// Numbas version: exam_results_page_options
{"showQuestionGroupNames": false, "duration": 0, "metadata": {"description": "<p>This quiz asks questions on basic techniques of differentation and some introductory applications.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "preventleave": true, "showresultspage": "oncompletion", "showfrontpage": true, "browse": true, "allowregen": true}, "name": "NUMBAS - CW -Differentiation", "percentPass": 0, "showstudentname": true, "timing": {"allowPause": true, "timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}}, "feedback": {"intro": "", "showactualmark": true, "showanswerstate": true, "allowrevealanswer": true, "showtotalmark": true, "feedbackmessages": [], "advicethreshold": 0}, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Kamila's copy of Power rule 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Kamila Yusufu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2554/"}], "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Power rule</p>"}, "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x^{\\var{b2}}+\\var{c1}\\)</p>", "rulesets": {}, "parts": [{"showCorrectAnswer": true, "type": "gapfill", "variableReplacements": [], "prompt": "<p>\\(\\frac{df}{dx}=\\)&nbsp;[[0]]</p>", "gaps": [{"showpreview": true, "variableReplacements": [], "vsetrangepoints": 5, "checkvariablenames": false, "marks": 1, "answer": "{a1}*{a2}*x^{{a2}-1}+{b1}*{b2}*x^{{b2}-1}", "showCorrectAnswer": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "expectedvariablenames": [], "checkingtype": "absdiff", "scripts": {}, "checkingaccuracy": 0.001}], "marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "functions": {}, "preamble": {"css": "", "js": ""}, "advice": "<p>Apply&nbsp;the rule:</p>\n<p style=\"padding-left: 30px;\">\\(y=ax^n\\,\\,\\,then\\,\\,\\,\\frac{dy}{dx}=nax^{n-1}\\)</p>\n<p>In this example</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x^{\\var{b2}}+\\var{c1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\var{a2}*\\var{a1}x^{\\var{a2}-1}+\\var{b2}*\\var{b1}x^{\\var{b2}-1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dy}{dx}=\\simplify{{a2}*{a1}x^{{a2}-1}+{b2}*{b1}x^{{b2}-1}}\\)</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"b2": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..5#1)", "name": "b2", "description": ""}, "c1": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(3..18#1)", "name": "c1", "description": ""}, "b1": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(2..14#1)", "name": "b1", "description": ""}, "a2": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(5..12#1)", "name": "a2", "description": ""}, "a1": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(2..10#1)", "name": "a1", "description": ""}}, "tags": [], "variable_groups": [], "type": "question"}, {"name": "Power rule 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "parts": [{"variableReplacements": [], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "prompt": "<p>\\(\\frac{df}{dx}=\\)&nbsp;[[0]]</p>", "type": "gapfill", "gaps": [{"variableReplacements": [], "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "type": "jme", "answer": "-{a1}*{a2}/x^{{a2}+1}+(1/{a3})x^{1/{a3}-1}", "scripts": {}, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showpreview": true, "checkingtype": "absdiff", "showCorrectAnswer": true, "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}]}], "metadata": {"description": "<p>Power rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "<p>\\(f(x)=\\frac{\\var{a1}}{x^{\\var{a2}}}+\\sqrt[\\var{a3}]{x}\\)</p>\n<p>\\(f(x)=\\var{a1}x^{-\\var{a2}}+{x}^{\\frac{1}{\\var{a3}}}\\)</p>\n<p>\\(\\frac{df}{dx}=-\\var{a2}*\\var{a1}x^{-\\var{a2}-1}+\\frac{1}{\\var{a3}}{x}^{\\frac{1}{\\var{a3}}-1}\\)</p>\n<p>\\(\\frac{df}{dx}=-\\simplify{{a2}*{a1}x^{-{a2}-1}}+\\frac{1}{\\var{a3}}{x}^{\\simplify{{1-{a3}}/{a3}}}\\)</p>\n<p></p>", "ungrouped_variables": ["a1", "a2", "a3"], "tags": [], "functions": {}, "rulesets": {}, "variables": {"a2": {"group": "Ungrouped variables", "templateType": "randrange", "description": "", "name": "a2", "definition": "random(2..8#1)"}, "a3": {"group": "Ungrouped variables", "templateType": "randrange", "description": "", "name": "a3", "definition": "random(3..6#1)"}, "a1": {"group": "Ungrouped variables", "templateType": "randrange", "description": "", "name": "a1", "definition": "random(6..18#1)"}}, "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\frac{\\var{a1}}{x^{\\var{a2}}}+\\sqrt[\\var{a3}]{x}\\)</p>", "type": "question"}, {"name": "Slope of a curve at a point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "preamble": {"js": "", "css": ""}, "statement": "<p>Calculate the slope of the curve</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\var{a}x^3-\\var{b}x^2+\\var{c}x+\\var{d}\\)</p>\n<p>at the point where \\(x=\\var{f}\\).&nbsp;</p>", "parts": [{"prompt": "<p>Input your answer correct to one decimal place.</p>\n<p style=\"padding-left: 30px;\">\\(slope = \\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "mustBeReduced": false, "precisionPartialCredit": 0, "minValue": "3*{a}*{f}^2-2*{b}*{f}+{c}", "customMarkingAlgorithm": "", "allowFractions": false, "marks": 1, "precisionType": "dp", "precision": "1", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "3*{a}*{f}^2-2*{b}*{f}+{c}", "mustBeReducedPC": 0, "unitTests": []}], "variableReplacements": [], "showCorrectAnswer": true, "sortAnswers": false, "unitTests": [], "customMarkingAlgorithm": "", "marks": 0}], "metadata": {"description": "<p>Slope of a curve at a point</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"c": {"description": "", "templateType": "randrange", "name": "c", "definition": "random(5..12#1)", "group": "Ungrouped variables"}, "f": {"description": "", "templateType": "randrange", "name": "f", "definition": "random(0..4#1)", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "randrange", "name": "b", "definition": "random(2..10#1)", "group": "Ungrouped variables"}, "d": {"description": "", "templateType": "randrange", "name": "d", "definition": "random(10..20#1)", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "randrange", "name": "a", "definition": "random(2..6#1)", "group": "Ungrouped variables"}}, "ungrouped_variables": ["a", "b", "c", "d", "f"], "advice": "<p>\\(f(x)=\\var{a}x^3-\\var{b}x^2+\\var{c}x+\\var{d}\\)</p>\n<p>The equation for the slope of a curve is found by differentiating the function.</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=3*\\var{a}x^2-2*\\var{b}x+\\var{c}\\)</p>\n<p>To find the slope at a particular point we simply insert the x-coordinate value into this equation.</p>\n<p style=\"padding-left: 30px;\">Slope = \\(3*\\var{a}*\\var{f}^2-2*\\var{b}*\\var{f}+\\var{c}\\)</p>\n<p style=\"padding-left: 30px;\">Slope = \\(\\simplify{3*{a}*{f}^2-2*{b}*{f}+{c}}\\)</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "functions": {}, "rulesets": {}, "type": "question"}, {"name": "Turning points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "preamble": {"css": "", "js": ""}, "parts": [{"type": "gapfill", "marks": 0, "prompt": "<p>Input the smaller of the two \\(x\\)&nbsp;values.&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(x=\\)&nbsp;[[0]]</p>\n<p>Input the larger of the two \\(x\\) values.