### All AP Calculus AB Resources

## Example Questions

### Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If f(1) = 12, *f*' is continuous, and the integral from 1 to 4 of f'(x)dx = 16, what is the value of f(4)?

**Possible Answers:**

16

28

27

12

4

**Correct answer:**

28

You are provided f(1) and are told to find the value of f(4). By the FTC, the following follows:

(integral from 1 to 4 of f'(x)dx) + f(1) = f(4)

16 + 12 = 28

### Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Find the limit.

lim as n approaches infiniti of ((4n^{3}) – 6n)/((n^{3}) – 2n^{2 }+ 6)

**Possible Answers:**

–6

1

nonexistent

0

4

**Correct answer:**

4

lim as n approaches infiniti of ((4n^{3}) – 6n)/((n^{3}) – 2n^{2 }+ 6)

Use L'Hopitals rule to find the limit.

lim as n approaches infiniti of ((4n^{3}) – 6n)/((n^{3}) – 2n^{2 }+ 6)

lim as n approaches infiniti of ((12n^{2}) – 6)/((3n^{2}) – 4n + 6)

lim as n approaches infiniti of 24n/(6n – 4)

lim as n approaches infiniti of 24/6

The limit approaches 4.

### Example Question #1 : Interpretations And Properties Of Definite Integrals

If a particle's movement is represented by , then when is the velocity equal to zero?

**Possible Answers:**

**Correct answer:**

The answer is seconds.

now set because that is what the question is asking for.

seconds

### Example Question #2 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

A particle's movement is represented by

At what time is the velocity at it's greatest?

**Possible Answers:**

**Correct answer:**

The answer is at 6 seconds.

We can see that this equation will look like a upside down parabola so we know there will be only one maximum.

Now we set to find the local maximum.

seconds

### Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the domain of ?

**Possible Answers:**

**Correct answer:**

because the denominator cannot be zero and square roots cannot be taken of negative numbers

If ,

then at , what is 's instantaneous rate of change?

**Possible Answers:**

**Correct answer:**

The answer is 8.

### Example Question #1 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function in Cartestian coordinates?

**Possible Answers:**

**Correct answer:**

First, mulitply both sides by r.

Then, use the identities and .

The answer is .

### Example Question #1 : Interpretations And Properties Of Definite Integrals

What is the average value of the function from to ?

**Possible Answers:**

**Correct answer:**

The average function value is given by the following formula:

, evaluated from to .

### Example Question #1 : Interpretations And Properties Of Definite Integrals

If

then find .

**Possible Answers:**

**Correct answer:**

We see the answer is 0 after we do the quotient rule.

### Example Question #1 : Interpretations And Properties Of Definite Integrals

If , then which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

According to the Fundamental Theorem of Calculus, if we take the derivative of the integral of a function, the result is the original function. This is because differentiation and integration are inverse operations.

For example, if , where is a constant, then .

We will apply the same principle to this problem. Because the integral is evaluated from 0 to , we must apply the chain rule.

The answer is .