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Remember the rule:

\n

If $y = ax^n$ then $\\frac{dy}{dx} = anx^{n-1}$

\n

\n

Differentiate each of the following:

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i)

\n

$y = x^{\\var{pow[0]}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{\\var{pow[0]}-1} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = \\var{num[0]}x^{\\var{pow[1]}}$

\n

$\\frac{dy}{dx} = \\var{ans21}x^{(\\var{pow[1]}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{num[1]}x^{\\var{pow[2]}}$

\n

$\\frac{dy}{dx} = \\var{ans31}x^{(\\var{pow[2]}-1)} = \\var{ans31}x^{\\var{ans32}}$

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$y = x^{\\var{pow[0]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = \\var{num[0]}x^{\\var{pow[1]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}x^{\\var{pow[2]}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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Remember the rule:

\n

If $y = ax^n$ then $\\frac{dy}{dx} = anx^{n-1}$

\n

\n

Differentiate each of the following:

", "variable_groups": [], "preamble": {"js": "", "css": ""}, "functions": {}, "advice": "

i)

\n

$y = x^{\\frac{1}{\\var{p1}}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{(\\frac{1}{\\var{p1}}-1)} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = x^{\\frac{\\var{p2[0]}}{\\var{p2[1]}}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{p2[0]}}{\\var{p2[1]}}x^{(\\frac{\\var{p2[0]}}{\\var{p2[1]}}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{num[0]}x^{\\var{p3}}$

\n

$\\frac{dy}{dx} = \\var{ans31}x^{(\\var{p3}-1)} = \\var{ans31}x^{\\var{ans32}}$

\n

iv)

\n

$y = \\var{num[1]}\\sqrt(x) = \\var{num[1]}x^{\\frac{1}{2}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{num[1]}}{2}x^{(\\frac{1}{2}-1)} = \\var{ans41}x^{\\var{ans42}}$

\n

v)

\n

$y = \\frac{\\var{n51}}{x^{\\var{n52}}} = \\var{n51}x^{-\\var{n52}}$

\n

$\\frac{dy}{dx} = \\var{ans51}x^{(-\\var{n52}-1)} = \\var{ans51}x^{\\var{ans52}}$

\n

vi)

\n

$y = \\sqrt(x^\\var{n6}) = x^{\\frac{\\var{n6}}{2}}$

\n

$\\frac{dy}{dx} = \\frac{\\var{n6}}{2}x^{(\\frac{\\var{n6}}{2}-1)} = \\var{ans61}x^{\\var{ans62}}$

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$y = \\var{num[0]}x^{\\var{p3}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}\\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

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Hint: What is $\\sqrt x$ as a power of $x$

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$y = \\frac{\\var{n51}}{x^{\\var{n52}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "extendBaseMarkingAlgorithm": true, "steps": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

Hint: What is $\\frac{1}{x^2}$ as a power of $x$

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$y = \\sin (x) => \\frac{dy}{dx} = \\cos (x)$

\n

$y = \\sin (ax) => \\frac{dy}{dx} = a\\cos (ax)$

\n

$y = \\cos (x) => \\frac{dy}{dx} = -\\sin (x)$

\n

$y = \\cos (ax) => \\frac{dy}{dx} = -a\\sin (ax)$

\n

\n

Differentiate each of the following:

\n

Note: Put the value of the answer of the sine/cosine function in brackets, i.e. write $\\sin2x$ as $\\sin(2x)$

", "variable_groups": [], "preamble": {"js": "", "css": ""}, "functions": {}, "advice": "

i)

\n

$y = \\sin \\var{n1[0]}x$

\n

$\\frac{dy}{dx} = \\var{n1[0]}cos(\\var{n1[0]}x)$

\n

ii)

\n

$y = \\cos \\var{n1[1]}x$

\n

$\\frac{dy}{dx} = -\\var{n1[1]}sin(\\var{n1[1]}x)$

\n

iii)

\n

$y = \\var{n1[3]}\\cos \\var{n1[4]}x$

\n

$\\frac{dy}{dx} = (-1 \\times \\var{n1[3]} \\times \\var{n1[4]})sin(\\var{n1[4]}x) = \\var{ans31}sin(\\var{ans32}x)$

\n

iv)

