std]{{-c1}/{d1}*x} \$

\n

#### First case: $y= \\simplify[std]{{-a1}/{b1}*x}$

\n

Substituting this into equation (2) gives:

\n

\$\\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \\implies \\simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}\$

\n

Hence $x=\\var{m}$ or $x = \\var{-m}$. The corresponding stationary points are:

\n

\$\\left(\\var{m},\\simplify[std]{-{a1*m}/{b1}}\\right) \\text{ and } \\left(\\var{-m},\\simplify[std]{{a1*m}/{b1}}\\right) \$

\n

#### Second case: $y= \\simplify[std]{{-c1}/{d1}*x}$

\n

Substituting this into equation (2) gives:

\n

\$\\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}\$

\n

There can be no more stationary points as this equation has no real solution.

"}, {"name": "Evaluate double integrals with numerical limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "b", "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": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "name": "d", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "name": "g", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"prompt": "

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

\n

$I =$ [[0]]

", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c*b*(a-1)+(4*d*b*b/4)*(a*a-1)}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "

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

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "answersimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 4, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "

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

\n

$I=$ [[0]]

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

Evaluate the following double integrals.

\n

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

", "tags": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Two double integrals with numerical limits

#### (a)

\n

We proceed to evaluate the double-integral:

\n

\\begin{align}
I &= \\int^\\var{a}_1 \\int^\\var{b}_0 \\left(\\var{c}+\\simplify[std]{{4*d}*x*y} \\right) \\; \\mathrm{d}x \\, \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left[\\simplify[std]{{c}x+{2*d}*y*x^2} \\right]_{x=0}^\\var{b} \\; \\mathrm{d}y \\\\
&= \\int^\\var{a}_1 \\left(\\simplify[std]{{c*b}+{2*d*b^2}*y} \\right) \\; \\mathrm{d}y \\\\
&= \\left[\\simplify[std]{{c*b}y+{d*b^2}*y^2} \\right]^\\var{a}_1 \\; \\mathrm{d}y \\\\
&= \\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{align}

\n

#### (b)

\n

\\begin{align}
I &= \\int^\\pi_0 \\int^\\var{h}_0 \\simplify[std]{y^{f}sin({g}x)} \\; \\mathrm{d}y \\, \\mathrm{d}x \\\\
&= \\int^\\pi_0 \\left[\\simplify[std]{(1/{f+1})*y^{f+1}*sin({g}x)}\\right]_{y=0}^\\var{h} \\; \\mathrm{d}x \\\0.5em] &= \\int^\\pi_0 \\simplify[std]{({h}^{f+1}/{f+1})*sin({g}x)} \\; \\mathrm{d}x \\\\[0.5em] &= \\simplify[std]{({h}^{f+1}/{f+1})}\\left[\\simplify[std]{-1/{g}*cos({g}x)}\\right]^\\pi_0 \\\\[0.5em] &= -\\simplify[std]{({h}^{f+1}/{g*(f+1)})} \\left(\\simplify[std]{{(-1)^g}}-1 \\right) \\\\[0.5em] &= \\simplify[fractionnumbers]{{-{h}^({f+1})*((-1)^{g}-1)/({g*(f+1)})}} \\end{align} "}, {"name": "Resolve a double integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variable_groups": [], "variables": {"ans": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(upper-lower,3)", "description": "", "name": "ans"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "m"}, "fun": {"templateType": "anything", "group": "Ungrouped variables", "definition": "latex(\n switch(\n t=1,\n '\\\\sin',\n t=2,\n '\\\\cos',\n '\\\\exp'\n )\n +'(x^{'+m+'}+'+a+')'\n)", "description": "", "name": "fun"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "t"}, "upper": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-cos(1+a),t=2,sin(1+a),exp(1+a))", "description": "", "name": "upper"}, "lower": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-cos(a),t=2,sin(a),exp(a))", "description": "", "name": "lower"}}, "ungrouped_variables": ["a", "upper", "lower", "m", "t", "ans", "fun"], "functions": {}, "parts": [{"prompt": " I=\\;[[0]] \n Input your answer to 3 decimal places. ", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "showCorrectAnswer": true, "allowFractions": false, "mustBeReduced": false, "correctAnswerFraction": false, "minValue": "ans", "maxValue": "ans", "precision": "3", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "strictPrecision": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": " Evaluate the following repeated integral: \n \\[ I = \\int_0^1 \\; \\mathrm{d}x \\; \\int_0^{\\simplify[all]{x^{m-1}}}\\var{m} \\var{fun} \\; \\mathrm{d}y \

", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Calculate a repeated integral of the form $\\displaystyle I=\\int_0^1\\;dx\\;\\int_0^{x^{m-1}}mf(x^m+a)dy$

\n

The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.

