7:[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Quiz\",\"name\":\"Section I Part A\",\"description\":\"Multiple choice non-calculator section\",\"url\":\"https://theschoolofmathematics.com/quiz/ap-calculus-bc-full-practice-exam-5-section-i-part-a\",\"timeRequired\":\"PT60M\",\"numberOfQuestions\":23,\"isAccessibleForFree\":true,\"about\":{\"@type\":\"Thing\",\"name\":\"Full Practice Exam 5\"}}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://theschoolofmathematics.com\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Quizzes\",\"item\":\"https://theschoolofmathematics.com/quiz\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Section I Part A\",\"item\":\"https://theschoolofmathematics.com/quiz/ap-calculus-bc-full-practice-exam-5-section-i-part-a\"}]}"}}],["$","div",null,{"className":"bg-gradient-to-b from-white via-gray-50 to-white","children":["$","div",null,{"className":"container mx-auto px-4 py-8 md:py-12","children":[["$","div",null,{"className":"mb-12","children":["$undefined",["$","div",null,{"className":"mb-8","children":["$","h1",null,{"className":"text-4xl md:text-5xl lg:text-6xl font-bold text-gray-900 mb-4","children":"Section I Part A"}]}],["$","div",null,{"className":"bg-white rounded-2xl shadow-lg p-6 md:p-8 border border-gray-200","children":[["$","div",null,{"className":"text-lg md:text-xl text-gray-700 mb-8 leading-relaxed prose prose-lg max-w-none","dangerouslySetInnerHTML":{"__html":"

Multiple choice non-calculator section

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AP Calculus BC Full Practice Exam 5 Section I Part A

"},"excerpt":{"rendered":"

Multiple choice non-calculator section

"},"featured_media":0,"featured_media_url":"$undefined","acf":{"duree_estimee":60,"niveau_difficulte":"$undefined","categorie":"Full Practice Exam 5","nombre_questions":30,"score_minimum":70,"ordre_questions":"Fixe","questions":[{"id":"cmpz9r8fn07cvjglle4o66445","texte_question":"

What is the instantaneous rate of change for $$f(x)=\\frac{x^3-1}{x-1}$$ at $$x=2$$?

","type_question":"QCM","explication":"

Factor: $$x^3-1=(x-1)(x^2+x+1)$$ So $$f(x)=x^2+x+1$$ Differentiate: $$f'(x)=2x+1$$ Evaluate at $$x=2$$: $$f'(2)=5$$ Final answer: $$5$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$3$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$4$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$5$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"

$$6$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8fq07d1jgllwy1nbe8i","texte_question":"

Evaluate $$\\int_1^4 \\frac{2}{x^2}\\,dx$$

","type_question":"QCM","explication":"

Rewrite: $$2x^{-2}$$ Integrate: $$\\int 2x^{-2}dx=-2x^{-1}$$ Evaluate: $$[-2/x]_1^4=-\\frac12+2=\\frac32$$ Final answer: $$\\frac32$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$\\frac12$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac34$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

$$\\frac32$$

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8fu07d7jgllwu8i59vt","texte_question":"

What is the slope of the curve defined by $$x^2+xy+y^2-3x+2y=0$$ at the point $$(1,1)$$?

","type_question":"QCM","explication":"

Differentiate implicitly: $$2x+y+xy'+2yy'-3+2y'=0$$ Group terms: $$(x+2y+2)y'=3-2x-y$$ Solve: $$y'=\\frac{3-2x-y}{x+2y+2}$$ Evaluate at $$(1,1)$$: $$y'=\\frac{3-2-1}{1+2+2}=0$$ Final answer: $$0$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$-1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$0$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

$$2$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8fy07ddjgllaqi628qh","texte_question":"

Evaluate $$\\int_0^1 \\sqrt{x}(x^2-2x+1)\\,dx$$

","type_question":"QCM","explication":"

Rewrite: $$x^{1/2}(x-1)^2$$ Expand: $$x^{5/2}-2x^{3/2}+x^{1/2}$$ Integrate: $$\\int_0^1 x^{5/2}dx=\\frac{2}{7}$$ $$\\int_0^1 2x^{3/2}dx=\\frac{4}{5}$$ $$\\int_0^1 x^{1/2}dx=\\frac{2}{3}$$ So $$\\frac{2}{7}-\\frac{4}{5}+\\frac{2}{3}$$ With common denominator $$105$$: $$\\frac{30-84+70}{105}=\\frac{16}{105}$$ Final answer: $$\\frac{16}{105}$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$\\frac{2}{7}$$

","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac{8}{15}$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac{16}{105}$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"

$$\\frac49$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8g107djjglltluyd748","texte_question":"

The radius of a sphere is increasing at a rate of $$3$$ inches per minute. At what rate is the volume increasing when the surface area is $$16\\pi$$?

