Use the product rule to differentiate the function $y=2t^2 \\cos (2t)$.
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\displaystyle \\frac{dy}{dt}= $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"4t*cos(2t)+2t^2*sin(2t)\", \"You're on the right track, but it looks like you have made a couple of mistakes. First, double check that you have correctly differentiated $\\\\cos(\\\\cdot)$. Next, note that while the log tables give the rule for $\\\\cos(\\\\textbf{$x$})$ or $\\\\cos(\\\\textbf{$t$})$, here you have $\\\\cos(\\\\color{red}{\\\\textbf{$2$}}t)$. What extra step do you need to correctly differentiate this?\"],\n [\"4t*cos(2t)-2t^2*sin(2t)\", \"Almost there! Note that while the log tables give the rule for $\\\\cos(\\\\textbf{$x$})$ or $\\\\cos(\\\\textbf{$t$})$, here you have $\\\\cos(\\\\color{red}{\\\\textbf{$2$}}t)$. What extra step do you need to correctly differentiate this?\"],\n [\"4t*cos(2t)+4t^2*sin(2t)\", \"Almost there! Double check that you have correctly differentiated $\\\\cos(\\\\cdot)$ - in particular, note the signs.\"],\n [\"4t*cos(2t)-4t^2*sin(t)\", \"Very close. Remember, the angle of a trig. function never changes when you differentiate!\"],\n [\"4t*cos(2t)+4t^2*sin(t)\", \"You're on the right track, but need to watch a couple of things. First, remember that the angle of a trig. function never changes when you differentiate. Next, double check that you have correctly differentiated $\\\\cos(\\\\cdot)$.\"],\n [\"4t*cos(2t)+2t^2*sin(t)\", \"You're on the right track, but need to watch a couple of things. First, remember that the angle of a trig. function never changes when you differentiate. Next, double check that you have correctly differentiated $\\\\cos(\\\\cdot)$. Finally, note that while the log tables give the rule for $\\\\cos(\\\\textbf{$x$})$ or $\\\\cos(\\\\textbf{$t$})$, here you have $\\\\cos(\\\\color{red}{\\\\textbf{$2$}}t)$. What extra step do you need to correctly differentiate this?\"],\n [\"4t*cos(2t)-2t^2*sin(t)\", \"You're on the right track, but need to watch a couple of things. First, remember that the angle of a trig. function never changes when you differentiate. Also, note that while the log tables give the rule for $\\\\cos(\\\\textbf{$x$})$ or $\\\\cos(\\\\textbf{$t$})$, here you have $\\\\cos(\\\\color{red}{\\\\textbf{$2$}}t)$. What extra step do you need to correctly differentiate this?\"],\n [\"-8t*sin(2t)\", \"The function you need to differentiate is $2t^2 \\\\times \\\\cos(2t)$ i.e. one function of $t$ multiplied by another function of $t$. Therefore, you need the product rule.\"],\n [\"8t*sin(2t)\", \"The function you need to differentiate is $2t^2 \\\\times \\\\cos(2t)$ i.e. one function of $t$ multiplied by another function of $t$. Therefore, you need the product rule.\"],\n [\"4t*sin(2t)\", \"The function you need to differentiate is $2t^2 \\\\times \\\\cos(2t)$ i.e. one function of $t$ multiplied by another function of $t$. Therefore, you need the product rule.\"],\n [\"-4t*sin(2t)\", \"The function you need to differentiate is $2t^2 \\\\times \\\\cos(2t)$ i.e. one function of $t$ multiplied by another function of $t$. Therefore, you need the product rule.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$f'(x) = $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"(3x^2+1/x^2)*e^x\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"],\n [\"(x^3-1/x)*e^x+3x^2*e^x\", \"Almost there! Double check how to correctly differentiate $\\\\frac{1}{x}$.