Worksheet: The Differentiability of a Function

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

Q1:

Let ๐‘“ ( ๐‘ฅ ) = ๏ณ 5 ๐‘Ž + ๐‘ ๐‘ฅ ๐‘ฅ < โˆ’ 2 , 5 ๐‘ฅ = โˆ’ 2 , ๐‘Ž ๐‘ฅ โˆ’ 3 ๐‘ ๐‘ฅ > โˆ’ 2 . ๏Šจ i f i f i f Determine the values of ๐‘Ž and ๐‘ so that ๐‘“ is continuous at ๐‘ฅ = โˆ’ 2 . What can be said of the differentiability of ๐‘“ at this point?

  • A ๐‘Ž = โˆ’ 4 , ๐‘ = 1 , differentiable at ๐‘ฅ = โˆ’ 2
  • B ๐‘Ž = 5 , ๐‘ = โˆ’ 5 , differentiable at ๐‘ฅ = โˆ’ 2
  • C ๐‘Ž = โˆ’ 4 , ๐‘ = 1 , not differentiable at ๐‘ฅ = โˆ’ 2
  • D ๐‘Ž = 5 , ๐‘ = โˆ’ 5 , not differentiable at ๐‘ฅ = โˆ’ 2

Q2:

Discuss the continuity and differentiability of the function ๐‘“ at ๐‘ฅ = 0 given ๐‘“ ( ๐‘ฅ ) = ๏ฎ โˆ’ 9 ๐‘ฅ โˆ’ 6 ๐‘ฅ < 0 , ๐‘ฅ โˆ’ 9 ๐‘ฅ โˆ’ 6 ๐‘ฅ โ‰ฅ 0 . i f i f ๏Šจ

  • AThe function is not continuous, so it is not differentiable at ๐‘ฅ = 0 .
  • BThe function is not continuous but differentiable at ๐‘ฅ = 0 .
  • CThe function is continuous but not differentiable at ๐‘ฅ = 0 .
  • DThe function is continuous and differentiable at ๐‘ฅ = 0 .

Q3:

Discuss the differentiability of a function ๐‘“ at ๐‘ฅ = โˆ’ 4 given ๐‘“ ( ๐‘ฅ ) = ๏ญ 8 ๐‘ฅ + 7 ๐‘ฅ < โˆ’ 4 , 2 ๐‘ฅ + 5 ๐‘ฅ โ‰ฅ โˆ’ 4 . i f i f

  • A ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ ( โˆ’ 4 ) is undefined.
  • B ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ is continuous at ๐‘ฅ = โˆ’ 4 .
  • C ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ โ€ฒ ( โˆ’ 4 ) = ๐‘“ โ€ฒ ( โˆ’ 4 ) ๏Šฐ ๏Šฑ .
  • D ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ โ€ฒ ( โˆ’ 4 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 4 ) ๏Šฐ ๏Šฑ .

Q4:

Discuss the differentiability of the function ๐‘“ ( ๐‘ฅ ) at ๐‘ฅ = 1 given ๐‘“ ( ๐‘ฅ ) = ( 6 ๐‘ฅ โˆ’ 6 ) | 6 ๐‘ฅ โˆ’ 6 | .

  • AThe function is not differentiable at ๐‘ฅ = 1 as ๐‘“ ( ๐‘ฅ ) is discontinuous at that point.
  • BThe function is not differentiable at that point as ๐‘“ โ€ฒ ( 1 ) โ‰  ๐‘“ โ€ฒ ( 1 ) ๏Šฑ ๏Šฐ .
  • CThe function is differentiable at ๐‘ฅ = 1 as ๐‘“ ( ๐‘ฅ ) is continuous at that point.
  • DThe function is differentiable at that point as ๐‘“ โ€ฒ ( 1 ) = ๐‘“ โ€ฒ ( 1 ) ๏Šฑ ๏Šฐ .

Q5:

Discuss the differentiability of the function ๐‘“ at ๐‘ฅ = 1 given ๐‘“ ( ๐‘ฅ ) = ๏ฎ 2 ๐‘ฅ + 8 ๐‘ฅ < 1 , ๐‘ฅ + 9 ๐‘ฅ โ‰ฅ 1 . i f i f ๏Šจ

  • A ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 1 because ๐‘“ is discontinuous at ๐‘ฅ = 1 .
  • B ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) โ‰  ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .
  • C ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 1 because ๐‘“ ( 1 ) is undefined.
  • D ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = 1 .
  • E ๐‘“ ( ๐‘ฅ ) is discontinuous but differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) = ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .

