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AP Calculus AB : Asymptotic and Unbounded Behavior

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Example Questions

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Example Question #46 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int\frac{1}{x^2}dx$

Possible Answers:

$-x^{-1}+C$

3x^3+C

(1/2)ln(x)+C

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

2ln(x)+C

Correct answer:

$-x^{-1}+C$

Explanation:

We 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. Keep in mind that $\frac{1}{x^{2}}$ is the same as $x^{-2}$ . As a note, we cannot forget the constant of integration which would be lost during the differentiation.

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Example Question #47 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int(3x^7+4x^4)dx$

Possible Answers:

$\frac{1}{8}x^8+\frac{1}{5}x^5+C$

$\frac{3}{8}x^8+\frac{4}{5}x^5+C$

$\frac{8}{3}x^8+\frac{5}{4}x^5+C$

8x^8+5x^5+C

3x^6+4x^3+C

Correct answer:

$\frac{3}{8}x^8+\frac{4}{5}x^5+C$

Explanation:

First, we remember that the integral of a sum is the same as the sum of the integrals, so we can split the sum into seperate integrals and solve them individually. We then evaluate each 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 #251 : Ap Calculus Ab

Evaluate the following indefinite integral.

$\int\frac{1}{2}x^2dx$

Possible Answers:

3x^3+C

$\frac{1}{6}x^3+C$

$\frac{1}{2}x^3+C$

6x^3+C

$\frac{1}{2}x^3+C$

Correct answer:

$\frac{1}{6}x^3+C$

Explanation:

First, we know that we can pull the constant $\frac{1}{2}$ 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 #21 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int \frac{3}{5}x^3dx$

Possible Answers:

$\frac{3}{20}x^4+C$

$\frac{9}{5}x^4+C$

$\frac{20}{3}x^4+C$

$\frac{1}{20}x^4+C$

$\frac{3}{15}x^4+C$

Correct answer:

$\frac{3}{20}x^4+C$

Explanation:

First, we know that we can pull the constant $\frac{3}{5}$ 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 #22 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int\sqrt{x}dx$

Possible Answers:

$3x^{-1/2}+C$

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

ln(x)+C

e^x+C

$\frac{1}{3}x^{\frac{1}{2}}+C$

Correct answer:

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

Explanation:

We evaluate the integral according to this equation:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$ . Keep in mind that $\sqrt{x}$ is the same as $x^{1/2}$ . 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 #23 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int2e^{2x}dx$

Possible Answers:

$e^{2x}+C$

e^x+C

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

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

2e^x+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 #24 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

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

Possible Answers:

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

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

$2x^{11}+C$

$2x^{13}+C$

$72x^{13}+C$

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 #25 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int e^xdx$

Possible Answers:

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

1/ln(x)+C

$e^{-x}+C$

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 #26 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int 3y^2 dy$

Possible Answers:

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

6y^3+C

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

y^3+C

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

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 #27 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

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

Possible Answers:

sec(x)+C

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

-tan(x)+C

tan(x)+C

tan(x)sec(x)

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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AP Calculus AB : Asymptotic and Unbounded Behavior

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #46 : Functions, Graphs, And Limits

Example Question #47 : Functions, Graphs, And Limits

Example Question #251 : Ap Calculus Ab

Example Question #21 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #22 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #23 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #24 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #25 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #26 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #27 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

All AP Calculus AB Resources

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