# Worksheet: The Differentiability of a Function

In this worksheet, we will practice determining whether a function is differentiable and identifying the relation between a function’s differentiability and its continuity.

**Q2: **

Discuss the continuity and differentiability of the function at given

- AThe function is not continuous, so it is not differentiable at .
- BThe function is continuous and differentiable at .
- CThe function is not continuous but differentiable at .
- DThe function is continuous but not differentiable at .

**Q3: **

Discuss the differentiability of a function at given

- A is not differentiable at because .
- B is differentiable at because is continuous at .
- C is not differentiable at because is undefined.
- D is differentiable at because .

**Q4: **

Discuss the differentiability of the function at given .

- AThe function is not differentiable at as is discontinuous at that point.
- BThe function is not differentiable at that point as .
- CThe function is differentiable at as is continuous at that point.
- DThe function is differentiable at that point as .

**Q5: **

Discuss the differentiability of the function at given

- A is not differentiable at because is discontinuous at .
- B is discontinuous but differentiable at because .
- C is continuous but not differentiable at because .
- D is differentiable at .
- E is not differentiable at because is undefined.

**Q6: **

Suppose What can be said of the differentiability of at ?

- AThe function is not continuous but differentiable at because .
- BThe function is not differentiable at because .
- CThe function is differentiable at as but is not continuous.
- DThe function is not differentiable at because is discontinuous at .
- EThe function is differentiable at because is undefined.

**Q7: **

What can be said of the differentiability of at ?

- A is continuous but not differentiable at because .
- B is not differentiable at because is undefined.
- C is differentiable at .
- D is not differentiable at because is discontinuous at that point.

**Q8: **

Discuss the differentiability of the function at .

- AThe function is differentiable at because exists.
- BThe function is not differentiable at because is not continuous at that point.
- CThe function is differentiable at because exists.
- DThe function is not differentiable at because does not exist.

**Q9: **

Let If and is continuous at , determine the values of and . What can be said of the differentiability of at this point?

- A , , differentiable at
- B , , not differentiable at
- C , , not differentiable at
- D , , differentiable at

**Q10: **

Find the values of and given the function is differentiable at where

- A ,
- B ,
- C ,
- D ,

**Q11: **

Find the values of and given the function is differentiable at where

- A ,
- B ,
- C ,
- D ,

**Q12: **

Let Determine the values of and so that is continuous at . What can be said of the differentiability of at this point?

- A , , is differentiable at
- B , , is not differentiable at
- C , , is differentiable at
- D , , is not differentiable at

**Q13: **

Discuss the differentiability of the function at , given

- A is not differentiable at because is discontinuous at .
- B is discontinuous but differentiable at because .
- C is not differentiable at because is undefined.
- D is differentiable at .
- E is continuous but not differentiable at because .

**Q14: **

Given that is a continuous function, find the value of . What can be said of the differentiability of at ?

- A , is differentiable at
- B , is differentiable at
- C , is not differentiable at
- D , is not differentiable at

**Q15: **

Suppose What can be said of the differentiability of at ?

- AThe function is differentiable at as but is not continuous.
- BThe function is not differentiable at because is undefined.
- CThe function is not differentiable at .
- DThe function is not differentiable at because is continuous at .
- EThe function is continuous but not differentiable at because .

**Q16: **

Discuss the differentiability of the function at given

- A is discontinuous but differentiable at because .
- B is continuous but not differentiable at because .
- C is differentiable at .
- D is not differentiable at because is discontinuous at .

**Q17: **

Suppose What can be said of the differentiability of at ?

- AThe function is not continuous, so it is not differentiable at .
- BThe function is not continuous but differentiable at because .
- CThe function is continuous and differentiable at because .
- DThe function is continuous but not differentiable at because .

**Q18: **

Suppose Find the value of a such that the function is continuous at , and then discuss the differentiability of at .

- A , not differentiable at
- B , differentiable at
- C , differentiable at
- D , not differentiable at

**Q19: **

Find the values of and and discuss the differentiability of the function at given is continuous and

- A , , and is not differentiable at .
- B , , and is not differentiable at .
- C , , and is differentiable at .
- D , , and is differentiable at .
- E , , and is not differentiable at .

**Q20: **

What can be said of the differentiability of at ?

- A is not differentiable at because is undefined.
- B is differentiable at .
- C is not differentiable at because does not exist.
- D is not differentiable at because is not continuous.

**Q21: **

Discuss the differentiability of the function at given

- AThe function is not differentiable at because is discontinuous at .
- BThe function is discontinuous, but differentiable at because .
- CThe function is continuous, but not differentiable at because .
- DThe function is differentiable at .

**Q22: **

Suppose What can be said of the differentiability of at ?

- AThe function is not differentiable at because is undefined.
- BThe function is differentiable at .
- CThe function is continuous but not differentiable at because .
- DThe function is discontinuous but differentiable at because .
- EThe function is not differentiable at because is discontinuous at .

**Q23: **

Let Determine the value of so that is continuous at . What can be said of the differentiability of at this point?

- A , differentiable at
- B , differentiable at
- C , not differentiable at
- D , not differentiable at

**Q24: **

Discuss the differentiability of the function at given

- A is differentiable at because is continuous at .
- B is not differentiable at because is discontinuous at .
- C is not differentiable at because is undefined.
- D is discontinuous but differentiable at because .
- E is continuous but not differentiable at because .

**Q25: **

Suppose What can be said of the differentiability of at ?

- AThe function is not continuous, so it is not differentiable at .
- BThe function is not continuous but differentiable at because .
- CThe function is continuous and differentiable at because .
- DThe function is continuous but not differentiable at because .