\n

$y = \\var{n1[5]}\\sin x$

\n

$\\frac{dy}{dx} = (\\var{n1[5]} \\times 1)cos(x) = \\var{ans41}cos(x)$

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$y = \\sin (\\var{n1[0]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\cos (\\var{n1[1]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{n1[3]} \\cos( \\var{n1[4]}x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{n1[5]} \\sin (x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = e^{\\var{num[0]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = e^{\\var{num1}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num[1]}e^{\\var{num2}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[2]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[3]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\ln \\var{num[4]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

"}], "variables": {"num2": {"definition": "random(-5..-2)", "templateType": "anything", "name": "num2", "group": "Ungrouped variables", "description": ""}, "num": {"definition": "shuffle(2..8)[0..5]", "templateType": "anything", "name": "num", "group": "Ungrouped variables", "description": ""}, "num1": {"definition": "random(0.5..4.5)", "templateType": "anything", "name": "num1", "group": "Ungrouped variables", "description": ""}, "ans1": {"definition": "num[0]", "templateType": "anything", "name": "ans1", "group": "Ungrouped variables", "description": ""}, "ans3": {"definition": "num2*num[1]", "templateType": "anything", "name": "ans3", "group": "Ungrouped variables", "description": ""}, "ans2": {"definition": "num1", "templateType": "anything", "name": "ans2", "group": "Ungrouped variables", "description": ""}}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "metadata": {"licence": "None specified", "description": ""}, "rulesets": {}, "statement": "

Remember the rules:

\n

$y = e^ x => \\frac{dy}{dx} = e^ x$

\n

$y = e^ {ax} => \\frac{dy}{dx} = ae^ {ax}$

\n

$y = \\ln (x) => \\frac{dy}{dx} = \\frac{1}{x}$

\n

$y = \\ln (ax) => \\frac{dy}{dx} = \\frac{1}{x}$

\n

Differentiate each of the following:

i)

\n

$y = e^{\\var{num[0]}x}$

\n

$\\frac{dy}{dx} = \\var{ans1}e^{\\var{num[0]}x}$

\n

ii)

\n

$y = e^{\\var{num1}x}$

\n

$\\frac{dy}{dx} = \\var{ans2}e^{\\var{num1}x}$

\n

iii)

\n

$y = \\var{num[1]}e^{\\var{num2}x}$

\n

$\\frac{dy}{dx} = (\\var{num[1]} \\times \\var{num2})e^{\\var{num2}x} = \\var{ans3}e^{\\var{num2}x}$

\n

iv)

\n

$y = \\ln \\var{num[2]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

\n

v)

\n

$y = \\ln \\var{num[3]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

\n

vi)

\n

$y = \\ln \\var{num[4]}x$

\n

$\\frac{dy}{dx} = \\frac{1}{x}$

", "tags": ["rebelmaths"], "type": "question"}, {"name": "Differentiation 5 Mix questions of the basics in differentiation ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "parts": [{"adaptiveMarkingPenalty": 0, "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "unitTests": [], "gaps": [{"useCustomName": false, "scripts": {}, "checkVariableNames": false, "showFeedbackIcon": true, "customName": "", "type": "jme", "marks": 1, "answer": "{ans11}x^({ans12}) + {ans13}+ {ans14}x^({ans15})", "checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "failureRate": 1, "checkingType": "absdiff", "unitTests": [], "showPreview": true, "showCorrectAnswer": false, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "customMarkingAlgorithm": "", "vsetRange": [0, 1], "valuegenerators": [{"name": "x", "value": ""}]}], "showFeedbackIcon": true, "showCorrectAnswer": true, "customName": "", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "prompt": "

$y = \\var{num1[0]}x^\\var{num1[1]} + \\var{num1[2]}x + x^\\var{num1[3]} + \\var{num1[4]}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

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$y = \\var{num2[0]}e^x + \\var{num2[1]}x + \\ln\\var{num2[2]}x + \\var{num2[3]}\\cos (x)$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = x^\\var{num3[0]} + x^\\var{num3[1]} + \\var{num3[2]}x + \\var{num3[3]}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num5[0]}x^\\var{num5[1]} + \\var{num5[2]}x - \\frac{1}{\\var{num5[3]}x^\\var{num5[4]}} + \\frac{1}{\\sqrt x} - \\var{num5[5]}$

\n

$\\frac{dy}{dx} =$ [[0]]

\n

N.B. Put a multipication between the number and the cosine and sine functions in your answer!!