\"],\n [\"3x^2*e^x\", \"There are a couple of things to watch here. Firstly, note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Secondly, double check how to correctly differentiate $\\\\frac{1}{x}$.\"],\n [\"(x^3-1/x)*e^x+(3x^2-ln(x))*e^x\", \"Almost there! Double check how to correctly differentiate $\\\\frac{1}{x}$.\"],\n [\"(x^3-1/x)*e^x+(3x^2-1/x^2)*e^x\", \"Almost there! Double check how to correctly differentiate $x^3 - \\\\frac{1}{x}$.\"],\n [\"(3x^2-1/x^2)*e^x\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"],\n [\"(3x^2+x^(-1))*e^x\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"],\n [\"(3x^2-x^(-1))*e^x\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"],\n [\"(x^3-1/x)*e^x+(3x^2-x^(-1))*e^x\", \"Almost there! Double check how to correctly differentiate $\\\\frac{1}{x}$.\"],\n [\"(3x^2+1/x^2)*x*e^(x-1)\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(x^3-1/x)*x*e^(x-1)+3x^2*e^x\", \"Double check how to correctly differentiate $\\\\frac{1}{x}$. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"3x^2*x*e^(x-1)\", \"There are a few things to watch here. Firstly, note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Secondly, double check how to correctly differentiate $\\\\frac{1}{x}$. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(x^3-1/x)*x*e^(x-1)+(3x^2-ln(x))*e^x\", \"A couple of things to check. Double check how to correctly differentiate $\\\\frac{1}{x}$. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(x^3-1/x)*x*e^(x-1)+(3x^2-1/x^2)*e^x\", \"A couple of things to check. Double check how to correctly differentiate $x^3 - \\\\frac{1}{x}$. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(3x^2-1/x^2)*x*e^(x-1)\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(3x^2+x^(-1))*x*e^(x-1)\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(3x^2-x^(-1))*x*e^(x-1)\", \"Note that the function you need to differentiate is $\\\\left( x^3 - \\\\frac{1}{x} \\\\right) \\\\times e^x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"],\n [\"(x^3-1/x)*x*e^(x-1)+(3x^2-x^(-1))*e^x\", \"Double check how to correctly differentiate $\\\\frac{1}{x}$. Also, be careful when differentiating $e^x$. You cannot use the power rule for this.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$f'(x) = $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"3+3*cos(x)+3x*e^(x-1)\", \"Be careful when you are differentiating $e^x$. You cannot use the power rule for this. The power rule is only used if you have a variable to the power of a number (e.g. $x^3$). Here, we have $e^x$ i.e. a number ($e$) to the power of a variable ($x$).\"],\n [\"3-3*cos(x)+3x*e^(x-1)\", \"There are two things to watch here. Firstly, check that you have correctly differentiated $\\\\sin(\\\\cdot)$. Secondly, be careful when you are differentiating $e^x$. You cannot use the power rule for this. The power rule is only used if you have a variable to the power of a number (e.g. $x^3$). Here, we have $e^x$ i.e. a number ($e$) to the power of a variable ($x$).\"],\n [\"3-3*cos(x)+3e^x\", \"Almost there. Check that you have correctly differentiated $\\\\sin(\\\\cdot)$.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))Find $\\displaystyle \\frac{dx}{dt}$.