</p>\n<p style=\"padding-left: 30px;\">\\(x=\\)&nbsp;[[1]]</p>", "variableReplacementStrategy": "originalfirst", "gaps": [{"marks": 1, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{a}", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "maxValue": "{a}", "correctAnswerStyle": "plain"}, {"marks": 1, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{b}", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "maxValue": "{b}", "correctAnswerStyle": "plain"}], "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "showCorrectAnswer": false}], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>The function \\(f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\\) &nbsp;has two turning points.</p>", "ungrouped_variables": ["a", "b", "c"], "advice": "<p>\\(f(x)=2x^3-\\simplify{3*({a}+{b})x^2+6*{a}*{b}x}+\\var{c}\\)</p>\n<p>To locate a turning point, differentite the function, set equal to zero and solve.</p>\n<p style=\"padding-left: 30px;\">\\(f'(x)=6x^2-\\simplify{6*({a}+{b})x+6*{a}*{b}}=0\\)</p>\n<p>Divide across by 6 to get the quadratic equation</p>\n<p style=\"padding-left: 30px;\">\\(x^2-\\simplify{({a}+{b})x+{a}*{b}}=0\\)</p>\n<p>This has factors</p>\n<p style=\"padding-left: 30px;\">\\((x-\\var{a})(x-\\var{b})=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(x-\\var{a}=0\\) &nbsp; &nbsp; or &nbsp; &nbsp; \\(x-\\var{b}=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\var{a}\\) &nbsp; &nbsp; or &nbsp; &nbsp; \\(x=\\var{b}\\)</p>", "metadata": {"description": "<p>Turning points of a cubic function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "rulesets": {}, "variable_groups": [], "variables": {"a": {"definition": "random(1..5#1)", "templateType": "randrange", "description": "", "name": "a", "group": "Ungrouped variables"}, "c": {"definition": "random(1..15#1)", "templateType": "randrange", "description": "", "name": "c", "group": "Ungrouped variables"}, "b": {"definition": "random(6..11#1)", "templateType": "randrange", "description": "", "name": "b", "group": "Ungrouped variables"}}, "type": "question"}, {"name": "Chain rule 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "functions": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>\\(f(x)=({\\var{a2}x^{\\var{a3}}+\\var{a4}})^\\var{a1}\\)</p>\n<p>Recall the chain rule: &nbsp; \\(\\frac{df}{dx}=\\frac{df}{du}.\\frac{du}{dx}\\)</p>\n<p style=\"padding-left: 30px;\">let \\(u=\\var{a2}x^{\\var{a3}}+\\var{a4}\\) &nbsp; &nbsp;then &nbsp;&nbsp;\\(f(x)=u^\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{du}=\\var{a1}u^{\\var{a1}-1}\\) &nbsp;and &nbsp;\\(\\frac{du}{dx}=\\var{a3}*\\var{a2}x^{\\var{a3}-1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\var{a1}u^{\\simplify{{a1}-1}}.\\simplify{{a2}*{a3}x^{{a3}-1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\simplify{{a1}*{a2}*{a3}x^{{a3}-1}}({\\var{a2}x^{\\var{a3}}+\\var{a4}})^{\\simplify{{a1}-1}}\\)</p>", "rulesets": {}, "parts": [{"prompt": "<p>\\(\\frac{df}{dx}=\\)[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({a1}*{a2}*{a3}x^{{a3}-1})({a2}x^{{a3}}+{a4})^{{a1}-1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=({\\var{a2}x^{\\var{a3}}+\\var{a4}})^\\var{a1}\\)</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a1": {"definition": "random(2..8#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "random(1..4#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "random(2..6#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a2", "description": ""}, "a4": {"definition": "random(1..14#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a4", "description": ""}}, "metadata": {"description": "<p>Chain rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "functions": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "a5"], "tags": [], "advice": "<p>\\(f(x)=(\\var{a1}x^\\var{a2}+\\var{a3})e^{\\var{a4}x+\\var{a5}}\\)</p>\n<p>Recall the product rule if \\(f(x)=u.v\\) where \\(u\\) and \\(v\\) are both functions of \\(x\\) then</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=v.