We want to find

\n

\$I=\\int_0^1 \\; dx \\; \\int_0^{\\simplify[all]{x^{m-1}}} \\var{m} \\var{fun} \\; \\mathrm{d}y \$

\n

The innermost integral gives:

\n

\$\\int_0^{\\simplify[all]{x^{m-1}}}\\var{m} \\var{fun} \\; \\mathrm{d}y = \\left[\\var{m}y \\; \\var{fun} \\right]_0^{\\simplify[all]{x^{m-1}}}=\\simplify[all]{{m}x^{m-1}} \\var{fun} \$

\n

So we have to find  $\\displaystyle I=\\int_0^1\\simplify[all]{{m}x^{m-1}} \\var{fun} \\; \\mathrm{d}x$.

\n

Note that if we use the substitution $u=\\simplify[all]{x^{m}+{a}}$ then it is easy to find this last definite integral and we find that:

\n

$I=\\var{ans}$ to 3 decimal places.

"}, {"name": "Double integral - limit is a polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"p2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p1=1,2,1)", "name": "p2", "description": ""}, "ans2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(m+n+2)*b1^(n+1)-m-1", "name": "ans2", "description": ""}, "ans3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p1=1,30*c1*c2+10*c1*d2+15*c2^2*d1+10*d1*d2*c2+3*d1*d2^2,6*c1*c2+2*d1*c2^3+2*d1*c2*d2^2)", "name": "ans3", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,6)", "name": "d", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "name": "m", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..3)", "name": "f", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..3 except 0)", "name": "b1", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "name": "b", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(p1=1,random(-6,-4,-2,2,4,6),random(-6,-3,3,6))", "name": "d1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "a", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,2)", "name": "c2", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2,-1,1,2)", "name": "d2", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "c1", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "name": "n", "description": ""}, "ans1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a*b*(g-f)+c*(g-f)*a^2/2+d*(g^2-f^2)*a/2", "name": "ans1", "description": ""}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,2)", "name": "p1", "description": ""}, "con": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(p2=1,3,15)", "name": "con", "description": ""}, "h1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "round(d1/(p1+1))", "name": "h1", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "f+random(2,4,6)", "name": "g", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-4..4 except 0)", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "p2", "c", "b", "d", "g", "f", "ans1", "ans2", "h1", "ans3", "m", "n", "p1", "b1", "c2", "c1", "d1", "d2", "con"], "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

\$I = \\int_0^{\\var{a}}\\;dx\\;\\int_{\\var{f}}^{\\var{g}}\\simplify[all]{({b}+{c}*x+{d}*y)}\\;dy\$

\n

$I=$ [[0]]

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans1", "maxValue": "ans1", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

\$I=\\var{(m+1)(m+n+2)}\\int _0^1\\;dx\\;\\int_x^{\\var{b1}}\\simplify[all]{{n+1}*x^{m}*y^{n}}\\;dy\$

\n

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

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans2", "maxValue": "ans2", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

\$I=\\var{con}\\int_{-1}^1\\;dx\\;\\int_0^{\\simplify[all]{{c2}+{d2}*x^{p2}}}\\simplify[all]{{c1}+{d1}*y^{p1}}\\;dy\$

\n

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

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans3", "maxValue": "ans3", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "

Calculate the following repeated integrals.

", "tags": [], "rulesets": {"std": ["all", "!collectnumbers", "!noleadingminus", "fractionNumbers"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

3 Repeated integrals of the form $\\int_a^b\\;dx\\;\\int_c^{f(x)}g(x,y)\\;dy$ where $g(x,y)$ is a polynomial in $x,\\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

#### a)

\n

\$I = \\int_0^{\\var{a}} \\; \\mathrm{d}x\\;\\int_{\\var{f}}^{\\var{g}}\\simplify[all]{({b}+{c}*x+{d}*y)} \\; \\mathrm{d}y\$

\n

Calculating the inner integral, we have:

\n

\\begin{align}
&= \\simplify[all,!noleadingminus,!collectNumbers]{{b} * {g} + {c} * {g} * x + {d / 2} * {g ^ 2} + {b} * { -f} + {c} * { -f} * x + {d / 2} * { -(f ^ 2)}} \\\\
&= \\simplify[all,!noleadingminus,!collectNumbers]{ {b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} + {c * g -(c * f)} * x}
\\end{align}

\n

The outer integral gives:

\n

\\begin{align}
I &= \\simplify[std]{DefInt({b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} + {c * g -(c * f)} * x,x,0,{a}) } \\\\
&= \\left[\\simplify[std]{{b * g -(b * f) + (d / 2) * (g ^ 2 -(f ^ 2))} * x + {(c * g -(c * f)) / 2} * x ^ 2}\\right]_0^{\\var{a}} \\\\
&= \\var{ans1}
\\end{align}

