# 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.

Q1:

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

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

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 continuous but not differentiable at because .
• C is differentiable at .
• D is not differentiable at because is undefined.
• E is discontinuous but differentiable at because .

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 does not exist.
• CThe function is not differentiable at because is not continuous at that point.
• DThe function is differentiable at because exists.

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, differentiable at
• B, not differentiable at
• C, not differentiable at
• D, 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 .