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AP Calculus AB : Comparing relative magnitudes of functions and their rates of change

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Example Questions

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Example Question #31 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

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

Possible Answers:

$\frac{1}{3}ln|x|+C$

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

3ln|x|+C

ln|x|+C

3e^x+C

Correct answer:

3ln|x|+C

Explanation:

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . But, in this case, IS equal to so a special condition of the rule applies. We must instead use $\int x^{-1}dx=ln|x|+C$ . Pull the constant "3" out front and evaluate accordingly. Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

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Example Question #32 : Functions, Graphs, And Limits

Evaluate the following definite integral.

$\int_{0}^{2}(x^2+3) dx$

Possible Answers:

3x^3+6x

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

26/3

5/3

Correct answer:

26/3

Explanation:

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 0 to 2. First, we use our inverse power rule to find the antiderivative. So, we have that $\int x^ndx=\frac{x^{n+1}}{n+1}$ . Once you find the antiderivative, we must remember that int_a^bf(z)dz=F(b)-F(a). where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have $(\frac{1}{3}x^3+3x)|_0^2=[\frac{1}{3}2^3+3(2)]-[\frac{1}{3}0^3+3(0)]=26/3$ .

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Example Question #41 : Functions, Graphs, And Limits

Evaluate the following definite integral.

$\int_{1}^{3}6xdx$

Possible Answers:

3x^2+C

6x^2

Correct answer:

Explanation:

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 3. First, we use our inverse power rule to find the antiderivative. So, we have that $\int x^ndx=\frac{x^{n+1}}{n+1}$ . Once you find the antiderivative, we must remember that int_a^bf(z)dz=F(b)-F(a). where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have (3x^2)|_1^3=3(3^2)-3(1^2)=24 .

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Example Question #42 : Functions, Graphs, And Limits

Evaluate the following definite integral.

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

Possible Answers:

$ln\ 4$

$ln\ 1$

2x^2

$ln\ 3$

Correct answer:

$ln\ 4$

Explanation:

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 4. First, we use our inverse power rule to find the antiderivative. So since is to the power of , we have that $\int x^{-1}=ln|x|$ . Once you find the antiderivative, we must remember that int_a^bf(z)dz=F(b)-F(a). where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have ln(x)|_1^4=ln(4)-ln(1)=ln(4) because we know that $ln\ 1=0$ .

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Example Question #43 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int 4xdx$

Possible Answers:

x^2+C

8x^2+C

2x^2+C

(1/2)x^2

Correct answer:

2x^2+C

Explanation:

First, we know that we can pull the constant "4" 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 #44 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

$\int (6x+3x^2+4)dx$

Possible Answers:

x^2+x^3+4x+C

$\frac{1}{3}x^2+x^3+\frac{1}{4}x+C$

3x^2+x^3+4x

6+3x+C

3x^2+x^3+4x+C

Correct answer:

3x^2+x^3+4x+C

Explanation:

First, we know that the integral of a sum is the same as the sum of the integrals, so if needed, we can split the three integrals up and evaluate them seperately. 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 #45 : Functions, Graphs, And Limits

Evalulate the following indefinite integral.

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

Possible Answers:

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

-1/x^2+C

$1/2 x^{-2}$

e^x

ln(x)+C

Correct answer:

ln(x)+C

Explanation:

Normally, we would evalute the indefinite integral according to the following equation:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C, n\ne1$ . However, in this case, n=-1 . Now we use our other rule that states the integral of $\frac{1}{x}$ is equal to $\ln (x)$ plus a constant. 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 #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 #48 : Functions, Graphs, And Limits

Evaluate the following indefinite integral.

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

Possible Answers:

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

6x^3+C

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

3x^3+C

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

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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AP Calculus AB : Comparing relative magnitudes of functions and their rates of change

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #31 : Functions, Graphs, And Limits

Example Question #32 : Functions, Graphs, And Limits

Example Question #41 : Functions, Graphs, And Limits

Example Question #42 : Functions, Graphs, And Limits

Example Question #43 : Functions, Graphs, And Limits

Example Question #44 : Functions, Graphs, And Limits

Example Question #45 : Functions, Graphs, And Limits

Example Question #46 : Functions, Graphs, And Limits

Example Question #47 : Functions, Graphs, And Limits

Example Question #48 : Functions, Graphs, And Limits

All AP Calculus AB Resources

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