\n$\\displaystyle \\frac{dx}{dt} = $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"4t+1/(2t)\", \"You're on the right track, but need to be careful when differentiating $\\\\ln 2t$. Remember that the derivative of $\\\\ln \\\\textbf{$t$}$ is $\\\\frac{1}{t}$. However, you need to differentiate $\\\\ln \\\\color{red}{\\\\textbf{$2$}}t$. What is the extra step you need to include in order to differentiate this correctly?\"],\n [\"4t+2/t\", \"Look back at how you differentiated $\\\\ln 2t$. Think of the $2t$ as being a single unit. If $\\\\frac{d}{dx} (\\\\ln x) = \\\\frac{1}{x}$, $\\\\frac{d}{dt} (\\\\ln t) = \\\\frac{1}{t}$ and $\\\\frac{d}{dA} (\\\\ln A) = \\\\frac{1}{A}$, what should you get for $\\\\frac{d}{dt} (\\\\ln 2t)$?\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$\\displaystyle \\frac{d^2 x}{dt^2}=$ [[0]]
\n$\\displaystyle \\left( \\frac{dx}{dt} \\right)^2 = $ [[1]]
\n$\\displaystyle \\frac{d^2 x}{dt^2} + \\left( \\frac{dx}{dt} \\right)^2= $ [[2]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"4+ln(t)\", \"The $\\\\textit{integral}$ of $\\\\frac{1}{t}$ is $\\\\ln t$. What is the $\\\\textit{derivative}$ of $\\\\frac{1}{t}$?\"],\n [\"4+1/t^2\", \"Almost there! Take another look at $\\\\frac{1}{t}$. Think of $\\\\frac{1}{t}$ as $\\\\frac{1}{t} = t^{-1}$. Now try the power rule...\"],\n [\"4\", \"How do you differentiate $\\\\frac{1}{t}$? Think of $\\\\frac{1}{t}$ as $\\\\frac{1}{t} = t^{-1}$. Now try the power rule...\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$f'(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "malrules:\n [\n [\"12*e^(6x)\", \"Note that the function you need to differentiate is $2x \\\\times e^{6x}$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"],\n [\"2*e^(6x)+2x*e^(6x)\", \"Almost there! What is the extra step you need when differentiating $e^{6x}$ (where the power is more than just $x$)?\"],\n [\"2*e^(6x)+12x*e^(6x-1)\", \"The power rule is only valid for a variable to the power of a number (e.g. $x^6$). Therefore, you cannot use the power rule for $e^{6x}$ (since the variable is in the power).\"],\n [\"2*e^(6x)+12x^2*e^(6x-1)\", \"The power rule is only valid for a variable to the power of a number (e.g. $x^6$). Therefore, you cannot use the power rule for $e^{6x}$ (since the variable is in the power).\"],\n [\"2*e^(6x)+12x*e^(5x)\", \"The power rule is only valid for a variable to the power of a number (e.g. $x^6$). Therefore, you cannot use the power rule for $e^{6x}$ (since the variable is in the power).\"],\n [\"2*e^(6x)+12x^2*e^(5x)\", \"The power rule is only valid for a variable to the power of a number (e.g. $x^6$). Therefore, you cannot use the power rule for $e^{6x}$ (since the variable is in the power).\"],\n [\"2e^(6x)+12x*e^x\", \"Remember, the power on the exponential never changes when you differentiate it!\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))First, determine $f'(x)$:
\n$f'(x) = $ [[0]]
\nNow, evaluate the derivative at the point $(\\pi, 1)$:
\n$f'(\\pi)=$ [[1]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"sin(x)*e^(cos(x)+1)\", \"Double check the rule for differentiating $\\\\cos x$.\"],\n [\"(cos(x)+1)*e^(cos(x))\", \"The power rule is only valid for a variable to the power of a number (e.g. $x^6$). Therefore, you cannot use the power rule for $e^{\\\\cos x + 1}$ (since the variable is in the power).\"],\n [\"e^(cos(x)+1)\", \"You're on the right track. The exponential function does not change when differentiated. However, when the power is more complicated than a single variable (e.g. $x$) on its own, what extra step do you need to perform?\"],\n [\"e^(-sin(x))\", \"Remember, the power on the exponential never changes when you differentiate it!\"],\n [\"e^(-sin(x)+1)\", \"Remember, the power on the exponential never changes when you differentiate it!