### All AP Calculus BC Resources

## Example Questions

### Example Question #1 : Fundamental Theorem Of Calculus With Definite Integrals

Find the result:

**Possible Answers:**

**Correct answer:**

Set . Then , and by the chain rule,

By the fundamental theorem of Calculus, the above can be rewritten as

### Example Question #1 : Fundamental Theorem Of Calculus

Evaluate :

**Possible Answers:**

**Correct answer:**

By the Fundamental Theorem of Calculus, we have that . Thus, .

### Example Question #2 : Fundamental Theorem Of Calculus

Evaluate when .

**Possible Answers:**

**Correct answer:**

Via the Fundamental Theorem of Calculus, we know that, given a function, .

Therefore .

### Example Question #3 : Fundamental Theorem Of Calculus

Evaluate when .

**Possible Answers:**

**Correct answer:**

Via the Fundamental Theorem of Calculus, we know that, given a function , . Therefore, .

### Example Question #2 : Fundamental Theorem Of Calculus With Definite Integrals

Suppose we have the function

What is the derivative, ?

**Possible Answers:**

**Correct answer:**

We can view the function as a function of , as so

where .

We can find the derivative of using the chain rule:

where can be found using the fundamental theorem of calculus:

So we get

### Example Question #9 : Fundamental Theorem Of Calculus

Given

, what is ?

**Possible Answers:**

None of the above.

**Correct answer:**

By the Fundamental Theorem of Calculus, for all functions that are continuously defined on the interval with in and for all functions defined by by , we know that .

Thus, for

,

.

Therefore,

### Example Question #10 : Fundamental Theorem Of Calculus

Given

, what is ?

**Possible Answers:**

None of the above.

**Correct answer:**

By the Fundamental Theorem of Calculus, for all functions that are continuously defined on the interval with in and for all functions defined by by , we know that .

Given

, then

.

Therefore,

.

### Example Question #3 : Fundamental Theorem Of Calculus With Definite Integrals

Evaluate

**Possible Answers:**

**Correct answer:**

Use the fundamental theorem of calculus to evaluate:

### Example Question #4 : Fundamental Theorem Of Calculus With Definite Integrals

**Possible Answers:**

**Correct answer:**

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints:

### Example Question #5 : Fundamental Theorem Of Calculus With Definite Integrals

**Possible Answers:**

**Correct answer:**

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints: