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AP Calculus AB : Functions, Graphs, and Limits

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Example Questions

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Example Question #1 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int 4 dx$

Possible Answers:

4x^2

4x+C

x+C

Correct answer:

4x+C

Explanation:

In order to evaluate the indefinite integral, ask yourself, "what expression do I differentiate to get 4". Next, use the power rule and increase the power of by 1. To start, we have $4x^{0}$ , so in the answer we have $4x^{1}$ . Next add a constant that would be lost in the differentiation. To check your work, differentiate your answer and see that it matches "4".

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Example Question #2 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

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

Possible Answers:

x^4-x^3+3x+C

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

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

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

4x^4-3x^3+3x+C

Correct answer:

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

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 $n \ne -1$ . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ . 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 #3 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

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

Possible Answers:

$\frac{1}{7}x^7+\frac{1}{4}x^4+x+C$

$\frac{1}{2}x^8+\frac{1}{5}x^5+\frac{1}{2}x^2+C$

${2}x^8+{5}x^5+{2}x^2+C$

$\frac{1}{2}x^8+\frac{1}{5}x^4+\frac{1}{2}x^2$

x^8+x^5+x^2+C

Correct answer:

$\frac{1}{2}x^8+\frac{1}{5}x^5+\frac{1}{2}x^2+C$

Explanation:

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Example Question #4 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int 3y dy$

Possible Answers:

$\frac{1}{2}y^2+C$

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

3y+C

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

3y^2+C

Correct answer:

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

Explanation:

Use the inverse Power Rule to evaluate the integral. Firstly, constants can be taken out of integrals, so we pull the 3 out front. Next, according to the inverse power rule, we know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ . 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 #5 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

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

Possible Answers:

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

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

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

2x^2-4x^4+x+C

2x^2-2x^4+x+C

Correct answer:

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

Explanation:

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Example Question #6 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int (1-x)dx$

Possible Answers:

x-x^2+C

x+2x^2+C

x+x^2+C

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

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

Correct answer:

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

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 $n \ne -1$ . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ 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 #7 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

$\int \frac{1}{2}xdx$

Possible Answers:

4x^2+C

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

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

2x^2+C

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

Correct answer:

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

Explanation:

Use the inverse Power Rule to evaluate the integral. Firstly, constants can be taken out of the integral, so we pull the 1/2 out front and then complete the integration according to the rule. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ . 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 #8 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

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

Possible Answers:

1+C

ln|x|+C

e^x+C

x+C

Correct answer:

ln|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 $\int x^{-1}dx=ln|x|+C$ . 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 #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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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 #1 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #2 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #3 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #4 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #5 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #6 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #7 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #8 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Example Question #31 : Functions, Graphs, And Limits

Example Question #32 : Functions, Graphs, And Limits

All AP Calculus AB Resources

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