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

## Example Questions

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

Evaluate the definite integral of the algebraic function.

integral (x+ √(x))dx from 0 to 1

11/12

10/12

1

0

5/12

11/12

Explanation:

Step 1: Rewrite the problem.

integral (x3+x1/2) 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

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

Evaluate the integral.

Integral from 1 to 2 of (1/x3) dx

0

3/8

1/2

–5/8

–3/8

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)

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

Evaluate the following indefinite integral.       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 , so in the answer we have .  Next add a constant that would be lost in the differentiation.  To check your work, differentiate your answer and see that it matches "4".

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

Evaluate the following indefinite integral.       Explanation:

Use the inverse Power Rule to evaluate the integral.  We know that for .  We see that this rule tells us to increase the power of by 1 and multiply by .  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       Explanation:

Use the inverse Power Rule to evaluate the integral.  We know that for .  We see that this rule tells us to increase the power of by 1 and multiply by .  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       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 for .  We see that this rule tells us to increase the power of by 1 and multiply by . Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       Explanation:

Use the inverse Power Rule to evaluate the integral.  We know that for .  We see that this rule tells us to increase the power of by 1 and multiply by .  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       Explanation:

Use the inverse Power Rule to evaluate the integral.  We know that for .  We see that this rule tells us to increase the power of by 1 and multiply by Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       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 for .  We see that this rule tells us to increase the power of by 1 and multiply by .  Next always add your constant of integration that would be lost in the differentiation.  Take the derivative of your answer to check your work.

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

Evaluate the following indefinite integral.       Use the inverse Power Rule to evaluate the integral.  We know that for . But, in this case, IS equal to so a special condition of the rule applies.  We must instead use .  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. 