### All AP Calculus AB Resources

## Example Questions

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

Evaluate the definite integral of the algebraic function.

integral (x^{3 }+ √(x))dx from 0 to 1

**Possible Answers:**

10/12

5/12

11/12

0

1

**Correct answer:**

11/12

Step 1: Rewrite the problem.

integral (x^{3}+x^{1/2})^{ }dx from 0 to 1

Step 2: Integrate

x^{4}/4 + 2x^{(2/3)}/3 from 0 to 1

Step 3: Plug in bounds and solve.

[1^{4}/4 + 2(1)^{(2/3)}/3] – [0^{4}/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/x^{3}) dx

**Possible Answers:**

0

1/2

–5/8

3/8

–3/8

**Correct answer:**

3/8

Integral from 1 to 2 of (1/x^{3}) 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 #3 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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

Evaluate the following indefinite integral.

**Possible Answers:**

**Correct answer:**

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.

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