\n

#### b)

\n

\$I=\\var{(m + 1) * (m + n + 2)} \\int_0^1 \\; \\mathrm{d}x \\int_x^{\\var{b1}}\\simplify[std]{({n + 1} * x ^ {m} * y ^ {n})} \\; \\mathrm{d}y \$

\n

Calculating the inner integral, we have:

\n

\\begin{align}
\\int_x^{\\var{b1}}\\simplify[std]{({n + 1} * x ^ {m} * y ^ {n})}dy &= \\left[x^{\\var{m}}y^{\\var{n+1}}\\right]_x^{\\var{b1}} \\\\
&= \\simplify{{b1 ^ (n + 1)}* x ^ {m} -(x ^ {m + n + 1})}
\\end{align}

\n

Finally the outer integral gives:

\n

\\begin{align}
I &= \\var{(m + 1) * (m + n + 2)}\\int_0^1\\simplify[std]{{b1^ (n + 1)} * x ^ {m} -(x ^ {m + n + 1})} \\; \\mathrm{d}x \\\\
&= \\simplify[std]{ {(m + 1) * (m + n + 2)} * ({b1 ^ (n + 1)} / {m + 1} -(1 / {m + n + 2})) } \\\\
&= \\var{ans2}
\\end{align}

\n

#### c)

\n

\$I=\\var{con}\\int_{-1}^1 \\; \\mathrm{d}x \\; \\int_0^{\\simplify[all]{{c2}+{d2}*x^{p2}}}\\simplify[all]{{c1}+{d1}*y^{p1}} \\; \\mathrm{d}y \$

\n

Calculating the inner integral, we have:

\n

\\begin{align}
\\int_0^{\\simplify[all]{{c2}+{d2}*x^{p2}}}\\simplify[all]{{c1}+{d1}*y^{p1}} \\; \\mathrm{d}y &= \\left[\\simplify[all]{{c1} * y + {d1 / (p1 + 1)} * y ^ {p1 + 1}}\\right]_0^{\\simplify[all]{{c2}+{d2}*x^{p2}}} \\\\
&= \\simplify[std]{{c1} * ({c2} + {d2} * x ^ {p2}) + {d1 / (p1 + 1)} * ({c2} + {d2} * x ^ {p2}) ^ {p1 + 1}} \\\\
&= \\simplify[std]{ {c1 * c2} + {c1 * d2} * x ^ {p2} + {p1 -1} * {h1} * ({c2 ^ 3} + {3 * c2 ^ 2 * d2} * x ^ {p2} + {3 * c2 * d2 ^ 2} * x ^ {2 * p2} + {d2} ^ 3 * x ^ {3 * p2}) + {2 -p1} * {h1} * ({c2 ^ 2} + {2 * c2 * d2} * x ^ {p2} + {d2 ^ 2} * x ^ {2 * p2})} \\\\
&= \\simplify[std,collectNumbers]{{c1 * c2 + (p1 -1) * h1 * c2 ^ 3 + (2 -p1) * h1 * c2 ^ 2} + {c1 * d2 + (p1 -1) * h1 * 3 * c2 ^ 2 * d2 + (2 -p1) * h1 * 2 * c2 * d2} * x ^ {p2} + {(p1 -1) * h1 * 3 * c2 * d2 ^ 2 + (2 -p1) * h1 * d2 ^ 2} * x ^ {2 * p2} + {(p1 -1) * h1 * d2 ^ 3} * x ^ {3 * p2}}
\\end{align}

\n

Finally the outer integral gives:

\n

\\begin{align}
I &= \\simplify[std]{{con} * DefInt({c1 * c2 + (p1 -1) * h1 * c2 ^ 3 + (2 -p1) * h1 * c2 ^ 2} + {c1 * d2 + (p1 -1) * h1 * 3 * c2 ^ 2 * d2 + (2 -p1) * h1 * 2 * c2 * d2} * x ^ {p2} + {(p1 -1) * h1 * 3 * c2 * d2 ^ 2 + (2 -p1) * h1 * d2 ^ 2} * x ^ {2 * p2} + {(p1 -1) * h1 * d2 ^ 3} * x ^ {3 * p2},x, -1,1)} \\\\
&= \\var{ans3}
\\end{align}

\n

\n

"}]}], "percentPass": 0, "duration": 0, "feedback": {"showactualmark": true, "allowrevealanswer": true, "advicethreshold": 0, "feedbackmessages": [], "showtotalmark": true, "showanswerstate": true, "intro": ""}, "navigation": {"onleave": {"action": "none", "message": "You haven't submitted all your answers on this page."}, "showfrontpage": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "reverse": true, "allowregen": true}, "name": "KaTeX theme test", "type": "exam", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}