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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #51 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int2e^{2x}dx$

Possible Answers:

e^x+C

$\frac{1}{2}e^x+C$

2e^x+C

$e^{2x}+C$

$\frac{1}{2}e^{2x}+C$

Correct answer:

$e^{2x}+C$

Explanation:

We know that the derivative of $e^{x}=e^{x}$ and the integral of $e^{x}=e^{x}$ . We must remember the chain rule and therefore keep the 2 in the exponent. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.

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Example Question #51 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int\frac{1}{6}x^{12}dx$

Possible Answers:

$72x^{13}+C$

$2x^{11}+C$

$\frac{6}{13}x^{13}+C$

$2x^{13}+C$

$\frac{1}{78}x^{13}+C$

Correct answer:

$\frac{1}{78}x^{13}+C$

Explanation:

First, we know that we can pull the constant $\frac{1}{6}$ out of the integral, and we then evaluate the integral according to this equation:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$ . From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.

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Example Question #41 : Asymptotic And Unbounded Behavior

Evaluate the following indefinite integral.

$\int e^xdx$

Possible Answers:

e^x+C

1/ln(x)+C

$\frac{1}{2}e^{-x}+C$

ln(x)+C

$e^{-x}+C$

Correct answer:

e^x+C

Explanation:

For this problem, we must simply remember that the integral of $e^{x}$ is $e^{x}$ , just like how the derivative of $e^{x}$ is $e^{x}$ . Just keep in mind that we need that constant of integration that would have been lost during differentiation.

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Example Question #54 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int 3y^2 dy$

Possible Answers:

6y^3+C

y^3+C

$\frac{1}{3}y^3+C$

$\frac{2}{3}y^3+C$

$\frac{3}{2}y^3+C$

Correct answer:

y^3+C

Explanation:

First, we know that we can pull the constant out of the integral, and we then evaluate the integral according to this equation:

$\int y^n dy=\frac{y^{n+1}}{n+1}+C, n\ne1$ . From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.

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Example Question #55 : Functions, Graphs, And Limits

$\int \sec x\tan x\ \mathrm{d}x=$

Possible Answers:

-tan(x)+C

sec(x)+C

tan(x)+C

tan(x)sec(x)

$\frac{1}{2}sec^2xtanx +C$

Correct answer:

sec(x)+C

Explanation:

The answer is sec(x)+C . The definition of the derivative of sec(x) is sec(x)tan(x) . Remember to add the to undefined integrals.

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Example Question #51 : Asymptotic And Unbounded Behavior

Evaluate the integral:

$\int (x+2) dx$

Possible Answers:

$\frac{x^2}{2}+2x+C$

x^2+2

x^2+2x

$\frac{x^2}{2}$

Correct answer:

$\frac{x^2}{2}+2x+C$

Explanation:

In order to find the antiderivative, add 1 to the exponent and divide by the exponent.

$\frac{x^{1+1}}{2}+\frac{2x^{0+1}}{1}=\frac{x^2}{2}+2x$

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Example Question #52 : Asymptotic And Unbounded Behavior

Evaluate:

$\int \frac{1}{x^4} dx$

Possible Answers:

$\frac{-1}{5x^5}$

$\frac{-5}{x^5}$

$\frac{-3}{x^3}$

$\frac{-1}{3x^3}$

Correct answer:

$\frac{-1}{3x^3}$

Explanation:

$\int x^{-4}= \frac{x^{-4+1}}{-3}= \frac{x^{-3}}{-3}=\frac{-1}{3x^3}$

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Example Question #58 : Functions, Graphs, And Limits

Evaluate:

$\int (sin^2x*cosx) dx$

Possible Answers:

$\frac{sin^3x}{3}-\frac{cos^2x}{2}$

cos^3x

$\frac{cos^3x}{3}$

$(\frac{1}{3})sin^3x+C$

Correct answer:

$(\frac{1}{3})sin^3x+C$

Explanation:

You should first know that the derivative of sin(x)=cos(x) .

Therefore, looking at the equation you can see that the antiderivative should involve something close to: $\square sin^3x+C$

Now to figure out what value represents the square take the derivative of sin^3x and set it equal to what the original integral contained.

3sin^2x*cosx = sin^2x*cosx

Since the derivative of sin^3x contains a 3 that the integral does not show, we know that the square is equal to 1/3 . Thus, the answer is $\frac{1}{3}sin^3x+C$ .

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Example Question #59 : Functions, Graphs, And Limits

Evaluate:

$\int cos(2x)dx$

Possible Answers:

-2sin(2x)

$(\frac{1}{2})sin(2x)+C$

$-\frac{1}{2}sin(2x)$

2sin(2x)

2cos(2x)

Correct answer:

$(\frac{1}{2})sin(2x)+C$

Explanation:

The antiderivative of cos(x) = sin(x) . The derivative of sin(2x) = 2cos(2x) . However, since there is no 2 in the original integral, we must divide sin(2x) by 2. Therefore, the answer is (1/2)sin(2x)+C.

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Example Question #31 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the integral:

$\int (x^3-2x^2+5x-1) dx$

Possible Answers:

${4x^4}-6x^3+5x+C$

3x^2-4x+5+C

$\frac{x^4}{4}-\frac{2x^3}{3}+\frac{5x^2}{2}-x+C$

x^4-6x^3+10x^2-x +C

Correct answer:

$\frac{x^4}{4}-\frac{2x^3}{3}+\frac{5x^2}{2}-x+C$

Explanation:

When taking the antiderivative add one to the exponent and then divide by the exponent.

$\frac{x^{3+1}}{4}-\frac{2*x^{2+1}}{3}+\frac{5x^{1+1}}{2}-\frac{1*x^{0+1}}{1}$

$\frac{x^4}{4}-\frac{2x^3}{3}+\frac{5x^2}{2}-x+C$

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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #51 : Functions, Graphs, And Limits

Example Question #51 : Functions, Graphs, And Limits

Example Question #41 : Asymptotic And Unbounded Behavior

Example Question #54 : Functions, Graphs, And Limits

Example Question #55 : Functions, Graphs, And Limits

Example Question #51 : Asymptotic And Unbounded Behavior

Example Question #52 : Asymptotic And Unbounded Behavior

Example Question #58 : Functions, Graphs, And Limits

Example Question #59 : Functions, Graphs, And Limits

Example Question #31 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

All AP Calculus AB Resources

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