// Numbas version: exam_results_page_options
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([[0]],[[1]])</p>\n<p>What are the coordinates of its minimum point? ([[2]],[[3]])</p>", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "tags": [], "advice": "<p>(i) A maximum point is a point where regardless if you move right or left, the height will decrease. &nbsp;A visual analogy would be a hill: if you're at the top of a hill, no matter which direction you go your height will decrease. &nbsp;So you're looking for a part of the graph which is 'like a hill', and in this graph the point is at $(\\var{xmax},&nbsp;\\var{ymax})$.</p>\n<p>(ii) A minimum point is the opposite of a maximum point (or an upside-down version of a maximum point, if you like). &nbsp;The analogy in this case would be a valley: no matter which direction you go your height will increase. &nbsp;In this graph, the minimum point is at $(\\var{xmin},&nbsp;\\var{ymin})$.</p>", "statement": "<p>Finding stationary&nbsp;points on a graph.</p>", "type": "question"}, {"name": "Differentiation: stationary points of a cubic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}], "advice": "<p>A stationary point is where the gradient is zero, i.e. when $f'$ is equal to zero. &nbsp;So first we determine&nbsp;$f'$ by differentiating.</p>\n<p>$f'(x) = \\simplify{3{a}x^2 + 2{b}x + {c}}$.</p>\n<p></p>\n<p></p>\n<p>When we are at a stationary point, $f'(x)$ has to equal 0, so we get the quadratic equation:</p>\n<p>$0 = f'(x) =&nbsp;\\simplify{3{a}x^2 + 2{b}x + {c}}$.</p>\n<p></p>\n<p></p>\n<p>To solve this, we can factorise the quadratic to get:</p>\n<p>$0 = (\\simplify{3x-{r1}}) (\\simplify{{a}x-{r2}})$.</p>\n<p></p>\n<p>So the $x$-coordinates are $\\var{x1}$ and $\\var{x2}$.</p>\n<p>To find 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"customMarkingAlgorithm": "", "correctAnswerFraction": false, "precisionPartialCredit": "0", "strictPrecision": false, "type": "numberentry", "extendBaseMarkingAlgorithm": true}, {"displayAnswer": "{type2}", "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "marks": 1, "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "matchMode": "regex", "type": "patternmatch", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "answer": "{type2}"}], "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Let $f(x)=\\simplify[all,!collectNumbers,!noleadingminus]{{a}x^3+{b}x^2+{c}x+{d}}$. Find the coordinates of the stationary points and determine whether they are a minimum or a maximum point.</p>\n<p>Give all answers to three significant figures.</p>\n<p></p>\n<p>Stationary Point 1:</p>\n<p>The coordinates are ([[0]],[[1]]) and it is a [[2]] point.</p>\n<p></p>\n<p>Stationary Point 2:</p>\n<p>The coordinates are ([[3]],[[4]]) and it is a [[5]] point.</p>\n<p></p>\n<p>(So the&nbsp;computer marks your answer correctly, make&nbsp;sure that Stationary Point 1 has a&nbsp;smaller $x$ coordinate than Stationary Point 2).</p>"}], "metadata": {"licence": "None specified", "description": "<p>A cubic is given which has two stationary points. &nbsp;Students are asked to determine the coordinates and the type of each stationary point. Calculator question. 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"showFeedbackIcon": true, "scripts": {}, "type": "jme", "showCorrectAnswer": true, "failureRate": 1, "vsetRange": [0, 1], "customMarkingAlgorithm": "", "expectedVariableNames": [], "answerSimplification": "all", "marks": 1, "variableReplacementStrategy": "originalfirst", "answer": "{c6}*{a6}*e^({a6}*t+{b6})", "unitTests": []}, {"showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "variableReplacements": [], "checkVariableNames": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "scripts": {}, "type": "jme", "showCorrectAnswer": true, "failureRate": 1, "vsetRange": [0, 1], "customMarkingAlgorithm": "", "expectedVariableNames": [], "answerSimplification": "all", "marks": 1, "variableReplacementStrategy": "originalfirst", "answer": "{a7}/({a7}*t+{b7})", "unitTests": []}, {"showPreview": true, "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "variableReplacements": [], "checkVariableNames": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "scripts": {}, "type": "jme", "showCorrectAnswer": true, "failureRate": 1, "vsetRange": [0, 1], "customMarkingAlgorithm": "", "expectedVariableNames": [], "answerSimplification": "all", "marks": 1, "variableReplacementStrategy": "originalfirst", "answer": "{a8}*{c8}/({a8}*t+{b8})", "unitTests": []}], "marks": 0, "prompt": "<p>Differentiate the following with respect to $t$.