\"],\n [\"-sin(x+1)*e^(cos(x+1))\", \"Be very careful when deciding what the argument of cosine is. The +1 is not part of the angle, as there are no brackets to group it with that $x$. The function is $e^{\\\\cos(x)+1}$.\"],\n [\"sin(x+1)*e^(cos(x+1))\", \"Be very careful when deciding what the argument of cosine is. The +1 is not part of the angle, as there are no brackets to group it with that $x$. The function is $e^{\\\\cos(x)+1}$. Also, double-check the rule for differentiating $\\\\cos x$.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$g'(x)=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "malrules:\n [\n [\"-3cos(x)*sin(x)\", \"Note that the function you need to differentiate is $3 \\\\sin x \\\\times \\\\cos x$ i.e. one function of $x$ multiplied by another function of $x$. Therefore, you need the product rule.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))First, find the derivative:
\n$f'(t) = $ [[0]]
\n\nNext, find the slope of the tangent when $\\displaystyle t = \\frac{\\pi}{4}$:
\nSlope = [[1]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"1/t*cos(t)-ln(sin(t))\", \"Note that the function you need to differentiate is $3 \\\\ln(\\\\cos t)$ not $3 \\\\ln \\\\times \\\\cos t$. This is not a product and so we do not need the product rule.\"],\n [\"1/t*cos(t)+ln(sin(t))\", \"Note that the function you need to differentiate is $3 \\\\ln(\\\\cos t)$ not $\\\\ln \\\\times \\\\cos t$. This is not a product and so we do not need the product rule.\"],\n [\"1/cos(t)\", \"You have used the correct basic rule for the natural log. However, when you are differentiating $\\\\ln(\\\\color{red}{f(t)})$ rather than just $\\\\ln(t)$, what extra step do you need to include?\"],\n [\"sin(t)/cos(t)\", \"You are almost there. Double check the rule for differentiating $\\\\cos t$.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))$\\displaystyle \\frac{dy}{dx}=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"sin(x)-12/x\", \"Almost there! Double check the rule for differentiating $\\\\cos x$.\"],\n [\"(cos(x)-12)*1/x-sin(x)*ln(x)\", \"The function you are asked to differentiate consists of two terms - one $\\\\textit{subtracted}$ from the other, not $\\\\textit{multiplied}$. (Note: If there are no brackets, a $+$ or a $-$ sign separates what came before it from what comes after it.)\"],\n [\"(cos(x)-12)*1/x+sin(x)*ln(x)\", \"The function you are asked to differentiate consists of two terms - one $\\\\textit{subtracted}$ from the other, not $\\\\textit{multiplied}$. (Note: If there are no brackets, a $+$ or a $-$ sign separates what came before it from what comes after it.)\"] \n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))multiple of the power of x
", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "power on the x
", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9 except a)", "description": "multiple of x
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "n", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\displaystyle \\frac{dy}{dx}=$ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"-({a*n}*x^{n-1}-{b})*cos({a}*x^{n}-{b}*x)\", \"Almost there! Double check the rule for differentiating $\\\\sin x$.\"],\n [\"cos({a}*x^{n}-{b}*x)\", \"You have used the correct rule for differentiating $\\\\sin x$. However, since the angle is more complicated than $x$, what extra step do you need to include?\"],\n [\"-cos({a}*x^{n}-{b}*x)\", \"Double check the rule for differentiating $\\\\sin x$. Also, since the angle is more complicated than $x$, what extra step do you need to include?\"],\n [\"cos({a*n}*x^{n-1}-{b})\", \"Remember, the angle in a trigonometric function never changes when you differentiate it!\"],\n [\"-cos({a*n}*x^{n-1}-{b})\", \"Remember, the angle in a trigonometric function never changes when you differentiate it! Also, double check the rule for differentiating $\\\\sin x$.\"],\n [\"cos({a}*x^{n}-{b}*x)+({a*n}*x^{n-1}-{b})*sin\", \"The function $\\\\sin (\\\\var{a}x^{\\\\var{n}} - \\\\var{b}x)$ is not a product, so you do not need the product rule.