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$y = \\var{num6[0]}\\sin \\var{num6[1]}x + \\var{num6[2]}\\cos \\var{num6[3]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

"}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"num1": {"description": "", "name": "num1", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..5]", "templateType": "anything"}, "ans51": {"description": "", "name": "ans51", "group": "Ungrouped variables", "definition": "num5[0]*num5[1]", "templateType": "anything"}, "ans61": {"description": "", "name": "ans61", "group": "Ungrouped variables", "definition": "num6[0]*num6[1]", "templateType": "anything"}, "num2": {"description": "", "name": "num2", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..4]", "templateType": "anything"}, "ans7": {"description": "", "name": "ans7", "group": "Ungrouped variables", "definition": "num7[0]*num7[1]*-1", "templateType": "anything"}, "ans32": {"description": "", "name": "ans32", "group": "Ungrouped variables", "definition": "num3[1]-1", "templateType": "anything"}, "num7": {"description": "", "name": "num7", "group": "Ungrouped variables", "definition": "shuffle(2..5)[0..2]", "templateType": "anything"}, "ans41": {"description": "", "name": "ans41", "group": "Ungrouped variables", "definition": "num4[0]*num4[1]", "templateType": "anything"}, "ans31": {"description": "", "name": "ans31", "group": "Ungrouped variables", "definition": "num3[0]-1", "templateType": "anything"}, "num3": {"description": "", "name": "num3", "group": "Ungrouped variables", "definition": "shuffle(3..8)[0..4]", "templateType": "anything"}, "num4": {"description": "", "name": "num4", "group": "Ungrouped variables", "definition": "shuffle(3..12)[0..7]", "templateType": "anything"}, "ans63": {"description": "", "name": "ans63", "group": "Ungrouped variables", "definition": "num6[4]*num6[5]", "templateType": "anything"}, "ans43": {"description": "", "name": "ans43", "group": "Ungrouped variables", "definition": "num4[2]*num4[3]", "templateType": "anything"}, "ans13": {"description": "", "name": "ans13", "group": "Ungrouped variables", "definition": "num1[2]", "templateType": "anything"}, "num5": {"description": "", "name": "num5", "group": "Ungrouped variables", "definition": "shuffle(3..10)[0..6]", "templateType": "anything"}, "ans45": {"description": "", "name": "ans45", "group": "Ungrouped variables", "definition": "num4[6]/2", "templateType": "anything"}, "ans11": {"description": "", "name": "ans11", "group": "Ungrouped variables", "definition": "num1[0]*num1[1]", "templateType": "anything"}, "ans52": {"description": "", "name": "ans52", "group": "Ungrouped variables", "definition": "num5[1]-1", "templateType": "anything"}, "ans54": {"description": "", "name": "ans54", "group": "Ungrouped variables", "definition": "(num5[4]*-1)-1", "templateType": "anything"}, "ans44": {"description": "", "name": "ans44", "group": "Ungrouped variables", "definition": "(num4[3]*-1)-1", "templateType": "anything"}, "ans62": {"description": "", "name": "ans62", "group": "Ungrouped variables", "definition": "num6[2]*num6[3]", "templateType": "anything"}, "ans42": {"description": "", "name": "ans42", "group": "Ungrouped variables", "definition": "num4[1]-1", "templateType": "anything"}, "num6": {"description": "", "name": "num6", "group": "Ungrouped variables", "definition": "shuffle(3..12)[0..8]", "templateType": "anything"}, "ans14": {"description": "", "name": "ans14", "group": "Ungrouped variables", "definition": "num1[3]", "templateType": "anything"}, "ans15": {"description": "", "name": "ans15", "group": "Ungrouped variables", "definition": "num1[3]-1", "templateType": "anything"}, "vs": {"description": "", "name": "vs", "group": "Ungrouped variables", "definition": "random('$\\pi$','e')", "templateType": "anything"}, "ans12": {"description": "", "name": "ans12", "group": "Ungrouped variables", "definition": "num1[1]-1", "templateType": "anything"}, "ans53": {"description": "", "name": "ans53", "group": "Ungrouped variables", "definition": "num5[4]/num5[3]", "templateType": "anything"}}, "tags": ["rebelmaths"], "advice": "

i)

\n

$y = \\var{num1[0]}x^\\var{num1[1]} + \\var{num1[2]}x + x^\\var{num1[3]} + \\var{num1[4]}$

\n

$\\frac{dy}{dx} = (\\var{num1[1]} \\times \\var{num1[0]})x^{\\var{num1[1]}-1} + \\var{num1[2]} + (1 \\times \\var{num1[3]})x^{\\var{num1[3]}-1}$

\n

$\\var{ans11}x^(\\var{ans12}) + \\var{ans13}+ \\var{ans14}x^\\var{ans15}$

\n

ii)