</p>\n<p>(i) $\\sin(\\simplify{{a1}*t+{b1}})$ [[0]]</p>\n<p>(ii) $\\var{c2} \\sin(\\simplify{{a2}*t+{b2}})$ [[1]]</p>\n<p>(iii) $\\cos(\\simplify{{a3}*t+{b3}})$ [[2]]</p>\n<p>(iv)&nbsp;$\\var{c4}\\cos(\\simplify{{a4}*t+{b4}})$ [[3]]</p>\n<p>(v)&nbsp;$ e^{\\simplify{{a5}*t+{b5}}}$ [[4]]</p>\n<p>(vi) $\\var{c6} e^{\\simplify{{a6}*t+{b6}}}$ [[5]]</p>\n<p>(vii) $\\ln(\\simplify{{a7}*t+{b7}})$ [[6]]</p>\n<p>(viii) $\\var{c8} \\ln(\\simplify{{a8}*t+{b8}})$ [[7]]</p>", "showFeedbackIcon": true, "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "tags": [], "advice": "<p>(Note, these are not 'model' answers, but explanations of how I obtain an&nbsp;answer.)</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(i) The `inside' is $\\simplify{{a1}*t+{b1}}$ and differentiating this we get $\\simplify{{a1}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{cos({a1}*t+{b1})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{a1}*cos({a1}*t+{b1})}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(ii) The `inside' is $\\simplify{{a2}*t+{b2}}$ and differentiating this we get $\\simplify{{a2}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{{c2}*cos({a2}*t+{b2})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{c2}*{a2}*cos({a2}*t+{b2})}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(iii) The `inside' is $\\simplify{{a3}*t+{b3}}$ and differentiating this we get $\\simplify{{a3}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{-sin({a3}*t+{b3})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{-{a3}*sin({a3}*t+{b3})}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(iv) The `inside' is $\\simplify{{a4}*t+{b4}}$ and differentiating this we get $\\simplify{{a4}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{-{c4}sin({a4}*t+{b4})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{-{c4}{a4}*sin({a4}*t+{b4})}$.</p>\n<p></p>\n<p></p>\n<p>(v) The `inside' is $\\simplify{{a5}*t+{b5}}$ and differentiating this we get $\\simplify{{a5}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{e^({a5}*t+{b5})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{a5}*e^({a5}*t+{b5})}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(vi) The `inside' is $\\simplify{{a6}*t+{b6}}$ and differentiating this we get $\\simplify{{a6}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{{c6}*e^({a6}*t+{b6})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{c6}*{a6}*e^({a6}*t+{b6})}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p>(vii) The `inside' is $\\simplify{{a7}*t+{b7}}$ and differentiating this we get $\\simplify{{a7}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{1/({a7}*t+{b7})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{a7}/({a7}*t+{b7})}$.</p>\n<p></p>\n<p></p>\n<p>(vii) The `inside' is $\\simplify{{a8}*t+{b8}}$ and differentiating this we get $\\simplify{{a8}}$.</p>\n<p>Differentiating the `outside' while keeping the inside the same gives $\\simplify{{c8}/({a8}*t+{b8})}$.</p>\n<p>Multiplying these two together gives the answer: $\\simplify{{a8}*{c8}/({a8}*t+{b8})}$.</p>\n<p></p>\n<p></p>", "statement": "<p>Differentiation using the Chain Rule</p>", "type": "question"}, {"name": "Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}], "rulesets": {}, "metadata": {"description": "<p>Differentiate the function $f(x)=(a + b x)^m &nbsp;e ^ {n x}$&nbsp;using the product and chain rule. Find $g(x)$ such that $f^{\\prime}(x)= (a + b x)^{m-1} &nbsp;e ^ {n x}g(x)$. Non-calculator. Advice is given.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"type": "gapfill", "prompt": "<p>$\\simplify{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$</p>\n<p>You are told that $\\simplify{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}$, for a polynomial $g(x)$.</p>\n<p></p>\n<p>You have to find $g(x)$.