","type_question":"QCM","explication":"

Surface area: $$4\\pi r^2=16\\pi$$ $$r^2=4$$ $$r=2$$ Volume: $$V=\\frac{4}{3}\\pi r^3$$ Differentiate: $$\\frac{dV}{dt}=4\\pi r^2\\frac{dr}{dt}$$ Substitute: $$\\frac{dV}{dt}=4\\pi(2^2)(3)=48\\pi$$ Final answer: $$48\\pi$$ Note: the source choices omitted $$\\pi$$, so the keyed choice below follows the source format.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$12$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$24$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$36$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

$$48$$

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8g707dpjgll2w45tmlg","texte_question":"

Evaluate $$\\int_0^{\\pi/2}\\sin x\\,e^{\\cos x}dx$$

","type_question":"QCM","explication":"

Let $$u=\\cos x$$ Then $$du=-\\sin x\\,dx$$ So $$\\int_0^{\\pi/2}\\sin x\\,e^{\\cos x}dx =-\\int_1^0 e^u\\,du =\\int_0^1 e^u\\,du$$ $$=e-1$$ Final answer: $$e-1$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$e-1$$

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"},{"texte":"

$$e$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$e^2-1$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$1-e$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8ga07dvjglll10z3o6q","texte_question":"

What is the average rate of change of $$f(x)=x^3-2x^2+x+2$$ over $$[0,3]$$?

","type_question":"QCM","explication":"

Average rate of change: $$\\frac{f(3)-f(0)}{3-0}$$ Compute: $$f(3)=27-18+3+2=14$$ $$f(0)=2$$ $$\\frac{14-2}{3}=4$$ Final answer: $$4$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$4$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$5$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$6$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$7$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8ge07e1jgllbbp63fs2","texte_question":"

If the graph of $$f''(x)$$ is a line with slope $$4$$, then $$f$$ could be which type of function?

","type_question":"QCM","explication":"

If $$f''(x)$$ is linear, then $$f''(x)$$ is quadratic, $$f''(x)$$ is cubic, and $$f(x)$$ is quartic. Final answer: quartic

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"linear","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"quadratic","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"cubic","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"quartic","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gh07e7jgllhbll9ibq","texte_question":"

Let $$f(x)=\\begin{cases}\\sqrt{x+2},&x\\ge-2\\\\x+4,&x<-2\\end{cases}$$. What is the average value of $$f$$ over $$[-4,2]$$?

","type_question":"QCM","explication":"

Average value: $$\\frac{1}{2-(-4)}\\int_{-4}^{2}f(x)\\,dx=\\frac16\\int_{-4}^{2}f(x)\\,dx$$ Split the integral: $$\\int_{-4}^{-2}(x+4)\\,dx+\\int_{-2}^{2}\\sqrt{x+2}\\,dx$$ First part: $$\\int_{-4}^{-2}(x+4)\\,dx=2$$ Second part: Let $$u=x+2$$, then from $$0$$ to $$4$$: $$\\int_0^4 \\sqrt{u}\\,du=\\left[\\frac{2}{3}u^{3/2}\\right]_0^4=\\frac{16}{3}$$ Total: $$2+\\frac{16}{3}=\\frac{22}{3}$$ Average value: $$\\frac16\\cdot\\frac{22}{3}=\\frac{11}{9}$$ Final answer: $$\\frac{11}{9}$$ Note: the source answer choices did not include the correct value.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$\\frac{5}{3}$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac{7}{3}$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$3$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac{11}{9}$$","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gk07edjgll08ary7hg","texte_question":"

Let $$f$$ be continuous on $$[1,9]$$. If $$f(1)=2$$ and $$f(9)=10$$, then the Mean Value Theorem guarantees that

","type_question":"QCM","explication":"

By the Mean Value Theorem, $$f''(c)=\\frac{f(9)-f(1)}{9-1}=\\frac{10-2}{8}=1$$ for at least one $$c$$ in $$(1,9)$$. Final answer: $$f''(c)=1$$ for at least one $$c$$ between $$1$$ and $$9$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$f(5)=6$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$f''(5)=1$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"$$$f''(c)=1$$ for at least one $$c$$ between $$1$$ and $$9$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$f(c)=0$$ for at least one $$c$$ between $$1$$ and $$9$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gn07ejjglltonza8eq","texte_question":"

Evaluate $$\\frac{d}{dx}\\int_0^{x^2}e^t\\,dt$$.

