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

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Example Question #21 : Functions, Graphs, And Limits

Evaluate the definite integral of the algebraic function.

integral (x³+ √(x))dx from 0 to 1

Possible Answers:

5/12

11/12

10/12

Correct answer:

11/12

Explanation:

Step 1: Rewrite the problem.

integral (x³+x^1/2)dx from 0 to 1

Step 2: Integrate

x⁴/4 + 2x^(2/3)/3 from 0 to 1

Step 3: Plug in bounds and solve.

[1⁴/4 + 2(1)^(2/3)/3] – [0⁴/4 + 2(0)^(2/3)/3] = (1/4) + (2/3) = (3/12) + (8/12) = 11/12

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Example Question #25 : Functions, Graphs, And Limits

Evaluate the integral.

Integral from 1 to 2 of (1/x³) dx

Possible Answers:

–3/8

3/8

–5/8

1/2

Correct answer:

3/8

Explanation:

Integral from 1 to 2 of (1/x³) dx

Integral from 1 to 2 of (x^-3) dx

Integrate the integral.

from 1 to 2 of (x^–2/-2)

(2^–2/–2) – (1^–2/–2) = (–1/8) – (–1/2)=(3/8)

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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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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 #21 : Functions, Graphs, And Limits

Example Question #25 : Functions, Graphs, And Limits

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

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