</p>\n<p>$g(x)=\\;$[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "customMarkingAlgorithm": "", "variableReplacements": [], "unitTests": [], "gaps": [{"type": "jme", "vsetRangePoints": 5, "showFeedbackIcon": true, "vsetRange": [0, 1], "showPreview": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "scripts": {}, "customMarkingAlgorithm": "", "variableReplacements": [], "unitTests": [], "checkVariableNames": false, "checkingType": "absdiff", "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "extendBaseMarkingAlgorithm": true, "failureRate": 1, "expectedVariableNames": [], "showCorrectAnswer": true, "answerSimplification": "all", "marks": "4"}], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "marks": 0}], "preamble": {"css": "", "js": ""}, "functions": {}, "variable_groups": [], "variables": {"a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(1..5)"}, "n": {"description": "", "name": "n", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)"}, "m": {"description": "", "name": "m", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8)"}, "s1": {"description": "", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)"}}, "tags": [], "advice": "<p></p>\n<p>$f(x)$ is the product of the two functions $\\simplify{({a} + {b}*x)^{m}}$ and $\\simplify{e ^ ({n} * x)}$, so we need to use the product rule.</p>\n<p></p>\n<p>Differentiating the first part, keeping the second half the same, gives the term: $\\simplify{{m} *{ b} * ({a} + {b} * x) ^ {m -1}} \\times \\simplify{e ^ ({n} * x)}$. &nbsp;</p>\n<p>Note that that we needed&nbsp;the chain rule to do this differentiation.</p>\n<p></p>\n<p></p>\n<p>Differentiating the second part, keeping the first half the same, gives the term: $\\simplify{{n} * e ^ ({n} * x)} \\times \\simplify{({a} + {b}x)^{m}}$.</p>\n<p>Again, we needed&nbsp;the chain rule to do this differentiation.</p>\n<p></p>\n<p>Hence, $\\simplify{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$.</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$= \\simplify{({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}$, (by doing some factorising)</p>\n<p></p>\n<p>Hence, $\\simplify{g(x) = {m * b + n * a} + {n * b} * x}$.</p>", "statement": "<p>Differentiate the following function $f(x)$.</p>", "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question"}, {"name": "Differentiation: product rule, stationary points of f(x)=e^x(ax^2+bx)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}], "parts": [{"customMarkingAlgorithm": "", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "mustBeReduced": false, "correctAnswerFraction": false, "precision": "3", "extendBaseMarkingAlgorithm": true, "minValue": "{x1}", "marks": "3", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "strictPrecision": false, "maxValue": "{x1}", "precisionType": "sigfig", "variableReplacements": [], "scripts": {}, "type": "numberentry", "showPrecisionHint": false, "unitTests": []}, {"notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "mustBeReduced": false, "correctAnswerFraction": false, "precision": "3", "extendBaseMarkingAlgorithm": true, "minValue": "y1", "marks": "2", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precisionPartialCredit": "0", "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "strictPrecision": false, "maxValue": "y1", "precisionType": "sigfig", "variableReplacements": [], "scripts": {}, "type": "numberentry", "showPrecisionHint": false, "unitTests": []}, {"notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "mustBeReduced": false, "correctAnswerFraction": false, "precision": "3", "extendBaseMarkingAlgorithm": true, "minValue": "x2", "marks": "3", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "strictPrecision": false, "maxValue": "x2", "precisionType": "sigfig", "variableReplacements": [], "scripts": {}, "type": "numberentry", "showPrecisionHint": false, "unitTests": []}, {"notationStyles": ["plain", "en", "si-en"], "customMarkingAlgorithm": "", "mustBeReduced": false, "correctAnswerFraction": false, "precision": "3", "extendBaseMarkingAlgorithm": true, "minValue": "y2", "marks": "2", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "precisionPartialCredit": "0", "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReducedPC": 0, "strictPrecision": false, "maxValue": "y2", "precisionType": "sigfig", "variableReplacements": [], "scripts": {}, "type": "numberentry", "showPrecisionHint": false, "unitTests": []}], "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "gapfill", "prompt": "<p>Let $f(x)=\\simplify{(e^x)({a}x^2+{b}x)}$. Find the coordinates of the stationary points.