\"],\n [\"-cos({a}*x^{n}-{b}*x)+({a*n}*x^{n-1}-{b})*sin\", \"The function $\\\\sin (\\\\var{a}x^{\\\\var{n}} - \\\\var{b}x)$ is not a product, so you do not need the product rule.\"],\n [\"cos({a}*x^{n}-{b}*x)+sin({a*n}*x^{n-1}-{b})\", \"The function $\\\\sin (\\\\var{a}x^{\\\\var{n}} - \\\\var{b}x)$ is not a product, so you do not need the product rule.\"],\n [\"-cos({a}*x^{n}-{b}*x)+sin({a*n}*x^{n-1}-{b})\", \"The function $\\\\sin (\\\\var{a}x^{\\\\var{n}} - \\\\var{b}x)$ is not a product, so you do not need the product rule.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))First, find the derivative:
\n$h'(t) = $ [[0]]
\nNow evaluate the derivative at the point $(1,0)$:
\n$h'(1)=$ [[1]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.7", "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"2\", \"Take a closer look at the function you are asked to differentiate. It consists of one function of $t$ (i.e. $t^2-9$) multiplied by another function of $t$ (i.e. $\\\\ln t$). Therefore you need the product rule.\"],\n [\"(t^2-9)/t+(2t-9)*ln(t)\", \"Double check your derivative of $t^2-9$.\"],\n [\"(t^2-9)/t-2t*ln(t)\", \"Double check the formula for the product rule.\"],\n [\"-(t^2-9)/t+2t*ln(t)\", \"Double check the formula for the product rule.\"]\n ]\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))power on the x
", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "multiple of x
", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..8 except a)", "description": "constant to be added
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\displaystyle \\frac{ds}{dx}= $ [[0]]
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "malrules:\n [\n [\"x-{a}x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\",\"Almost there! Don't forget to differentiate the first term (the $x$) as well!\"],\n [\"1-{a*n}*x^{n-1}*sin({a}x+{b})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule.\"],\n [\"1-{n}x^{n-1}*sin({a})\", \"There are two issues here. Firstly, take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule. Secondly, remember that the angle of a trigonometric function never changes when you differentiate!\"],\n [\"-{a}(x+x^{n})*sin({a}x+{b})+(1+{n}x^{n-1})*cos({a}x+{b})\", \"Be very careful when using the product rule how you decide what $u$ and $v$ are. If $s=(x+x^{\\\\var{n}})\\\\cos (\\\\var{a}x+\\\\var{b})$, then $u=x+x^{\\\\var{n}}$ and $v=\\\\cos (\\\\var{a}x+\\\\var{b})$. However, here, $s=x+x^{\\\\var{n}} \\\\cos (\\\\var{a}x+\\\\var{b})$ (i.e. there are no brackets). Therefore, $u=x^{\\\\var{n}}$ and $v=\\\\cos (\\\\var{a}x+\\\\var{b})$. The $x$ being added at the start is differentiated separately.\"],\n [\"-{a}(1+{n}x^{n-1})*sin({a}x+{b})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule.\"],\n [\"-(1+{n}x^{n-1})*sin({a})\", \"Take a closer look at the function you are asked to differentiate. Whenever you have something to do with $x$ multiplied by something else to do with $x$, you need the product rule. Also, remember that the angle of a trigonometric function never changes when you differentiate!\"],\n [\"1+{a}x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\", \"Almost there! Double check the rule for differentiating $\\\\cos x$.\"],\n [\"1-x^{n}*sin({a}x+{b})+{n}x^{n-1}*cos({a}x+{b})\", \"You have used the correct rule for differentiating $\\\\cos x$, but when the angle is more complicated than $x$, what extra step do you need to include?\"],\n [\"1-x^{n}*sin({a})+{n}x^{n-1}*cos({a}x+{b})\",\"Remember, the angle in a trigonometric function never changes when you differentiate!\"],\n [\"1+x^{n}*sin({a})+{n}x^{n-1}*cos({a}x+{b})\",\"Remember, the angle in a trigonometric function never changes when you differentiate! Also, double check the rule for differentiating $\\\\cos x$.\"] \n ]\n \nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))