// Numbas version: exam_results_page_options
{"name": "Maths Support: Questions on differentiation", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>Diagnostic testing on differentiation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 50.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Leicester: Differentiation 1", "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/"}], "functions": {}, "tags": ["differentiation", "fractional powers", "negative powers", "powers"], "advice": "", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"prompt": "\n      <p>$f(x)=\\var{a}x^{\\var{b}}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 1.0, "answer": "{a*b}*x^{b-1}", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$f(x)=\\var{a}x^{\\var{b}}+\\var{c}x^{1/\\var{d}}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a*b}*x^{b-1}+{c}/{d}*x^({1-d}/{d})", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$\\displaystyle f(x)=\\frac{\\var{a}}{x^{1/\\var{c}}}+ \\frac{\\var{b}}{x^{-1/\\var{d}}}+\\var{ee}&nbsp;$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std,!noLeadingMinus", "marks": 1.0, "answer": "-{a}/{c}*x^(-{c+1}/{c})+{b}/{d}x^({1-d}/{d})", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "\n  <p>Differentiate the following functions $f(x)$ with respect to $x$.</p>\n    <p>Input all numbers as fractions or integers, not as decimals.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "random(2..9)", "name": "c"}, "b": {"definition": "random(2..9)", "name": "b"}, "d": {"definition": "random(2..9)", "name": "d"}, "f": {"definition": "random(2..9)", "name": "f"}, "ee": {"definition": "random(0..10)", "name": "ee"}}, "metadata": {"notes": "", "description": "<p>Differentiate $\\displaystyle ax^b, ax^b+cx^{1/d}, \\frac{a}{x^{1/c}}+\\frac{b}{x^{-1/d}}+c$ with respect to $x$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Leicester: Differentiation 2", "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/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "ee", "ans1", "tol"], "tags": ["differentiation", "Differentiation", "fractional powers", "gradient at a point", "negative powers", "powers"], "preamble": {"css": "", "js": ""}, "advice": "<p>We have \\[\\frac{df}{dx}=\\var{a*b}x^{\\var{b-1}}-\\var{c*d}x^{\\var{-d-1}}\\]</p>\n<p>The gradient at $x=\\var{g}$ is given by the value of $\\displaystyle \\frac{df}{dx}$ at $x=\\var{g}$ and we therefore have:</p>\n<p>Gradient = $\\var{a*b}\\times(\\var{g})^{\\var{b-1}}-\\var{c*d}\\times (\\var{g})^{\\var{-d-1}}= \\var{dpformat(ans1,3)}$ to 3 decimal places.&nbsp;</p>", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"prompt": "<p>\\[ f(x) = \\simplify{ {a}*x^{b} + {c}/(x^{d}) + {ee}} \\]</p>\n<p>Gradient at $x=\\var{g}\\;$ is [[0]]</p>\n<p>Input your answer to 3 decimal places.&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "ans1", "strictPrecision": true, "minValue": "ans1", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Find the gradient of the curve $y= f(x)$ at the given point.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..7)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(-3..3#0.01 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "ee": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "ee", "description": ""}, "ans1": {"definition": "precround(a*b*g^(b-1)-c*d*g^(-d-1),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}}, "metadata": {"notes": "", "description": "<p>Find the gradient of &nbsp;$ \\displaystyle ax^b+\\frac{c}{x^{d}}+f$ at $x=a$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Leicester: Differentiation 3", "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/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "r1", "r2", "mn", "d", "lg2", "lg1", "type1", "mx", "type2"], "tags": ["stationary points"], "preamble": {"css": "", "js": ""}, "advice": "<p>On differentiating we get $\\displaystyle \\frac{df}{dx}=\\simplify[std]{{3*a}x^2+{2*b}x+{c}}$.</p>\n<p>To find the stationary points we have to solve $\\displaystyle \\frac{df}{dx}=0$&nbsp;for $x$.</p>\n<p>So we have to solve $\\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$.</p>\n<p>Note that the quadratic factorises and the equation becomes $\\simplify[std]{({3a}x-{r1})(x-{r2})=0}$.</p>\n<p>Hence we have two stationary points: $x=\\simplify[std]{{r1}/{3a}}$ and $x=\\var{r2}$.</p>\n<p>To find out the types of these stationary points we look at the sign of $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{6a}*x+{2*b}}$ at &nbsp;the stationary points.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} \\lt 0 $ at a stationary point then it is a MAXIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} \\gt 0 $ at a stationary point then it is a MINIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} = 0 $ at a stationary point then we have to do more work!</p>\n<p>At $x=\\var{r2}$ we have $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{6*a*r2+2*b}}${lg1}$0$ hence is a {type1}.</p>\n<p>At $\\displaystyle x=\\simplify[std]{{r1}/{3a}}$ we have $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{2*r1+2*b}}${lg2}$0$ hence is a {type2}.