","type_question":"QCM","explication":"

By the Fundamental Theorem of Calculus and the chain rule, $$\\frac{d}{dx}\\int_0^{x^2}e^t\\,dt=e^{x^2}\\cdot\\frac{d}{dx}(x^2)=2xe^{x^2}$$ Final answer: $$2xe^{x^2}$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$e^{x^2}$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$2xe^{x^2}$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$e^x$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$2xe^x$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gq07epjgll7pygsns4","texte_question":"

Let $$f(x)=e^{2x}$$. If the rate of change of $$f$$ at $$x=c$$ is twice its rate of change at $$x=1$$, then $$c=$$?

","type_question":"QCM","explication":"

Derivative: $$f'(x)=2e^{2x}$$ Condition: $$2e^{2c}=2(2e^2)$$ $$e^{2c}=2e^2$$ Take ln: $$2c=2+\\ln 2$$ $$c=1+\\frac12\\ln 2\\approx1.346$$ Closest choice: $$\\frac32$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac32$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$3$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gt07evjglljyng52sc","texte_question":"

Let $$f$$ be differentiable on $$[0,8]$$ such that $$f(0)=1$$ and $$f(8)=5$$. If there are exactly two solutions to $$f(x)=3$$ over $$(0,8)$$, which must be true?

","type_question":"QCM","explication":"

Since the function crosses the value twice, Rolle's theorem implies at least one point where slope is zero. Final answer: derivative zero somewhere

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$f'(c)=0$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"local maximum","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$f''(c)=0$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"strictly increasing","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gw07f1jglly81uitgd","texte_question":"

The normal line to the curve $$y=\\sqrt{9-x^2}$$ at $$(0,3)$$ has slope

","type_question":"QCM","explication":"

Differentiate: $$y'=-\\frac{x}{\\sqrt{9-x^2}}$$ At $$x=0$$: $$y'=0$$ Tangent slope $$0$$ Normal slope undefined Final answer: undefined

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$0$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$-1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

undefined

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8gz07f7jgllzxl90ho4","texte_question":"

What are all values for $$k$$ such that $$\\int_{-1}^{k}x^2dx=1$$?

","type_question":"QCM","explication":"

Integrate: $$\\frac{x^3}{3}$$ Solve: $$\\frac{k^3+1}{3}=1$$ $$k^3=2$$ $$k=\\sqrt[3]{2}$$ Closest: $$2$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$0$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$-1\\ and\\ 1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"}]},{"id":"cmpz9r8hi07fdjgll014ce93q","texte_question":"

If the rate of change of $$y$$ is directly proportional to $$y$$, then it is possible that

","type_question":"QCM","explication":"

General solution: $$y=Ce^{kt}$$ Final answer: exponential

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$y=3e^{2t}$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$y=2t^2$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$y=\\ln(t+1)$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"$$$y=\\sqrt t$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8hr07fjjgllytqyhftz","texte_question":"

The graph of $$y=2x^3-3x^2+1$$ is decreasing for which interval?

","type_question":"QCM","explication":"

Derivative: $$6x^2-6x=6x(x-1)$$ Negative between: $$0

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$(-\\infty,0)$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$(0,1)$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$(1,\\infty)$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$(-\\infty,\\infty)$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8hz07fpjgll2i06iqub","texte_question":"

Determine the value for $$c$$ on $$[1,4]$$ that satisfies the mean value theorem for $$f(x)=x^2-2x$$

","type_question":"QCM","explication":"

Average slope: $$\\frac{f(4)-f(1)}{3}$$ Compute: $$\\frac{8-(-1)}{3}=3$$ Derivative: $$2x-2=3$$ $$x=\\frac52$$ Closest: $$3$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$3$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$4$$","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmpz9r8i807fvjglllcuvzynu","texte_question":"$18","type_question":"QCM","explication":"

Check each statement.

$$A)$$ This does not have to be true. A continuous function can pass through the points $$\\left(0,2\\right),\\left(1,5\\right),\\left(2,6\\right)$$ without being increasing on the entire interval.

$$B)$$ This does not have to be true. The three values alone do not force the graph to be concave down.

$$C)$$ This does not have to be true. The derivative at $$x=1$$ could be positive, zero, or negative depending on the shape of the function.

$$D)$$ The average rate of change of $$f$$ over $$[0,2]$$ is:

$$\\frac{f\\left(2\\right)-f\\left(0\\right)}{2-0}$$

Substitute the values:

$$\\frac{6-2}{2}$$

$$=\\frac{4}{2}$$

$$=2$$

So, this statement must be true.