</p>\n<p>Give all answers to three significant figures.</p>\n<p></p>\n<p>Stationary Point 1:</p>\n<p>The coordinates are ([[0]],[[1]]).</p>\n<p></p>\n<p>Stationary Point 2:</p>\n<p>The coordinates are ([[2]],[[3]]).</p>\n<p></p>\n<p>(So the&nbsp;computer marks your answer correctly, make&nbsp;sure that Stationary Point 1 has a&nbsp;smaller $x$ coordinate than Stationary Point 2).</p>", "unitTests": [], "showCorrectAnswer": true}], "variablesTest": {"maxRuns": 100, "condition": "x1 <> x2"}, "advice": "<p>A stationary point is where the gradient is zero, i.e. when $f'$ is equal to zero. &nbsp;So first we determine&nbsp;$f'$, using the product rule.</p>\n<p>$f'(x) = \\simplify{e^(x)({a}x^2 + {b}x)+(e^(x))*(2*{a}*x +{b})}$.</p>\n<p></p>\n<p></p>\n<p>When we are at a stationary point, $f'(x)$ has to equal 0, so:</p>\n<p>$0 = f'(x) =&nbsp;\\simplify{e^(x)({a}x^2 + {b}x)+(e^(x))*(2*{a}*x +{b})}$.</p>\n<p></p>\n<p></p>\n<p>To solve this equation, we can first divide both sides by $e^{x}$ to get:</p>\n<p>$0 = \\simplify{{a}x^2 + {b}x + 2*{a}*x +{b} }$.</p>\n<p></p>\n<p>After expanding brackets and simplifying this becomes the quadratic equation:</p>\n<p>$0 = \\simplify{{a}x^2 + {2a+b}x + {b}}$.</p>\n<p></p>\n<p>Using the quadratic formula, we solve this equation to find the $x$-coordinates, which are: $\\var{x1}$ and $\\var{x2}$.</p>\n<p></p>\n<p></p>\n<p>To find the $y$-coordinates, we plug these $x$-coordinates into the original function $f$ to get $\\var{y1}$ and $\\var{y2}$.</p>\n<p></p>\n<p></p>\n<p>After rounding appropriately, we get the answer.</p>", "variables": {"b": {"name": "b", "description": "", "group": "Ungrouped variables", "definition": "random(-4..-1)", "templateType": "anything"}, "y1": {"name": "y1", "description": "", "group": "Ungrouped variables", "definition": "e^(x1)(a*x1*x1+b*x1)", "templateType": "anything"}, "y2": {"name": "y2", "description": "", "group": "Ungrouped variables", "definition": "e^(x2)(a*x2*x2+b*x2)", "templateType": "anything"}, "x1": {"name": "x1", "description": "", "group": "Ungrouped variables", "definition": "(-(b+2a)-sqrt((b+2a)*(b+2a)-4*a*b))/(2a)", "templateType": "anything"}, "x2": {"name": "x2", "description": "", "group": "Ungrouped variables", "definition": "(-(b+2a)+sqrt((b+2a)*(b+2a)-4*a*b))/(2a)", "templateType": "anything"}, "a": {"name": "a", "description": "", "group": "Ungrouped variables", "definition": "random(1..3 except 0)", "templateType": "anything"}}, "statement": "<p>Determine the coordinates of the stationary points of a function.</p>", "ungrouped_variables": ["a", "b", "x1", "x2", "y1", "y2"], "metadata": {"description": "<p>A function of the form $e^{x}(ax^2+bx)$ is given which has two stationary points. &nbsp;Students are asked to determine the coordinates of each stationary point. Calculator. Advice is provided.</p>", "licence": "None specified"}, "tags": [], "rulesets": {}, "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "type": "question"}]}], "name": "Daniel's copy of CLE 6b", "percentPass": 0, "metadata": {"description": "<p>Questions on differentiation and stationary points.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "feedback": {"intro": "", "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true, "feedbackmessages": [], "showanswerstate": true, "advicethreshold": 0}, "showstudentname": true, "navigation": {"onleave": {"action": "none", "message": ""}, "allowregen": false, "browse": true, "reverse": true, "preventleave": true, "showfrontpage": true, "showresultspage": "oncompletion"}, "duration": 0, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}, "allowPause": true}, "type": "exam", "contributors": [{"name": "Daniel Meselu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1671/"}], "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [], "resources": []}