Q6:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ฎ ๐‘ฅ โˆ’ 7 ๐‘ฅ + 5 ๐‘ฅ โ‰ค โˆ’ 8 , 3 ๐‘ฅ + 4 ๐‘ฅ โˆ’ 4 ๐‘ฅ > โˆ’ 8 . ๏Šจ ๏Šจ i f i f What can be said of the differentiability of ๐‘“ at ๐‘ฅ = โˆ’ 8 ?

  • AThe function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ is discontinuous at ๐‘“ ( โˆ’ 8 ) .
  • BThe function ๐‘“ ( ๐‘ฅ ) is not continuous but differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ โ€ฒ ( โˆ’ 8 ) = ๐‘“ โ€ฒ ( โˆ’ 8 ) ๏Šฑ ๏Šฐ .
  • CThe function ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 8 as l i m l i m ๏— โ†’ ๏Šฑ ๏Šฎ ๏— โ†’ ๏Šฑ ๏Šฎ ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) but is not continuous.
  • DThe function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ โ€ฒ ( โˆ’ 8 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 8 ) ๏Šฑ ๏Šฐ .
  • EThe function ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ ( โˆ’ 8 ) is undefined.

Q7:

What can be said of the differentiability of ๐‘“ ( ๐‘ฅ ) = โˆš ๐‘ฅ + 4 ๐‘ฅ + 4 ๏Šจ at ๐‘ฅ = โˆ’ 2 ?

  • A ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 2 .
  • B ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ is discontinuous at that point.
  • C ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ ( โˆ’ 2 ) is undefined.
  • D ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ โ€ฒ ( โˆ’ 2 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 2 ) ๏Šฐ ๏Šฑ .

Q8:

Discuss the differentiability of the function ๐‘“ ( ๐‘ฅ ) = โˆ’ 4 ๐‘ฅ + 1 ๐‘ฅ at ๐‘ฅ = โˆ’ 7 .

  • AThe function is not differentiable at ๐‘ฅ = โˆ’ 7 because ๐‘“ โ€ฒ ( โˆ’ 7 ) does not exist.
  • BThe function is not differentiable at ๐‘ฅ = โˆ’ 7 because ๐‘“ ( ๐‘ฅ ) is not continuous at that point.
  • CThe function is differentiable at ๐‘ฅ = โˆ’ 7 because ๐‘“ ( โˆ’ 7 ) exists.
  • DThe function is differentiable at ๐‘ฅ = โˆ’ 7 because ๐‘“ โ€ฒ ( โˆ’ 7 ) exists.

Q9:

Let ๐‘“ ( ๐‘ฅ ) = ๏ฎ โˆ’ 4 ๐‘ + ๐‘š ๐‘ฅ ๐‘ฅ < 1 , ๐‘ ๐‘ฅ โˆ’ 4 ๐‘š ๐‘ฅ โ‰ฅ 1 . i f i f ๏Šจ If ๐‘“ ( 1 ) = 1 2 and ๐‘“ is continuous at ๐‘ฅ = 1 , determine the values of ๐‘š and ๐‘ . What can be said of the differentiability of ๐‘“ at this point?

  • A ๐‘š = โˆ’ 1 2 , ๐‘ = โˆ’ 6 , differentiable at ๐‘ฅ = 1
  • B ๐‘š = โˆ’ 4 , ๐‘ = โˆ’ 4 , differentiable at ๐‘ฅ = 1
  • C ๐‘š = โˆ’ 1 2 , ๐‘ = โˆ’ 6 , not differentiable at ๐‘ฅ = 1
  • D ๐‘š = โˆ’ 4 , ๐‘ = โˆ’ 4 , not differentiable at ๐‘ฅ = 1

Q10:

Find the values of ๐‘Ž and ๐‘ given the function ๐‘“ is differentiable at ๐‘ฅ = โˆ’ 1 where ๐‘“ ( ๐‘ฅ ) = ๏ฎ 9 ๐‘ฅ + 5 ๐‘ฅ < โˆ’ 1 , ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฅ โˆ’ 4 ๐‘ฅ โ‰ฅ โˆ’ 1 . i f i f ๏Šจ

  • A ๐‘Ž = โˆ’ 1 0 , ๐‘ = โˆ’ 8
  • B ๐‘Ž = โˆ’ 1 8 , ๐‘ = 0
  • C ๐‘Ž = โˆ’ 4 , ๐‘ = 1
  • D ๐‘Ž = โˆ’ 9 , ๐‘ = โˆ’ 9

Q11:

Find the values of ๐‘Ž and ๐‘ given the function ๐‘“ is differentiable at ๐‘ฅ = 1 where ๐‘“ ( ๐‘ฅ ) = ๏ญ โˆ’ ๐‘ฅ + 4 ๐‘ฅ โ‰ค 1 , โˆ’ 2 ๐‘Ž ๐‘ฅ โˆ’ ๐‘ ๐‘ฅ > 1 . ๏Šจ i f i f

  • A ๐‘Ž = โˆ’ 1 , ๐‘ = 5
  • B ๐‘Ž = 3 , ๐‘ = 6
  • C ๐‘Ž = 4 , ๐‘ = 5
  • D ๐‘Ž = 1 , ๐‘ = โˆ’ 5

Q12:

Let ๐‘“ ( ๐‘ฅ ) = ๏ณ ๐‘Ž ๐‘ฅ + 8 ๐‘ฅ โˆ’ 4 ๐‘ฅ < 1 , โˆ’ 4 ๐‘ฅ = 1 , ๐‘Ž + ๐‘ ๐‘ฅ ๐‘ฅ > 1 . ๏Šจ i f i f i f Determine the values of ๐‘Ž and ๐‘ so that ๐‘“ is continuous at ๐‘ฅ = 1 . What can be said of the differentiability of ๐‘“ at this point?

  • A ๐‘Ž = โˆ’ 1 0 , ๐‘ = 6 , ๐‘“ is differentiable at ๐‘ฅ = 1
  • B ๐‘Ž = โˆ’ 8 , ๐‘ = 4 , ๐‘“ is differentiable at ๐‘ฅ = 1
  • C ๐‘Ž = โˆ’ 1 0 , ๐‘ = 6 , ๐‘“ is not differentiable at ๐‘ฅ = 1
  • D ๐‘Ž = โˆ’ 8 , ๐‘ = 4 , ๐‘“ is not differentiable at ๐‘ฅ = 1

Q13:

Discuss the differentiability of the function ๐‘“ at ๐‘ฅ = 6 , given ๐‘“ ( ๐‘ฅ ) = ๏ฎ โˆ’ 7 ๐‘ฅ โˆ’ 8 โ‰ค ๐‘ฅ โ‰ค โˆ’ 2 , 8 ๐‘ฅ + 5 ๐‘ฅ โˆ’ 2 < ๐‘ฅ โ‰ค 6 . i f i f ๏Šจ

  • A ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 6 because ๐‘“ ( ๐‘ฅ ) is discontinuous at ๐‘ฅ = 6 .
  • B ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = 6 because ๐‘“ โ€ฒ ( 6 ) โ‰  ๐‘“ โ€ฒ ( 6 ) ๏Šฐ ๏Šฑ .
  • C ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 6 because ๐‘“ ( 6 ) is undefined.
  • D ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = 6 .
  • E ๐‘“ ( ๐‘ฅ ) is discontinuous but differentiable at ๐‘ฅ = 6 because ๐‘“ โ€ฒ ( 6 ) = ๐‘“ โ€ฒ ( 6 ) ๏Šฐ ๏Šฑ .

Q14:

Given that ๐‘“ ( ๐‘ฅ ) = ๏ฎ 8 ๐‘ฅ โˆ’ 8 ๐‘ฅ < โˆ’ 2 , ๐‘Ž ๐‘ฅ ๐‘ฅ โ‰ฅ โˆ’ 2 . i f i f ๏Šฉ is a continuous function, find the value of ๐‘Ž . What can be said of the differentiability of ๐‘“ at ๐‘ฅ = โˆ’ 2 ?

  • A ๐‘Ž = โˆ’ 3 , ๐‘“ is differentiable at ๐‘ฅ = โˆ’ 2
  • B ๐‘Ž = 3 , ๐‘“ is differentiable at ๐‘ฅ = โˆ’ 2
  • C ๐‘Ž = โˆ’ 3 , ๐‘“ is not differentiable at ๐‘ฅ = โˆ’ 2
  • D ๐‘Ž = 3 , ๐‘“ is not differentiable at ๐‘ฅ = โˆ’ 2

Q15:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ญ โˆ’ 6 ๐‘ฅ โˆ’ 4 ๐‘ฅ โ‰ค โˆ’ 1 , 3 ๐‘ฅ ๐‘ฅ > โˆ’ 1 . i f i f ๏Šจ What can be said of the differentiability of ๐‘“ at ๐‘ฅ = โˆ’ 1 ?