\n

$y = \\var{num2[0]}e^x + \\var{num2[1]}x + \\ln\\var{num2[2]}x + \\var{num2[3]}\\cos x$

\n

$\\frac{dy}{dx} = \\var{num2[0]}e^x + \\var{num2[1]} + \\frac{1}{x} - \\var{num2[3]}\\sin x$

\n

iii)

\n

$y = x^\\var{num3[0]} + x^\\var{num3[1]} + \\var{num3[2]}x + \\var{num3[3]}$

\n

$\\frac{dy}{dx} = (1 \\times \\var{num3[0]})x^{\\var{num3[0]}-1} + (1 \\times \\var{num3[1]})x^{\\var{num3[1]}-1} + \\var{num3[2]}$

\n

$\\var{num3[0]}x^\\var{ans31} + \\var{num3[1]}x^\\var{ans32}+ \\var{num3[2]}$

\n

iv)

\n

$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} = (\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + \\var{num4[2]}x^{-\\var{num4[3]}} - \\var{num4[5]} + \\var{num4[6]}x^{\\frac{1}{2}}$

\n

$(\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + (-\\var{num4[3]} \\times \\var{num4[2]})x^{-\\var{num4[3]}-1} - \\var{num4[5]} + (\\frac{1}{2} \\times \\var{num4[6]})x^{\\frac{1}{2}-1}$

\n

$\\var{ans41}x^\\var{ans42} - \\var{ans43}x^\\var{ans44} - \\var{num4[5]} + \\var{ans45}x^{-\\frac{1}{2}}$

\n

v)

\n

$y = \\var{num5[0]}x^\\var{num5[1]} + \\var{num5[2]}x - \\frac{1}{\\var{num5[3]}x^\\var{num5[4]}} + \\frac{1}{\\sqrt x} - \\var{num5[5]}$

\n

$\\frac{dy}{dx} = (\\var{num5[1]} \\times \\var{num5[0]})x^{\\var{num5[1]}-1} + \\var{num5[2]} - \\frac{1}{\\var{num5[3]}}x^{-\\var{num5[4]}} + x^{-\\frac{1}{2}}$

\n

$\\var{ans51}x^\\var{ans52} + \\var{num5[2]} + \\var{ans53}x^\\var{ans54} - \\frac{1}{2}x^{-\\frac{3}{2}}$

\n

vi)

\n

$y = \\var{num6[0]}\\sin \\var{num6[1]}x + \\var{num6[2]}\\cos \\var{num6[3]}x +\\var{num6[4]}e^{\\var{num6[5]}x} + \\var{num6[6]}\\ln\\var{num6[7]}x$

\n

$\\frac{dy}{dx} = (\\var{num6[0]} \\times \\var{num6[1]})\\cos \\var{num6[1]}x - (\\var{num6[2]} \\times \\var{num6[3]})\\sin \\var{num6[3]}x + (\\var{num6[4]} \\times \\var{num6[5]})e^{\\var{num6[5]}x} + \\var{num6[6]} \\times \\frac{1}{x}$

\n

$\\var{ans61}\\cos \\var{num6[1]}x - \\var{ans62}\\sin \\var{num6[3]}x + \\var{ans63}e^{\\var{num6[5]}x} + \\frac{\\var{num6[6]}}{x}$

\n

vii)

\n

$y = \\frac{\\var{num7[0]}}{e^{\\var{num7[1]}x}} + \\var{vs}$

\n

$\\frac{dy}{dx} = \\var{num7[0]}e^{-\\var{num7[1]}x} + \\var{vs}$

\n

$(\\var{num7[0]} \\times -\\var{num7[1]})e^{-\\var{num7[1]}x} + 0$

\n

$\\var{ans7}e^(-\\var{num7[1]}x)$

", "metadata": {"description": "", "licence": "None specified"}, "functions": {}, "ungrouped_variables": ["num1", "ans11", "ans12", "ans13", "ans14", "ans15", "num2", "num3", "ans31", "ans32", "num4", "ans41", "ans42", "ans43", "ans44", "ans45", "num5", "ans51", "ans52", "ans53", "ans54", "num6", "ans61", "ans62", "ans63", "vs", "num7", "ans7"], "preamble": {"js": "", "css": ""}, "statement": "

Remember the rules from questions 1 to 4:

\n

For each of the following find $\\frac{dy}{dx}$:

", "variable_groups": []}, {"name": "Differentiation 7 Quotient Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "tags": ["rebelmaths"], "preamble": {"js": "", "css": ""}, "statement": "

Remember the Quotient Rule:

\n

The quotient rule says that if $u$ and $v$ are functions of $x$ then
\$\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\$

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\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Find the derivative of $y = \\frac{\\var{n1[0]} \\sin{\\var{n1[1]}x}}{\\var{n1[2]}x^{\\var{n1[3]}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Differentiate  $y = \\frac{\\var{n2[0]}e^{\\var{n2[1]}t}}{\\cos \\var{n2[2]}t}$

\n

$\\frac{dy}{dt} =$ [[0]]

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\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "customMarkingAlgorithm": ""}], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "prompt": "

Find the slope of the curve $y = \\frac{\\var{n3[0]}x}{\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x}$ at the point $(\\var{num31},\\var{num32})$

\n

In formula:

\n

$\\frac{dy}{dx} =$ [[0]]

\n

In units (correct to 3 decimal places!!):

\n

$\\frac{dy}{dx} =$ Slope=  [[1]]

i)

\n

$y = \\frac{\\var{n1[0]} \\sin{\\var{n1[1]}x}}{\\var{n1[2]}x^{\\var{n1[3]}}}$

\n

$\\frac{dy}{dx} = \\frac{((\\var{n1[2]}x^\\var{n1[3]}*\\var{ans11}*\\cos(\\var{n1[1]}x))) - ((\\var{n1[0]}*\\sin(\\var{n1[1]}x) * \\var{ans13}x^\\var{ans12}))}{\\var{ans14}x^\\var{ans15}}$

\n

ii)

\n

$y = \\frac{\\var{n2[0]}e^{\\var{n2[1]}t}}{cos \\var{n2[2]}t}$

\n

$\\frac{dy}{dt} = \\frac{(\\cos(\\var{n2[2]}t) * \\var{ans21} * e^{(\\var{n2[1]}t))} + (\\var{n2[0]} * \\var{n2[2]} * e^{(\\var{n2[1]}t)} * \\sin(\\var{n2[2]}t))}{\\cos(\\var{n2[2]}t)*\\cos(\\var{n2[2]}t)}$

\n

iii)

\n

$y = \\frac{\\var{n3[0]}x}{\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x}$

\n

$\\frac{dy}{dx} = \\frac{(\\var{n3[0]}*(\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x)) - (\\var{n3[0]}x*(\\var{ans32}x^\\var{ans31} + \\var{n3[3]}))}{(\\var{n3[1]}x^{\\var{n3[2]}} + \\var{n3[3]}x)^2}$

\n

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Remember the Chain Rule:

\n

The Chain rule says that if $y$ is a function of $x$, then
$\\frac{dy}{dx} = \\frac{dy}{du}.\\frac{du}{dx}$

", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "advice": "

i)

\n

\$\\simplify[std]{y = ({a} * x^{m}+{b})^{n}}\$

\n

$\\frac{dy}{dx} = (\\var{a} \\times \\var{m} \\times \\var{n})x ^ {\\var{m}-1} \\times (\\var{a} \\times x^{\\var{m}}+(\\var{{b}}))^{\\var{n}-1}$

\n

\$\\simplify[std]{{a*m*n}x ^ {m-1} * ({a} * x^{m}+{b})^{n-1}}\$

\n

ii)

\n

\$\\simplify[std]{f(x) = ({a2} x^{m2}+{c2}x^2+{b2})^{n2}}\$

\n

$\\frac{dy}{dt} = \\var{n2}((\\var{a2} \\times \\var{m2})x ^ {\\var{m2}-1}+2 \\times \\var{c2}x) \\times (\\var{a2} \\times x^\\var{m2} + \\var{c2}x^2 + \\var{b2})^{\\var{n2}-1}$

\n

\$\\simplify[std]{{n2}({a2*m2}x ^ {m2-1}+{2*c2}x) * ({a2} * x^{m2}+{c2}x^2+{b2})^{n2-1}}\$

\n

iii)

\n

\$\\simplify[std]{y = sqrt({a3} * x^{m3}+{b3})}\$

\n

$\\frac{dy}{dx} = \\frac{((\\var{a3}*\\var{m3})x ^ {\\var{m3}-1})}{(2*\\sqrt(\\var{a3} * x^\\var{m3}+\\var{b3}))}$

\n

\$\\simplify[std]{({a3*m3}x ^ {m3-1})/(2*sqrt({a3} * x^{m3}+{b3}))}\$

\n

iv)

