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AP Calculus AB : Integrals

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Example Questions

Example Question #1 : Numerical Approximations To Definite Integrals

Differentiate $y=\frac{t+2}{t^{2}-4t-12}$

Possible Answers:

$y'=\frac{1}{2t-4}$

$y'=\frac{-1}{(t-6)^{2}}$

$y'=\frac{(t+2)-1}{(t^{2}-4t-12)^{2}}$

$y'=\frac{1}{(t-6)}$

$y'=\frac{1}{(t^{2}-4t-12)^{2}}$

Correct answer:

$y'=\frac{-1}{(t-6)^{2}}$

Explanation:

We see the answer is $y'=\frac{-1}{(t-6)^{2}}$ after we simplify and use the quotient rule.

$y=\frac{t+2}{t^{2}-4t-12}$ we could use the quotient rule immediatly but it is easier if we simplify first.

$y=\frac{t+2}{(t+2)(t-6)}$

$y=\frac{1}{(t-6)}$

$y=(t-6)^{-1}$

$y'=-(t-6)^{-2}$

$y'=\frac{-1}{(t-6)^{2}}$

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Example Question #2 : Numerical Approximations To Definite Integrals

Find $\lim_{x\rightarrow \infty }\frac{-2x^3+x^2-2}{x^2+10}$

Possible Answers:

$0$

$\infty$

$-2$

$1$

$-\infty$

Correct answer:

$-\infty$

Explanation:

When taking limits to infinity, we usually only consider the highest exponents. In this case, the numerator has $-2x^3$ and the denominator has $x^2$ . Therefore, by cancellation, it becomes $-2x$ as approaches infinity. So the answer is $-\infty$ .

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Example Question #1 : Calculus 3

Evaluate:

$\sum_{n=1}^{\infty }4(\frac{-1}{2})^{n-1}$

Possible Answers:

$\infty$

$\frac{8}{3}$

cannot be determined

Correct answer:

$\frac{8}{3}$

Explanation:

First, we can write out the first few terms of the sequence $a_{n}=$ $4(\frac{-1}{2})^{n-1}$ , where ranges from 1 to 3.

$a_{1}=4(\frac{-1}{2})^{1-1}=4$

$a_{2}=4(\frac{-1}{2})^{2-1}=-2$

$a_{3}=4(\frac{-1}{2})^{3-1}=1$

Notice that each term $\left ( n>1 \right )$ , is found by multiplying the previous term by $-\frac{1}{2}$ . Therefore, this sequence is a geometric sequence with a common ratio of $-\frac{1}{2}$ . We can find the sum of the terms in an infinite geometric sequence, provided that |r|<1 , where is the common ratio between the terms. Because $r=-\frac{1}{2}$ in this problem, $\left | r \right |$ is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

$S=\frac{a_{1}}{1-r}=\frac{4}{1-(\frac{-1}{2})}=\frac{4}{3/2}=\frac{8}{3}$

The answer is $\frac{8}{3}$ .

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Example Question #3 : Numerical Approximations To Definite Integrals

If $f(x)=4, f'(x)=3, g(x)=2, \ and\ g'(x)=1$ then find h'(x) .

$h(x)=\frac{2f}{g}$

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The answer is 1.

$h(x)=\frac{2f}{g}$

$h'(x)=\frac{2(f'(x)g(x)-f(x)g'(x))}{g^{2}}$

$h'(x)=\frac{2(3*2-4*1)}{4} = 1$

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Example Question #4 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at (4,1) on graph f(x)=(x-2)(x+4)

Possible Answers:

y=7x+8

y=x+40

y=10x-39

y=x-11

y=3x-7

Correct answer:

y=10x-39

Explanation:

The answer is 10x-39

f(x)=(x-2)(x+4)

$f(x)=x^{2}+2x-8$

$f'(x)=2x+2$

f'(x) = 2(4)+2=10

(This is the slope. Now use the point-slope formula)

y-1=10(x-4)

y-1=10x-40

y=10x-39

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Example Question #5 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at (1,1) in

$f(x)=ax^{2}+bx+c$

Possible Answers:

y=2ax-b+1

y=2ax-bx-2a+1

y=2ab+b-1

y=2a+b

y=2ax+bx-2a-b+1

Correct answer:

y=2ax+bx-2a-b+1

Explanation:

The answer is y=2ax+bx-2a-b+1

$f(x)=ax^{2}+bx+c$

$f'(x)=2ax+b$

$f'(1)=2a(1)+b = 2a+b$

(This is the slope. Now use the point-slope formula.)