  • AThe function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 because ๐‘“ ( ๐‘ฅ ) is continuous at ๐‘“ ( โˆ’ 1 ) .
  • BThe function ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = โˆ’ 1 because ๐‘“ โ€ฒ ( โˆ’ 1 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 1 ) ๏Šฑ ๏Šฐ .
  • C The function ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 1 as l i m l i m ๏— โ†’ ๏Šฑ ๏Šง ๏— โ†’ ๏Šฑ ๏Šง ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) โ‰  ๐‘“ ( ๐‘ฅ ) but is not continuous.
  • DThe function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 .
  • EThe function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 because ๐‘“ ( โˆ’ 1 ) is undefined.

Q16:

Discuss the differentiability of the function ๐‘“ at ๐‘ฅ = 1 given

  • A ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 1 because ๐‘“ ( ๐‘ฅ ) is discontinuous at ๐‘ฅ = 1 .
  • B ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) โ‰  ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .
  • C ๐‘“ ( ๐‘ฅ ) is discontinuous but differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) = ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .
  • D ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = 1 .

Q17:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ญ ๐‘ฅ โˆ’ 9 ๐‘ฅ โ‰ค 4 , ๐‘ฅ + 3 ๐‘ฅ > 4 . ๏Šจ i f i f What can be said of the differentiability of ๐‘“ at ๐‘ฅ = 4 ?

  • AThe function is not continuous but differentiable at ๐‘ฅ = 4 because l i m l i m ๏— โ†’ ๏Šช ๏— โ†’ ๏Šช ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( 4 ) .
  • BThe function is continuous and differentiable at ๐‘ฅ = 4 because l i m l i m ๏— โ†’ ๏Šช ๏— โ†’ ๏Šช ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( 4 ) .
  • CThe function is not continuous, so it is not differentiable at ๐‘ฅ = 4 .
  • DThe function is continuous but not differentiable at ๐‘ฅ = 4 because ๐‘“ โ€ฒ ( 4 ) โ‰  ๐‘“ โ€ฒ ( 4 ) ๏Šฑ ๏Šฐ .

Q18:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ฎ โˆ’ 8 ๐‘ฅ + 7 ๐‘ฅ โ‰ค 0 , ๐‘Ž โˆ’ 2 ๐‘ฅ ๐‘ฅ > 0 . ๏Šจ ๏Šจ i f i f Find the value of a such that the function ๐‘“ is continuous at ๐‘ฅ = 0 , and then discuss the differentiability of ๐‘“ at ๐‘ฅ = 0 .

  • A ๐‘Ž = โˆ’ 7 , differentiable at ๐‘ฅ = 0
  • B ๐‘Ž = 7 , not differentiable at ๐‘ฅ = 0
  • C ๐‘Ž = โˆ’ 7 , not differentiable at ๐‘ฅ = 0
  • D ๐‘Ž = 7 , differentiable at ๐‘ฅ = 0

Q19:

Find the values of ๐‘Ž and ๐‘ and discuss the differentiability of the function ๐‘“ at ๐‘ฅ = โˆ’ 1 given ๐‘“ is continuous and

  • A ๐‘Ž = 2 , ๐‘ = 9 , and ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 .
  • B ๐‘Ž = โˆ’ 2 , ๐‘ = โˆ’ 9 , and ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 1 .
  • C ๐‘Ž = โˆ’ 2 , ๐‘ = 9 , and ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 1 .
  • D ๐‘Ž = 2 , ๐‘ = โˆ’ 9 , and ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 .
  • E ๐‘Ž = 8 , ๐‘ = 4 , and ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 1 .

Q20:

What can be said of the differentiability of ๐‘“ ( ๐‘ฅ ) = 9 ๐‘ฅ + 8 ๐‘ฅ + 7 ๏Šจ at ๐‘ฅ = โˆ’ 2 ?

  • A ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ ( โˆ’ 2 ) ๏Ž˜ does not exist.
  • B ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ ( โˆ’ 2 ) is undefined.
  • C ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 2 because ๐‘“ ( ๐‘ฅ ) is not continuous.
  • D ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 2 .

Q21:

Discuss the differentiability of the function ๐‘“ at ๐‘ฅ = โˆ’ 4 given ๐‘“ ( ๐‘ฅ ) = ๏ฎ โˆ’ 6 ๐‘ฅ + 7 ๐‘ฅ โˆ’ 4 โˆ’ 4 โ‰ค ๐‘ฅ < โˆ’ 1 , โˆ’ 2 ๐‘ฅ โˆ’ 1 โ‰ค ๐‘ฅ โ‰ค 1 . ๏Šจ i f i f

  • A The function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ ( ๐‘ฅ ) is discontinuous at ๐‘ฅ = โˆ’ 4 .
  • B The function ๐‘“ ( ๐‘ฅ ) is continuous, but not differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ โ€ฒ ( โˆ’ 4 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 4 ) ๏Šฐ ๏Šฑ .
  • C The function ๐‘“ ( ๐‘ฅ ) is discontinuous, but differentiable at ๐‘ฅ = โˆ’ 4 because ๐‘“ โ€ฒ ( โˆ’ 4 ) = ๐‘“ โ€ฒ ( โˆ’ 4 ) ๏Šฐ ๏Šฑ .
  • D The function ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 4 .