Final answer:

The average rate of change of $$f$$ over $$[0,2]$$ is $$2$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$f$$ must be increasing on $$[0,2]$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$f$$ must be concave down on $$\\left(0,2\\right)$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$f'\\left(1\\right)$$>$$0$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

The average rate of change of $$f$$ over $$[0,2]$$ is $$2$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmpz9r8ik07g1jglljyt6h9iy","texte_question":"$19","type_question":"QCM","explication":"$1a","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$1$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$2$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$0$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

$$3$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"

$$4$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$3$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"}]},{"id":"cmpz9r8iq07gbjgll5gaed8pu","texte_question":"The function $$f$$ is continuous on the closed interval $$[0,2]$$. It is given that $$f(0)=0$$ and $$f(2)=4$$. If $$f'(x)>0$$ for all $$x$$ on $$[0,2]$$ and $$f''(x)<0$$ for all $$x$$ on $$(0,2)$$, then $$f(1)$$ could be","type_question":"QCM","explication":"Since $$f'(x)>0$$ on $$[0,2]$$, the function is increasing. Since $$f''(x)<0$$ on $$(0,2)$$, the function is concave down. A concave down function lies above its secant line, so at the midpoint,\r\n\r\n$$f(1)>\\frac{f(0)+f(2)}{2}=\\frac{0+4}{2}=2$$\r\n\r\nAmong the choices, the only possible value is\r\n\r\n$$3$$\r\n\r\nFinal answer: $$3$$","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$0$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$1$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$3$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"}]},{"id":"cmpz9r8iu07ghjgllzzx6smyl","texte_question":"The water level in a cylindrical tank is rising at a rate of $$2$$ inches per minute. If the radius of the tank is $$6$$ inches, what is the rate that water is entering the tank, in cubic inches per minute, when the volume is $$144\\pi$$ cubic inches?","type_question":"QCM","explication":"For a cylinder,\r\n\r\n$$V=\\pi r^2 h$$\r\n\r\nSince $$r=6$$ is constant,\r\n\r\n$$V=36\\pi h$$\r\n\r\nDifferentiate with respect to time:\r\n\r\n$$\\frac{dV}{dt}=36\\pi\\frac{dh}{dt}$$\r\n\r\nGiven\r\n\r\n$$\\frac{dh}{dt}=2$$\r\n\r\nso\r\n\r\n$$\\frac{dV}{dt}=36\\pi(2)=72\\pi$$\r\n\r\nThe given volume $$144\\pi$$ is not needed because the rate is constant for fixed radius.\r\n\r\nFinal answer: $$72\\pi$$","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$24\\pi$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$48\\pi$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$72\\pi$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$96\\pi$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"}]},{"id":"cmpz9r8iy07gnjgllpls154do","texte_question":"If $$f(x)=\\arctan(x^3)$$, then $$f'(1)=$$","type_question":"QCM","explication":"Use the chain rule:\r\n\r\n$$f'(x)=\\frac{1}{1+(x^3)^2}\\cdot 3x^2=\\frac{3x^2}{1+x^6}$$\r\n\r\nEvaluate at $$x=1$$:\r\n\r\n$$f'(1)=\\frac{3}{1+1}=\\frac{3}{2}$$\r\n\r\nFinal answer: $$\\frac{3}{2}$$","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$\\frac12$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac32$$","correcte":true,"explication":"Correct","imageUrl":"$undefined"},{"texte":"$$$\\frac34$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"},{"texte":"$$$\\frac14$$","correcte":false,"explication":"Incorrect","imageUrl":"$undefined"}]}]},"categories":["full-practice-exam-5"],"date":"2026-06-04T09:04:31.855Z","modified":"2026-06-04T09:04:31.855Z"}}],["$","div",null,{"className":"container mx-auto px-4 pb-12","children":["$","$L1b",null,{"targetType":"quiz","targetSlug":"ap-calculus-bc-full-practice-exam-5-section-i-part-a"}]}]]}]}]]

Section I Part A

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