\\frac{du}{dx}+u.\\frac{dv}{dx}\\)</p>\n<p style=\"padding-left: 30px;\">let \\(u=\\var{a1}x^\\var{a2}+\\var{a3}\\) &nbsp;and &nbsp;\\(v=e^{\\var{a4}x+\\var{a5}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{du}{dx}=\\var{a2}*\\var{a1}x^{\\var{a2}-1}\\) &nbsp;and &nbsp;\\(\\frac{dv}{dx}=\\var{a4}*e^{\\var{a4}x+\\var{a5}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=e^{\\var{a4}x+\\var{a5}}*\\var{a2}*\\var{a1}x^{\\var{a2}-1}+(\\var{a1}x^\\var{a2}+\\var{a3})*\\var{a4}*e^{\\var{a4}x+\\var{a5}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\simplify{e^({a4}x+{a5})*{a1}*{a2}x^{{a2}-1}+({a1}x^{a2}+{a3})*{a4}*e^({a4}x+{a5})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=(\\simplify{{a1}*{a4}x^{a2}+{a1}*{a2}x^{{a2}-1}+{a3}*{a4}})\\simplify{e^({a4}x+{a5})}\\)</p>", "rulesets": {}, "parts": [{"prompt": "<p>\\(\\frac{df}{dx}=\\)[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({a1}*{a4}x^{a2}+{a1}*{a2}x^{{a2}-1}+{a3}*{a4})*e^({a4}x+{a5})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Differentiate the function</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=(\\var{a1}x^\\var{a2}+\\var{a3})e^{\\var{a4}x+\\var{a5}}\\)</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a1": {"definition": "random(2..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "random(1..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a2", "description": ""}, "a5": {"definition": "random(4..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a5", "description": ""}, "a4": {"definition": "random(2..8#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a4", "description": ""}}, "metadata": {"description": "<p>Product rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Quotient rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "f", "d"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>\\(f(x)=\\frac{\\var{a}x^{\\var{b}}+\\var{f}}{\\var{c}cos(\\var{d}x)}\\)</p>\n<p>Recall the quotient rule: if \\(y=\\frac{u}{v}\\) where \\(u\\) and \\(v\\) are both functions of \\(x\\) then</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}-u\\frac{dv}{dx}}{v^2}\\)</p>\n<p>Let &nbsp; \\(u=\\var{a}x^{\\var{b}}+\\var{f}\\) &nbsp; and &nbsp; \\(v=\\var{c}cos(\\var{d}x)\\)</p>\n<p>then &nbsp; \\(\\frac{du}{dx}=\\var{b}*\\var{a}x^{\\var{b}-1}\\) &nbsp; and &nbsp; \\(\\frac{dv}{dx}=-\\var{d}*\\var{c}sin(\\var{d}x)\\)</p>\n<p>Putting these results together as shown in the rule gives:</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\frac{(\\var{c}cos(\\var{d}x))*\\var{b}*\\var{a}x^{\\var{b}-1}-(\\var{a}x^{\\var{b}}+\\var{f})*(-\\var{d}*\\var{c}sin(\\var{d}x))}{(\\var{c}cos(\\var{d}x))^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{df}{dx}=\\frac{\\simplify{({c}*cos({d}x))*{b}*{a}x^{{b}-1}+({a}x^{{b}}+{f})*({c}*{d}*sin({d}x))}}{(\\var{c}*cos(\\var{d}x))^2}\\)</p>", "rulesets": {}, "parts": [{"prompt": "<p>\\(\\frac{df}{dx} = \\)&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}*cos({d}*x)*{a}*{b}*x^({b}-1)+({a}*x^{b}+{f})*{c}*{d}*sin({d}*x))/({c}*cos({d}*x))^2", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Differentiate the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\frac{\\var{a}x^{\\var{b}}+\\var{f}}{\\var{c}cos(\\var{d}x)}\\)</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..6#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(2..7#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d", "description": ""}, "f": {"definition": "random(1..11#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "f", "description": ""}}, "metadata": {"description": "<p>Quotient rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Rate of change", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "ungrouped_variables": ["a", "b"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Rate of change problem involving velocity &amp; acceleration</p>"}, "statement": "<p>A missile is launched straight up in the air. The height of the missile,&nbsp;\\(h\\)&nbsp;metres, above the ground \\(t\\) seconds after the launch button is pressed is given by:</p>\n<p style=\"padding-left: 30px;\">\\(h=\\var{a}t-4.9t^2\\)</p>", "rulesets": {}, "variable_groups": [], "functions": {}, "preamble": {"css": "", "js": ""}, "advice": "<p>\\(h=\\var{a}t-4.9t^2\\)</p>\n<p>Recall that speed is the rate of change of position with respect to time &nbsp; i.e. \\(v=\\frac{dh}{dt}\\)</p>\n<p style=\"padding-left: 30px;\">\\(v=\\frac{dh}{dt}=\\var{a}-2*4.9t\\)</p>\n<p>when \\(t=\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(v=\\var{a}-2*4.9*\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(v=\\simplify{{a}-9.8*{b}}m/s\\)</p>\n<p></p>\n<p>The missile will reach its maximum height when its speed = 0. &nbsp; i.e.&nbsp;\\(v=\\frac{dh}{dt}=\\var{a}-2*4.9t=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a}=9.8t\\)</p>\n<p style=\"padding-left: 30px;\">\\(t=\\var{a}/9.8\\)</p>\n<p>The maximum height reached will occur when \\(t=\\simplify{{a}/9.8}\\)</p>\n<p style=\"padding-left: 30px;\">\\(h=\\var{a}*\\left(\\simplify{{a}/9.8}\\right)-4.9*\\left(\\simplify{{a}/9.8}\\right)^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(h=\\simplify{{a}^2/19.6}\\)</p>\n<p style=\"padding-left: 30px;\">\\(h=\\simplify{{{a}/{19.6}^0.5}^2}\\)</p>\n<p></p>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"b": {"templateType": "randrange", "description": "", "definition": "random(3..10#1)", "name": "b", "group": "Ungrouped variables"}, "a": {"templateType": "randrange", "description": "", "definition": "random(100..300#5)", "name": "a", "group": "Ungrouped variables"}}, "tags": [], "parts": [{"showCorrectAnswer": true, "type": "gapfill", "variableReplacements": [], "prompt": "<p>Calculate the speed of the missile (m/s) \\(\\var{b}\\) seconds after launch. Give your answer correct to one decimal place.</p>\n<p style=\"padding-left: 30px;\">\\(v = \\)&nbsp;[[0]]m/s</p>\n<p>What is the maximum height achieved by this missile?&nbsp;Give your answer correct to one decimal place.</p>\n<p style=\"padding-left: 30px;\">\\(h = \\)&nbsp;[[1]]m</p>", "gaps": [{"precisionPartialCredit": 0, "variableReplacements": [], "precisionType": "dp", "correctAnswerFraction": false, "showPrecisionHint": false, "strictPrecision": false, "allowFractions": false, "scripts": {}, "minValue": "{a}-9.8*{b}", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "precision": "1", "marks": 1, "maxValue": "{a}-9.8*{b}", "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionPartialCredit": 0, "variableReplacements": [], "precisionType": "dp", "correctAnswerFraction": false, "showPrecisionHint": false, "strictPrecision": false, "allowFractions": false, "scripts": {}, "minValue": "{a}^2/19.6", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "precision": "1", "marks": 1, "maxValue": "{a}^2/19.6", "precisionMessage": "You have not given your answer to the correct precision."}], "marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "type": "question"}]}], "type": "exam", "contributors": [{"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}, {"name": "Kamila Yusufu", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2554/"}, {"name": "Ha Nee Yeoum", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2565/"}], "extensions": [], "custom_part_types": [], "resources": []}