### All AP Calculus AB Resources

## Example Questions

### Example Question #1 : Relationship Between Differentiability And Continuity

The function is differentiable at the point . List which of the following statements** must** be true about :

**1)** The limit exists.

_________________________________________________________

**2) **

_________________________________________________________

**3)**

_________________________________________________________

**4) **

_________________________________________________________

**5) **

**Possible Answers:**

1, and 5

All must be true.

1,3,4,and 5

1, 3, and 5

1, 2, and 4

**Correct answer:**

1, 2, and 4

**1)** If a function is differentiable, then by definition of differentiability the limit defined by,

exists. Therefore **(1) is required** by definition of differentiability. _______________________________________________________________

**2)** If a function is differentiable at a point then it must also be continuous at that point. (This is not conversely true).

For a function to be continuous at a point we must have:

Therefore **(2) and (4) are required. **

-----------------------------------------------------------------------------------------

**3) **

This is not required, the left side of the equation is the definition of a derivative at a point for a function . The derivative at a point does not have to equal to the function value at that point, it is equal to the** slope** at that point. Therefore** 3 does not have to be true. **

However, we can note that it is possible for a function and its' derivative to be equal for a given point. Sine and cosine, for instance will intersect periodically. Another example would be the exponential function which has itself as its' derivative .

______________________________________________________________

**4) See 2**

_______________________________________________________________

**5) **

Again, the function does not have to approach the same limit as its' derivative. It is possible for a function to behave in this manner, such as in the case of sine and its' derivative cosine, which will both have the same limit at points where they intersect.

### Example Question #2 : Relationship Between Differentiability And Continuity

When the limit fails to exist,

**Possible Answers:**

The function is not continuous at .

The function is not differentiable at .

The function is not defined at .

None of the above necessarily

**Correct answer:**

The function is not differentiable at .

By definition of differentiability, when the limit exists. When exists, we say the function is 'differentiable at '.

### Example Question #3 : Relationship Between Differentiability And Continuity

Which of the following functions is differentiable at , but not continuous there?

**Possible Answers:**

They are all differentiable and continuous at

**Correct answer:**

They are all differentiable and continuous at

All of the functions are differentiable at . If you examine the graph of each of the functions, they are all defined at , and do not have a corner, cusp, or a jump there; they are all smooth and connected (Not necessarily everywhere, just at ). Additionally it is not possible to have a function that is differentiable at a point, but not continuous at that same point; differentiablity implies continuity.

### Example Question #4 : Relationship Between Differentiability And Continuity

For which of the following functions does a limit exist at , but not a y-value?

**Possible Answers:**

**Correct answer:**

To answer the question, we must find an equation which satisfies two criteria:

(1) it must have limits on either side of that approach the same value *and *(2) it must have a hole at .

Each of the possible answers provide situations which demonstrate each combination of (1) and (2). That is to say, some of the equations include *both *a limit and a y-value at , *neither,* or,in the case of the piecewise function, a y-value and a limit that does not exist.

In the function, , the numerator factors to

while the denominator factors to . As a result, the graph of this

function resembles that for , but with a hole at . Therefore, the limit

at exists, even though the y-value is undefined at .