\n

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\frac{dy}{dx} = -(\\var{a4}e^{(\\var{a4}x)}+{2*\\var{b4}}x) \\times sin(e^{(\\var{a4}x)} +\\var{b4}x^2+\\var{c4})$

\n

\n

\$\\simplify[std]{-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})}\$

\n

v)

\n

\$\\simplify[std]{y = ln(({a5}x+{b5})^{m5})}\$

\n

$\\frac{dy}{dx} = \\frac{({\\var{m5} \\times \\var{a5}})}{(\\var{a5}x+\\var{b5})}$

\n

\$\\simplify[std]{({m5*a5})/({a5}x+{b5})}\$

\n

\n

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\$\\simplify[std]{y = ({a} * x^{m}+{b})^{n}}\$

\n

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

\n

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\$\\simplify[std]{y = ({a2} * x^{m2}+{c2}x^2+{b2})^{n2}}\$

\n

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

\n

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\$\\simplify[std]{y = sqrt({a3} * x^{m3}+{b3})}\$

\n

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

\n

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\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

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

\n

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\$\\simplify[std]{f(x) = ln(({a5}x+{b5})^{m5})}\$

\n

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

\n

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$y = \\var{num1}x^{\\var{pow1}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\frac{\\var{n21}}{x^{\\var{n22}}}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{n3[0]} \\cos \\var{n3[1]}x$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "{ans4}e^({n42}x)"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$y = \\var{n41}e^{\\var{n42}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

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$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "({n6[0]}x^(1/2)/x) + ({ans61}x^{-(1/2)} * ln({n6[1]}x))"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find the derivative of y with respect to x given  $y = \\var{n6[0]}\\sqrt x \\ln{\\var{n6[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "t"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "((cos({n7[2]}t) * {ans71} * e^({n7[1]}t)) + ({n7[0]} * {n7[2]} * e^({n7[1]}t) * sin({n7[2]}t)))/(cos({n7[2]}t)*cos({n7[2]}t))"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Differentiate  $y = \\frac{\\var{n7[0]}e^{\\var{n7[1]}t}}{\\cos \\var{n7[2]}t}$

\n

$\\frac{dy}{dt} =$ [[0]]

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}, {"customName": "", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "variableReplacements": [], "useCustomName": false, "showCorrectAnswer": true, "gaps": [{"checkingAccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "adaptiveMarkingPenalty": 0, "type": "jme", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRangePoints": 5, "customName": "", "variableReplacements": [], "unitTests": [], "failureRate": 1, "useCustomName": false, "showCorrectAnswer": true, "showPreview": true, "scripts": {}, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "answer": "-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})"}], "adaptiveMarkingPenalty": 0, "scripts": {}, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

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

", "variableReplacementStrategy": "originalfirst", "sortAnswers": false}], "ungrouped_variables": ["num1", "pow1", "ans11", "ans12"], "tags": ["rebelmaths"], "advice": "

i)

\n

$y = \\var{num1}x^{\\var{pow1}}$

\n

$\\frac{dy}{dx} = \\var{ans11}x^{(\\var{pow1}-1)} = \\var{ans11}x^{\\var{ans12}}$

\n

ii)

\n

$y = \\frac{\\var{n21}}{x^{\\var{n22}}} = \\var{n21}x^{-\\var{n22}}$

\n

$\\frac{dy}{dx} = \\var{ans21}x^{(-\\var{n22}-1)} = \\var{ans21}x^{\\var{ans22}}$

\n

iii)

\n

$y = \\var{n3[0]}\\cos \\var{n3[1]}x$

\n

$\\frac{dy}{dx} = (-1 \\times \\var{n3[0]} \\times \\var{n3[1]})sin(\\var{n3[1]}x) = \\var{ans31}sin(\\var{ans32}x)$

\n

iv)

\n

$y = \\var{n41}e^{\\var{n42}x}$

\n

$\\frac{dy}{dx} = (\\var{n41} \\times \\var{n42})e^{\\var{n42}x} = \\var{ans3}e^{\\var{n42}x}$

\n

v)

\n

$y = \\var{num4[0]}x^\\var{num4[1]} + \\frac{\\var{num4[2]}}{x^\\var{num4[3]}} + \\var{num4[4]} - \\var{num4[5]}x + \\var{num4[6]} \\sqrt x$

\n

$\\frac{dy}{dx} = (\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + \\var{num4[2]}x^{-\\var{num4[3]}} - \\var{num4[5]} + \\var{num4[6]}x^{\\frac{1}{2}}$

\n

$(\\var{num4[1]} \\times \\var{num4[0]})x^{\\var{num4[1]}-1} + (-\\var{num4[3]} \\times \\var{num4[2]})x^{-\\var{num4[3]}-1} - \\var{num4[5]} + (\\frac{1}{2} \\times \\var{num4[6]})x^{\\frac{1}{2}-1}$

\n

$\\var{ans41}x^\\var{ans42} - \\var{ans43}x^\\var{ans44} - \\var{num4[5]} + \\var{ans45}x^{-\\frac{1}{2}}$

\n

vi)

\n

$y = \\var{n6[0]}\\sqrt x \\ln{\\var{n6[1]}x}$

\n

$\\frac{dy}{dx} = (\\frac{\\var{n6[0]}x^{\\frac{1}{2}}}{x}) + (\\var{ans61}x^{-\\frac{1}{2}} \\times ln(\\var{n6[1]}x))$

\n

vii)

\n

$y = \\frac{\\var{n7[0]}e^{\\var{n7[1]}t}}{cos \\var{n7[2]}t}$

\n

$\\frac{dy}{dt} = \\frac{(\\cos(\\var{n7[2]}t) * \\var{ans71} * e^{(\\var{n7[1]}t))} + (\\var{n7[0]} * \\var{n7[2]} * e^{(\\var{n7[1]}t)} * \\sin(\\var{n7[2]}t))}{\\cos(\\var{n7[2]}t)*\\cos(\\var{n7[2]}t)}$

\n

viii)

\n

\$\\simplify[std]{y = cos(e^({a4}x) +{b4}x^2+{c4})}\$

\n

$\\frac{dy}{dx} = -(\\var{a4}e^{(\\var{a4}x)}+{2*\\var{b4}}x) \\times sin(e^{(\\var{a4}x)} +\\var{b4}x^2+\\var{c4})$

\n

\n

\$\\simplify[std]{-({a4}e^({a4}x)+{2*b4}x)*sin(e^({a4}x) +{b4}x^2+{c4})}\$

", "statement": "

Remember all the rules from the previous questions and use them to solve the following:

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Find the derivative of $y = \\var{n1[0]}x^{\\var{n1[1]}}\\sin{\\var{n1[2]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
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Find the derivative of y with respect to x given  $y = \\var{n2[0]}\\sqrt x \\ln{\\var{n2[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
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Differentiate $y = x^{\\var{n3[0]}}\\cos{\\var{n3[1]}x}$

\n

$\\frac{dy}{dx} =$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(x^{n3[0]}*{-n3[1]}*sin({n3[1]}x)) + ({n3[0]}x^{ans31} * cos({n3[1]}x))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

Determine the rate of change of voltage, given $V = \\var{n4[0]}t \\sin{\\var{n4[1]}t}$ volts when $t = \\var{num4} seconds.$

\n

In formula:

\n

$\\frac{dV}{dt} =$ [[0]]

\n

In units (correct to 2 decimal places!!):

\n

$\\frac{dV}{dt} =$  [[1]] Volts per second

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

\n\n\n\n\n\n\n\n\n\n\n\n
 $u =$_________________ $v =$_________________ $\\frac{du}{dx} =$_________________ $\\frac{dv}{dx} =$_________________
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({ans41}t*cos({n4[1]}t)) + ({n4[0]}* sin({n4[1]}t))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "{ans42}", "strictPrecision": false, "minValue": "{ans42}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

Remember the Product Rule:

\n

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)}\$

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On differentiating we get $\\displaystyle \\frac{df}{dx}=\\simplify[std]{{3*a}x^2+{2*b}x+{c}}$.

\n

To find the stationary points we have to solve $\\displaystyle \\frac{df}{dx}=0$ for $x$.

\n

So we have to solve $\\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$.

\n

Note that the quadratic factorises and the equation becomes $\\simplify[std]{({3a}x-{r1})(x-{r2})=0}$.

\n

Hence we have two stationary points: $x=\\simplify[std]{{r1}/{3a}}$ and $x=\\var{r2}$.

\n

To find out the types of these stationary points we look at the sign of $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{6a}*x+{2*b}}$ at  the stationary points.

\n

If  $\\displaystyle \\frac{d^2f}{dx^2} \\lt 0$ at a stationary point then it is a MAXIMUM.

\n

If  $\\displaystyle \\frac{d^2f}{dx^2} \\gt 0$ at a stationary point then it is a MINIMUM.