</p>\n<p>&nbsp;</p>", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"prompt": "\n      <p>$x$-coordinate of the stationary point giving a minimum: [[0]]</p>\n        <p>$x$-coordinate of the stationary point giving a maximum: [[1]]</p>\n      ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{mx*r2}+{(1-mx)}*{(r1)}/{(3*a)}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{mn*r2}+{(1-mn)}*{(r1)}/{(3*a)}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "\n  <p>Find the coordinates of the stationary points of the function and classify their types:&nbsp;</p>\n    <p>$f(x)=\\simplify[all,!collectNumbers]{{a}x^3+{b}x^2+{c}x+{d}}$</p>\n    <p>&nbsp;</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-2..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "r1*r2", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "round(-(3*a*r2+r1)/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "r1": {"definition": "random(-4..4#2 except 0)-3*a*r2", "templateType": "anything", "group": "Ungrouped variables", "name": "r1", "description": ""}, "r2": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "r2", "description": ""}, "mn": {"definition": "if(3a*r2+b<0,1,0)", "templateType": "anything", "group": "Ungrouped variables", "name": "mn", "description": ""}, "type2": {"definition": "if(mx=1, 'maximum','minimum')", "templateType": "anything", "group": "Ungrouped variables", "name": "type2", "description": ""}, "lg2": {"definition": "if(mx=0,'$\\\\gt$','$\\\\lt$')", "templateType": "anything", "group": "Ungrouped variables", "name": "lg2", "description": ""}, "lg1": {"definition": "if(mx=0,'$\\\\lt$','$\\\\gt$')", "templateType": "anything", "group": "Ungrouped variables", "name": "lg1", "description": ""}, "type1": {"definition": "if(mx=0, 'maximum','minimum')", "templateType": "anything", "group": "Ungrouped variables", "name": "type1", "description": ""}, "mx": {"definition": "if(3a*r2+b<0,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "mx", "description": ""}, "d": {"definition": "random(-10..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"notes": "", "description": "<p>Find the stationary points of a cubic which has 2 turning points.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Leicester: Differentiation 4", "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/"}], "functions": {}, "tags": ["chain rule", "differentiation", "quotient rule"], "advice": "\n  <p>c) Let $\\displaystyle \\simplify[std]{u={u}*e^({c}x)+{1-u}*ln({c}*x)}$</p>\n  <p>So we have: $\\displaystyle \\frac{du}{dx}=\\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}$ and&nbsp;$\\displaystyle f(u)=\\simplify[std]{{t[0]}*sin(u)+{t[1]}*cos(u)}\\Rightarrow \\frac{df}{du}=&nbsp;\\simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}$.</p>\n  <p>By the Chain Rule \\[\\begin{eqnarray*} \\frac{df}{dx}&amp;=&amp;\\frac{df}{du}\\times\\frac{du}{dx}\\\\&amp;=&amp;\\simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}\\times\\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}\\\\&amp;=&amp;\\simplify[std]{{t[0]}*({u}*{c}e^({c}x)+{1-u}/x)*cos({u}*e^({c}x)+{1-u}*ln({c}*x))-{t[1]}*({u}*{c}e^({c}x)+{1-u}/x)*sin({u}*e^({c}x)+{1-u}*ln({c}*x))}\\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[all]{(x^{a}-{b})/(x^{-c}+{d})}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.5, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "all", "marks": 1.0, "answer": "({a+c}x^{a-c-1}+{a*d}x^{a-1}-{b*c}x^{-c-1})/(x^{-c}+{d})^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[std]{(e^({a}x)-e^({b}x))/( e^({c}x)+e^({d}x))}$</p>\n      <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers, not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std,!noLeadingMinus", "marks": 1.0, "answer": "({a-c}e^({a+c}x)+{a-d}e^({a+d}x)+{c-b}e^({b+c}x)+{d-b}e^({b+d}x))/(e^({c}x)+e^({d}x))^2", "type": "jme"}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n      <p>$f(x)=\\simplify[std]{{t[0]}*sin({u}*e^({c}x)+{1-u}*ln({c}*x))+{t[1]}*cos({u}*e^({c}x)+{1-u}*ln({c}*x))}$.</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "gaps": [{"checkingaccuracy": 0.001, "vsetrange": [0.5, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{t[0]}*({u}*{c}e^({c}x)+{1-u}/x)*cos({u}*e^({c}x)+{1-u}*ln({c}*x))-{t[1]}*({u}*{c}e^({c}x)+{1-u}/x)*sin({u}*e^({c}x)+{1-u}*ln({c}*x))", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "\n    <p>Differentiate the following functions $f(x)$ with respect to $x$.</p>\n    <p>Input all numbers as fractions or integers, not as decimals.</p>\n    \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-5..5 except 0)", "name": "a"}, "c": {"definition": "random(1..7 except d)", "name": "c"}, "b": {"definition": "random(-5..5 except [a,0])", "name": "b"}, "d": {"definition": "random(2..9 except 0)", "name": "d"}, "f": {"definition": "random(2..9)", "name": "f"}, "ee": {"definition": "random(0..10)", "name": "ee"}, "u": {"definition": "random(0,1)", "name": "u"}, "t": {"definition": "switch(v=0,[1,0],[0,1])", "name": "t"}, "v": {"definition": "random(0,1)", "name": "v"}}, "metadata": {"notes": "", "description": "<p>Examples on differentiation using the quotient rule and chain rule.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}