// Numbas version: exam_results_page_options {"name": "Integration", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": "0", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Integration: Polynomial Functions - Population Growth", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Evi Papadaki", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18113/"}], "tags": [], "metadata": {"description": "

Integrating a polynomial functions which describe the rate of change of a population over time to find and use an equation that describes the total population according to time.

", "licence": "None specified"}, "statement": "

The growth rate, $r$ individuals per year, of a population in year $t$ after the start of observation is given by
$\\simplify {r=2*{a}t+{b}}$

", "advice": "

a) The function for the growth rate, $r$, is the derivative of the function that describes the total population, $n$, in respect of time, $t$. So, to find an expression of the function $n$ we need to integrate $r$ in respect of $t$.

Remember, indefinite integration (integration without limits) is the reverse of differentiation.

Therefore,

\\[ \\begin{split} \\simplify{n=int({2*a}*t+{b},t)} &\\,= \\simplify{ {2*a}/2*t^2+{b}*t+c} \\\\ &\\,= \\simplify{{a}*t^2+{b}*t+c} \\end{split} \\]

To find the exact expression that describes the total population we need to find the value of $c$. To do so, we use the fact that when $t=0$ the total population is $\\var{c}$.

\\[ \\begin{split} \\simplify[!all]{{a}*0^2+{b}*0+c} &= \\var{c} \\\\ c&= \\var{c} \\end{split} \\]

Thus, the expression that describes the total population in year $t$ is:

\\[ n= \\simplify{{a}*t^2+{b}*t+{c}} \\]

b) The function $n= \\simplify{{a}*t^2+{b}*t+{c}}$ gives us the total population at year t. Thus, to find the population at year $\\var{t_1}$, I need to substitute the value $t=\\var{t_1}$ in the formula.

\\[ \\begin{split} n&= \\simplify[!all]{{a}*{t_1}^2+{b}*{t_1}+{c}} \\\\ &= \\simplify[!all]{{a}*{t_1^2}+{b}*{t_1}+{c}} \\\\ &= \\simplify[!all]{{a*t_1^2}+{b*t_1}+{c}}\\\\ &=\\var{ansB} \\end{split} \\]

Therefore, when $t=\\var{t_1}$, the population is $n=\\var{ansB}$.

If there are $\\var{c}$ individuals in the population in year $t=0$, find an expression for the total population, $n$ individuals, in year $t$.

$n=$[[0]]

Give your answer in the form $n=at^2+bt+c$.

Use your expression to find the total population at year $t=\\var{t_1}$.

When $t=\\var{t_1}$, $n=$[[0]]

Calculating area under curves of the form $ax^2+bx$ and $ax^4+bx^3+cx^2+dx+e$ in a contextualised problem.

", "licence": "None specified"}, "statement": "", "advice": "

A cross-section is a cutting made across something, here, the fins of a fish. The curve y models the outline of the cross-section. The area of the cross-section can be found by calculating the definite integral between the two boundaries. Therefore,

a) The area of the dorsal fin is:

\\[ \\begin{split} \\simplify[!noLeadingMinus]{defint({-3*a}*x^2+{2*b}*x, x ,0 ,{x_upper})} &\\,= \\left[ \\simplify[!all]{{-3*a}/3 x^3+{2*b}/2 x^2}\\right]_0^\\var{x_upper} \\\\ &\\,= \\left[ \\simplify[all, !noLeadingMinus]{{-a}x^3+{b}x^2}\\right]_0^\\var{x_upper} \\\\ &\\,= \\left[ \\simplify[ !noLeadingMinus]{{-a}*{x_upper}^3+{b}*{x_upper}^2}\\right]-\\left[ \\simplify[ !noLeadingMinus]{{-a}*0^3+{b}*0^2}\\right] \\\\ &\\,= \\left[ \\simplify{{-a*{x_upper}^3}+{b*x_upper^2}}\\right]-\\left[ \\simplify{{-a}*0+{b}*0}\\right] \\\\ &\\,= \\var{ans_a} cm^2\\end{split}\\]

b)Similarly, the area of the pelvic fin is:

\\[ \\begin{split} \\simplify[!noLeadingMinus]{defint(-x^4+{4*c}*x^3-{3*d}*x^2+{2*k}*x-{f}, x ,1 ,3.5)} &\\,= \\left[ \\simplify[!all]{-{1/5}* x^5+{4*c}/4*x^4-{3*d}/3*x^3+{2*k}/2*x^2-{f}*x }\\right]_1^{3.5} \\\\ &\\,= \\left[ \\simplify[unitFactor]{-{1/5}* x^5+{4*c/4}*x^4-{3*d/3}*x^3+{2*k/2}*x^2-{f}*x }\\right]_1^{3.5} \\\\ &\\,= \\left[ \\simplify[unitFactor]{-{1/5}* 3.5^5+{4*c/4}*3.5^4-{3*d/3}*3.5^3+{2*k/2}*3.5^2-{f}*3.5 }\\right] - \\left[ \\simplify[unitFactor]{-{1/5}* 1^5+{4*c/4}*1^4-{3*d/3}*1^3+{2*k/2}*1^2-{f}*1 }\\right] \\\\ &\\,= \\left[ \\simplify[unitFactor, fractionNumbers]{-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} }\\right] - \\left[ \\simplify[unitFactor, fractionNumbers]{-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }\\right] \\\\ &\\,= \\left[ \\simplify{-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} }\\right] - \\left[ \\simplify{-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }\\right] \\\\ &\\,= \\simplify{ {-{1*3.5^5/5}+{4*c*3.5^4/4}-{3*d*3.5^3/3}+{2*k*3.5^2/2}-{f*3.5} } - {-{1* 1^5/5}+{4*c*1^4/4}-{3*d*1^3/3}+{2*k*1^2/2}-{f*1} }} \\\\ &\\,= \\var{ansb} cm^2 \\text{(2 d.p.)} \\end{split}\\]

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0.1..0.5 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "Random(0.6..1 #0.15)", "description": "", "templateType": "anything", "can_override": false}, "x_upper": {"name": "x_upper", "group": "Ungrouped variables", "definition": "(2b)/(3a)", "description": "", "templateType": "anything", "can_override": false}, "ans_a": {"name": "ans_a", "group": "Ungrouped variables", "definition": "precround (-a*x_upper^3+b*x_upper^2,2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Unnamed group", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Unnamed group", "definition": "random(6..9)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Unnamed group", "definition": "random (15..20)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Unnamed group", "definition": "random (10..15)", "description": "", "templateType": "anything", "can_override": false}, "ansb": {"name": "ansb", "group": "Unnamed group", "definition": "precround(0.2*(1-3.5^5)+c*(3.5^4-1)+d*(1-3.5^3)+k*(3.5^2-1)+f*(1-3.5),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "x_upper=ceil(x_upper)", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x_upper", "ans_a"], "variable_groups": [{"name": "Unnamed group", "variables": ["c", "d", "k", "f", "ansb"]}], "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": "

The cross-section of the dorsal fin of a fish can be modelled by the curve $\\simplify[all,!noLeadingMinus]{y=-3*{a}x^2+2*{b}x}$ for $0\\le\\ x\\le \\var{x_upper}$, where all units are measured in centimetres.

Calculate the area of the dorsal fin of this fish.

Give your answer rounded to 2 decimal places, if needed.

[[0]] $\\mathrm{cm}^2$

The cross-section of the pelvic fin of this fish is modelled by the curve \\[ \\simplify[all,!noLeadingMinus]{ y=-x^4+4*{c}x^3-3*{d}x^2+2*{k}x-{f}}\\] for $1\\le\\ x\\le3.5$.

Calculate the area of the pelvic fin of the fish.

Give your answer rounded to 2 decimal places, if needed.

[[0]] $\\mathrm{cm}^2$

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ansb}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "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": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Evi Papadaki", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18113/"}], "extensions": [], "custom_part_types": [], "resources": []}