\n

If  $\\displaystyle \\frac{d^2f}{dx^2} = 0$ at a stationary point then we have to do more work!

\n

At $x=\\var{r2}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{6*a*r2+2*b}}${lg1}$0$ hence is a {type1}.

\n

At $\\displaystyle x=\\simplify[std]{{r1}/{3a}}$ we have $\\displaystyle \\frac{d^2f}{dx^2} = \\simplify{{2*r1+2*b}}${lg2}$0$ hence is a {type2}.

\n

", "parts": [{"type": "gapfill", "prompt": "

$f(x)=\\simplify[all,!collectNumbers,!noleadingminus]{{a}x^3+{b}x^2+{c}x+{d}}$

\n

$f'(x)=$ [[2]]

\n

$f''(x)=$ [[3]]

\n

\n

Find when $f'(x)=0$, hence find:

\n

$x$-coordinate of the stationary point giving a minimum $=$ [[0]]

\n

$x$-coordinate of the stationary point giving a maximum $=$ [[1]]

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Finding the stationary points of a cubic with two turning points

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Find the coordinates of the stationary points of the function.

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{eqnline(a,b,x2,y2)}

\n

The above graph shows a graph of a quadratic equation, it is your task to find this equation.

\n

You are given the two points of the curve with the x axis, $(\\var{b},0)$ and $(\\var{a},0)$, and the $y$-intercept at $(0,\\var{c})$ as indicated on the diagram.

", "preamble": {"css": "", "js": ""}, "functions": {"eqnline": {"type": "html", "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n \n// The line is described by the equation \n// y = a*x + b\n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-13,22,13,-22],\n axis: false,\n showNavigation: false,\n grid: true\n});\n \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,2],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,2],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n// mark the two given points - one on the y-axis, and one at (x2,y2)\n\n\n\n\nboard.create('functiongraph',[function(x){ return (x-a)*(x-b);},-13,13]);\n\nreturn div;", "parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"]]}}, "rulesets": {}, "parts": [{"type": "gapfill", "gaps": [{"checkingtype": "absdiff", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "vsetrangepoints": 5, "scripts": {}, "marks": 1, "expectedvariablenames": [], "type": "jme", "showpreview": true, "showFeedbackIcon": true, "variableReplacements": [], "answer": "x^2-({a}+{b})x+{a}{b}", "showCorrectAnswer": true, "answersimplification": "all", "checkvariablenames": false, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

Write the equation of the graph in the diagram.

\n

$y=\\;$[[0]]

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Find the coordinates of the turning point of this quadratic

\n

$x=$[[0]]

\n

$y=$[[1]]

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Students enter equation and turning point

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "x2", "b", "y2", "c"], "advice": "

We know that the graph crosses the $x$-axis at both $(\\var{a},0)$ and $(\\var{b},0)$. Since this is a quadratic, we know our equations has two roots, and by the previous observation, they are at $\\var{a}$ and $\\var{b}$. Hence we can write our equation as $\\simplify{y=(x-{a})(x-{b})}$ which simplifies to $\\simplify{y=x^2-({a}+{b})x+({a}*{b})}$.

\n

\n

To find the coefficients of the turning point of the quadratic, we know the x-coordinate of the turning point will correspond to the solution to $dy/dx=0$. So we get $\\simplify{2x-({a}+{b})}=0$ hence $\\simplify{x=({a}+{b})/2}$. We substitute this value of x back into the equation of the quadratic to find the corresponding y-coordinate.

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Input the smaller of the two \$$x\$$ values.

\n

\$$x=\$$ [[0]]

\n

Input the larger of the two \$$x\$$ values.

\n

\$$x=\$$ [[1]]

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The function \$$f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\$$  has two turning points.

", "ungrouped_variables": ["a", "b", "c"], "advice": "

\$$f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\$$

\n

To locate a turning point, differentite the function, set equal to zero and solve.

\n

\$$f'(x)=6x^2-\\simplify{6*({a}+{b})x+6*{a}*{b}}=0\$$

\n

Divide across by 6 to get the quadratic equation

\n

\$$x^2-\\simplify{({a}+{b})x+{a}*{b}}=0\$$

\n

This has factors

\n

\$$(x-\\var{a})(x-\\var{b})=0\$$

\n

\$$x-\\var{a}=0\$$     or     \$$x-\\var{b}=0\$$

\n

\$$x=\\var{a}\$$     or     \$$x=\\var{b}\$$