y-1=(2a+b)(x-1)

y-1=2ax+bx-2a-b

y=2ax+bx-2a-b+1

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Example Question #6 : Numerical Approximations To Definite Integrals

If $f(x)=\cos x$ then

$f^{'}\left ( \frac{\pi }{2} \right )=$

Possible Answers:

$\frac{1}{2}$

$\frac{2^{1/2}}{-2}$

Correct answer:

Explanation:

The answer is .

f(x)=cos(x)

f'(x)=-sin(x)

We know that

$\frac{\pi }{2}=90^{\circ}$

so,

$f'\left ( \frac{\pi }{2}\right )=-sin(90^{\circ})=-1$

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Example Question #7 : Numerical Approximations To Definite Integrals

$\lim_{x\rightarrow 0}\frac{2\sin(x^2)}{x\tan(x)}=$

Possible Answers:

1/2

1/4

does not exist

Correct answer:

Explanation:

When we let x = 0 in our original limit, we obtain the 0/0 indeterminate form. Therefore, we can apply L'Hospital's Rule, which requires that we take the derivative of the numerator and denominator separately.

$\lim_{x\rightarrow 0}\frac{2\sin(x^2)}{x\tan(x)}=\lim_{x\rightarrow 0}\frac{\frac{\mathrm{d} }{\mathrm{d} x}2\sin(x^2)}{\frac{\mathrm{d} }{\mathrm{d} x}x\tan(x)}$

Apply the Chain Rule in the numerator and the Product Rule in the denominator.

$\lim_{x\rightarrow 0}\frac{4x\cos(x^2)}{\tan(x)+x\sec^2(x)}$

If we again substitute x = 0, we still obtain the 0/0 indeterminate form. Thus, we can apply L'Hospital's Rule one more time.

$\lim_{x\rightarrow 0}\frac{4x\cos(x^2)}{\tan(x)+x\sec^2(x)}=\lim_{x\rightarrow 0}\frac{\frac{\mathrm{d} }{\mathrm{d} x}4x\cos(x^2)}{\frac{\mathrm{d} }{\mathrm{d} x}(\tan(x)+x\sec^2(x))}$

$=\lim_{x\rightarrow 0}\frac{4\cos(x^2)+4x(2x)(-\sin(x^2))}{\sec^2(x)+(\sec^2(x)+x(2\sec^2x\tan x))}$

If we now let x = 0, we can evaluate the limit.

$=\lim_{x\rightarrow 0}\frac{4\cos(x^2)+4x(2x)(-\sin(x^2))}{\sec^2(x)+(\sec^2(x)+x(2\sec^2x\tan x))}=\frac{4\cos(0)+0}{\sec^2(0)+\sec^2(0)+0}$

$=\frac{4}{1+1}=2$

The answer is 2.

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Example Question #11 : Numerical Approximations To Definite Integrals

Differentiate $y=3\csc x+5\sin x$

Possible Answers:

$\cot x(3\csc x-5)$

$-3\csc x\cot x+5\cos x$

$3csc^{2}x-5cosx$

$3sec^{2}x-5cosx$

$3\csc x\cot x-5\cos x$

Correct answer:

$-3\csc x\cot x+5\cos x$

Explanation:

The answer is $-3\csc x\cot x+5\cos x$

We simply differentiate by parts, remembering our trig rules.

$y=3\csc x+5\sin x$

$y^{'}= -3\csc x\cot x+5\cos x$

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Example Question #12 : Numerical Approximations To Definite Integrals

If $f(x)=1, f'(x)=2, g(x)=3, \ and\ g'(x)=4$ then find h'(x) .

h(x)=fg

Possible Answers:

Correct answer:

Explanation:

The answer is 10.

h(x)=fg

h'(x)=f'(x)g(x)+f(x)g'(x)

h'(x)=(2)(3)+(1)(4) =10

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AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Numerical Approximations To Definite Integrals

Example Question #2 : Numerical Approximations To Definite Integrals

Example Question #1 : Calculus 3

Example Question #3 : Numerical Approximations To Definite Integrals

Example Question #4 : Numerical Approximations To Definite Integrals

Example Question #5 : Numerical Approximations To Definite Integrals

Example Question #6 : Numerical Approximations To Definite Integrals

Example Question #7 : Numerical Approximations To Definite Integrals

Example Question #11 : Numerical Approximations To Definite Integrals

Example Question #12 : Numerical Approximations To Definite Integrals

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