Q22:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ฎ 4 ๐‘ฅ โˆ’ 7 ๐‘ฅ < 1 , 2 ๐‘ฅ โˆ’ 5 ๐‘ฅ โ‰ฅ 1 . i f i f ๏Šจ What can be said of the differentiability of ๐‘“ at ๐‘ฅ = 1 ?

  • A The function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 1 because ๐‘“ is discontinuous at ๐‘ฅ = 1 .
  • B The function ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) โ‰  ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .
  • C The function ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = 1 because ๐‘“ ( 1 ) is undefined.
  • D The function ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = 1 .
  • E The function ๐‘“ ( ๐‘ฅ ) is discontinuous but differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) = ๐‘“ โ€ฒ ( 1 ) ๏Šฐ ๏Šฑ .

Q23:

Let ๐‘“ ( ๐‘ฅ ) = ๏ฎ 1 2 ๐‘ฅ โˆ’ 6 ๐‘ฅ < 2 , ๐‘Ž ๐‘ฅ + 6 ๐‘ฅ โ‰ฅ 2 . i f i f ๏Šจ Determine the value of ๐‘Ž so that ๐‘“ is continuous at ๐‘ฅ = 2 . What can be said of the differentiability of ๐‘“ at this point?

  • A ๐‘Ž = 1 2 , differentiable at ๐‘ฅ = 2
  • B ๐‘Ž = 3 , not differentiable at ๐‘ฅ = 2
  • C ๐‘Ž = 1 2 , not differentiable at ๐‘ฅ = 2
  • D ๐‘Ž = 3 , differentiable at ๐‘ฅ = 2

Q24:

Discuss the differentiability of the function ๐‘“ at ๐‘ฅ = โˆ’ 8 given ๐‘“ ( ๐‘ฅ ) = ๏ฎ 6 ๐‘ฅ โˆ’ 9 โ‰ค ๐‘ฅ โ‰ค โˆ’ 8 , 6 ๐‘ฅ + 6 ๐‘ฅ โˆ’ 8 < ๐‘ฅ โ‰ค 5 . i f i f ๏Šจ

  • A ๐‘“ ( ๐‘ฅ ) is differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ ( ๐‘ฅ ) is continuous at ๐‘ฅ = โˆ’ 8 .
  • B ๐‘“ ( ๐‘ฅ ) is continuous but not differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ โ€ฒ ( โˆ’ 8 ) โ‰  ๐‘“ โ€ฒ ( โˆ’ 8 ) ๏Šฐ ๏Šฑ .
  • C ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ ( โˆ’ 8 ) is undefined.
  • D ๐‘“ ( ๐‘ฅ ) is not differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ ( ๐‘ฅ ) is discontinuous at ๐‘ฅ = โˆ’ 8 .
  • E ๐‘“ ( ๐‘ฅ ) is discontinuous but differentiable at ๐‘ฅ = โˆ’ 8 because ๐‘“ โ€ฒ ( โˆ’ 8 ) = ๐‘“ โ€ฒ ( โˆ’ 8 ) ๏Šฐ ๏Šฑ .

Q25:

Suppose ๐‘“ ( ๐‘ฅ ) = ๏ญ ๐‘ฅ โˆ’ 1 5 ๐‘ฅ โ‰ค 1 , 2 ๐‘ฅ โˆ’ 1 6 ๐‘ฅ > 1 . ๏Šจ i f i f What can be said of the differentiability of ๐‘“ at ๐‘ฅ = 1 ?

  • AThe function is not continuous but differentiable at ๐‘ฅ = 1 because l i m l i m ๏— โ†’ ๏Šง ๏— โ†’ ๏Šง ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( 1 ) .
  • BThe function is continuous but not differentiable at ๐‘ฅ = 1 because l i m l i m ๏— โ†’ ๏Šง ๏— โ†’ ๏Šง ๏Žช ๏Žฉ ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( ๐‘ฅ ) = ๐‘“ ( 1 ) .
  • CThe function is not continuous, so it is not differentiable at ๐‘ฅ = 1 .
  • DThe function is continuous and differentiable at ๐‘ฅ = 1 because ๐‘“ โ€ฒ ( 1 ) = ๐‘“ โ€ฒ ( 1 ) ๏Šฑ ๏Šฐ .

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