$\\simplify[std]{f(x) = sin({b}x + {a}) * e ^ ({n} * x)}$
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\n\t\t\tClick on Show steps for more information, you will not lose any marks by doing so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "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)}\\]
Differentiate the following function $f(x)$ using the product rule.
", "tags": ["calculus", "Calculus", "checked2015", "derivative of a product", "differentiating a product", "differentiating exponential functions", "differentiating trigonometric functions", "differentiation", "mas1601", "MAS1601", "product rule", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t31/07/2012:
\n\t\tChecked calculation.
\n\t\tAdded tags.
\n\t\tAllowed no penalty on looking at Show steps.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Differentiate $ \\sin(ax+b) e ^ {nx}$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tThe 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)}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = sin({a} + {b} * x)}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {b} * cos({a} + {b} * x)}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = e ^ ({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * e ^ ({n} * x)}\\]
\n\t \n\t \n\t \n\tHence on substituting into the product rule above we get:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*}\\frac{df}{dx} &=& \\simplify[std]{{b} * cos({a} + {b} * x) * e ^ ({n} * x) + {n} * sin({a} + {b} * x) * e ^ ({n} * x)}\\\\\n\t \n\t &=&\\simplify[std]{({b}cos({a}+{b}x)+{n}sin({a}+{b}x))e^({n}x)}\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t"}, {"name": "Integration: Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..b-1)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*b+r", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "vsetrange": [1, 2], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$
\n\t\t\t$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\n\t\t\tInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\tClick on Show steps to get more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
Input the constant of integration as $C$.
2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\tMessage about Show steps included. Also another message about including the constant of integration.
\n\t\tChanged checking range from 0 to 1 to 1 to 2.
\n\t\tImproved display.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tUsing
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]
Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$
\n\t\t\t$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\n\t\t\tInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\tClick on Show steps to get more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"type": "information", "prompt": "The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
Input the constant of integration as $C$.
2/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\tMessage about Show steps included. Also another message about including the constant of integration.
\n\t\tChanged checking range from 0 to 1 to 1 to 2.
\n\t\tImproved display.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tUsing
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]
Using $\\cos^2\\theta+\\sin^2\\theta=1$ to evaluate expressions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The following questions require some familiarity with trigonometric identities.
\n", "advice": "We will use the following Pythagorean identity \\[\\cos^2\\theta+\\sin^2\\theta=1.\\]
\nFor part a)
\nGiven $\\simplify{(cos^2(x)+sin^2(x))^{m1}+{m2}}$ we can replace $\\cos^2(x)+\\sin^2(x)$ with $1$. So our expression is $\\simplify{1^{m1}+{m2}}$. Therefore our expression simplifies to $\\var{m2+1}$.
\n| $\\simplify{{m1}+{m2}(2-sin^2(y)-cos^2(y))}$ | \n$=$ | \n$\\simplify{{m1}+{m2}(2-(sin^2(y)+cos^2(y)))}$ | \n
| \n | $=$ | \n$\\simplify[!collectNumbers]{{m1}+{m2}(2-1)}$ | \n
| \n | $=$ | \n$\\simplify[!collectNumbers]{{m1}+{m2}}$ | \n
| \n | $=$ | \n$\\var{m1+m2}$ | \n
| $\\simplify{{m1^2}cos^4(z)+{2*m1^2}cos^2(z)sin^2(z)+{m1^2}sin^4(z)}$ | \n$=$ | \n$\\simplify{{m1^2}(cos^4(z)+2cos^2(z)sin^2(z)+sin^4(z))}$ | \n
| \n | $=$ | \n$\\simplify{{m1^2}(cos^2(z)+sin^2(z))^2}$ | \n
| \n | $=$ | \n$\\var{m1^2}\\times 1^2$ | \n
| \n | $=$ | \n$\\var{m1^2}$ | \n
For part b)
\nRearranging the Pythagorean identity $\\cos^2\\theta+\\sin^2\\theta=1$ for $\\sin\\theta$ gives the equation \\[\\sin\\theta=\\pm\\sqrt{1-\\cos^2\\theta}\\]
\nRecall that $\\sin\\theta$ is the $y$ value of a point on the unit circle, whether $\\sin\\theta$ is taken as the postive square root or as the negative square root depends on the whether the point on the circle is on the top semicircle (positive $y$ value) or the bottom semicircle (negative $y$ value).
\nSince we are told $\\theta$ is in the first or second quadrant, the $y$ value must be postive, that is $\\sin\\theta=\\sqrt{1-\\cos^2\\theta}$. Therefore our expresson simplifies as follows
\n| $\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}$ | \n$=$ | \n$\\simplify{{n}sin(theta)-{n}sin(theta)}$ | \n
| \n | $=$ | \n$0$ | \n
Since we are told $\\theta$ is in the third or fourth quadrant, the $y$ value must be negative, that is $\\sin\\theta=-\\sqrt{1-\\cos^2\\theta}$ or equivalently $-\\sin\\theta=\\sqrt{1-\\cos^2\\theta}$. Therefore our expresson simplifies as follows
\n\n| $\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}$ | \n$=$ | \n$\\simplify{{n}sin(theta)-{n}sin(theta)}$ | \n
| \n | $=$ | \n$0$ | \n
Rearranging the Pythagorean identity $\\cos^2\\theta+\\sin^2\\theta=1$ for $\\cos\\theta$ gives the equation \\[\\cos\\theta=\\pm\\sqrt{1-\\sin^2\\theta}\\]
\nRecall that $\\cos\\theta$ is the $x$ value of a point on the unit circle, whether $\\cos\\theta$ is taken as the postive square root or as the negative square root depends on the whether the point on the circle is on the right semicircle (positive $x$ value) or the left semicircle (negative $x$ value).
\nSince we are told $\\theta$ is in the first or fourth quadrant, the $x$ value must be postive, that is $\\cos\\theta=\\sqrt{1-\\sin^2\\theta}$. Therefore our expresson simplifies as follows
\n| $\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}$ | \n$=$ | \n$\\simplify{{n}cos(theta)-{n}cos(theta)}$ | \n
| \n | $=$ | \n$0$ | \n
Since we are told $\\theta$ is in the second or third quadrant, the $x$ value must be negative, that is $\\cos\\theta=-\\sqrt{1-\\sin^2\\theta}$ or equivalently $-\\cos\\theta=\\sqrt{1-\\sin^2\\theta}$. Therefore our expresson simplifies as follows
\n\n| $\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}$ | \n$=$ | \n$\\simplify{{n}cos(theta)-{n}cos(theta)}$ | \n
| \n | $=$ | \n$0$ | \n
The expression
\n\\[\\simplify{(cos^2(x)+sin^2(x))^{m1}+{m2}}\\] \\[\\simplify{{m1}+{m2}(2-sin^2(y)-cos^2(y))}\\] \\[\\simplify{{m1^2}cos^4(z)+{2*m1^2}cos^2(z)sin^2(z)+{m1^2}sin^4(z)}\\]
\ncan be simplified to [[0]].
\nNote: For this question, your answer should be a number.
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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": "The expression
\n\\[\\simplify{{n}sin(theta)-{n}sqrt(1-cos^2(theta))}\\] \\[\\simplify{{n}sin(theta)+{n}sqrt(1-cos^2(theta))}\\]\\[\\simplify{{n}cos(theta)-{n}sqrt(1-sin^2(theta))}\\] \\[\\simplify{{n}cos(theta)+{n}sqrt(1-sin^2(theta))}\\]
\ncan be simplified to [[0]] for $\\theta$ in the first or second quadrant. third or fourth quadrant. first or fourth quadrant. second or third quadrant.
\nNote: For this question, your answer should be a number.
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "warnifunattempted", "message": "Try and complete this question before you move on
"}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "contributors": [{"name": "Joanne Withnall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4442/"}], "extensions": [], "custom_part_types": [], "resources": []}