AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

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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}{(t-6)}

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

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

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

y'=\frac{(t+2)-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}}

Example Question #1 : Numerical Approximations To Definite Integrals

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

Possible Answers:

-2

-\infty

\infty

0

1

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.

Example Question #6 : Calculus 3

Evaluate:

Possible Answers:

cannot be determined

Correct answer:

Explanation:

First, we can write out the first few terms of the sequence , where  ranges from 1 to 3.

Notice that each term , is found by multiplying the previous term by . Therefore, this sequence is a geometric sequence with a common ratio of . We can find the sum of the terms in an infinite geometric sequence, provided that , where  is the common ratio between the terms. Because  in this problem,  is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

The answer is .

 

Example Question #2 : Numerical Approximations To Definite Integrals

If  then find .

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

Example Question #1 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at  on graph 

Possible Answers:

Correct answer:

Explanation:

The answer is 

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

f'(x)=2x+2

 

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

 

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:

Correct answer:

Explanation:

The answer is 

 

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.)

 

Example Question #6 : Numerical Approximations To Definite Integrals

If  then 

Possible Answers:

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

\frac{1}{2}

Correct answer:

Explanation:

The answer is .

 

 

We know that

so,

 

Example Question #11 : Integrals

Possible Answers:

does not exist

1

1/2

1/4

2

Correct answer:

2

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.

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

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.

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

The answer is 2.

Example Question #11 : Integrals

Differentiate

Possible Answers:

3sec^{2}x-5cosx

3csc^{2}x-5cosx

Correct answer:

Explanation:

The answer is 

We simply differentiate by parts, remembering our trig rules.

Example Question #12 : Numerical Approximations To Definite Integrals

If   then find .

Possible Answers:

Correct answer:

Explanation:

The answer is 10.

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