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 𝑓(𝑥)=5𝑎+𝑏𝑥𝑥<2,5𝑥=2,𝑎𝑥3𝑏𝑥>2.ififif Determine the values of 𝑎 and 𝑏 so that 𝑓 is continuous at 𝑥=2. What can be said of the differentiability of 𝑓 at this point?

  • A 𝑎 = 4 , 𝑏 = 1 , not differentiable at 𝑥=2
  • B 𝑎 = 5 , 𝑏 = 5 , not differentiable at 𝑥=2
  • C 𝑎 = 4 , 𝑏 = 1 , differentiable at 𝑥=2
  • D 𝑎 = 5 , 𝑏 = 5 , differentiable at 𝑥=2

Q2:

Discuss the continuity and differentiability of the function 𝑓 at 𝑥=0 given 𝑓(𝑥)=9𝑥6𝑥<0,𝑥9𝑥6𝑥0.ifif

  • AThe function is not continuous, so it is not differentiable at 𝑥=0.
  • BThe function is continuous and differentiable at 𝑥=0.
  • CThe function is not continuous but differentiable at 𝑥=0.
  • DThe function is continuous but not differentiable at 𝑥=0.

Q3:

Discuss the differentiability of a function 𝑓 at 𝑥=4 given 𝑓(𝑥)=8𝑥+7𝑥<4,2𝑥+5𝑥4.ifif

  • A 𝑓 ( 𝑥 ) is not differentiable at 𝑥=4 because 𝑓(4)𝑓(4).
  • B 𝑓 ( 𝑥 ) is differentiable at 𝑥=4 because 𝑓 is continuous at 𝑥=4 .
  • C 𝑓 ( 𝑥 ) is not differentiable at 𝑥=4 because 𝑓(4) is undefined.
  • D 𝑓 ( 𝑥 ) is differentiable at 𝑥=4 because 𝑓(4)=𝑓(4).

Q4:

Discuss the differentiability of the function 𝑓(𝑥) at 𝑥=1 given 𝑓(𝑥)=(6𝑥6)|6𝑥6|.

  • AThe function is not differentiable at 𝑥=1 as 𝑓(𝑥) is discontinuous at that point.
  • BThe function is not differentiable at that point as 𝑓(1)𝑓(1).
  • CThe function is differentiable at 𝑥=1 as 𝑓(𝑥) is continuous at that point.
  • DThe function is differentiable at that point as 𝑓(1)=𝑓(1).

Q5:

Discuss the differentiability of the function 𝑓 at 𝑥=1 given 𝑓(𝑥)=2𝑥+8𝑥<1,𝑥+9𝑥1.ifif

  • A 𝑓 ( 𝑥 ) is not differentiable at 𝑥=1 because 𝑓 is discontinuous at 𝑥=1.
  • B 𝑓 ( 𝑥 ) is discontinuous but differentiable at 𝑥=1 because 𝑓(1)=𝑓(1).
  • C 𝑓 ( 𝑥 ) is continuous but not differentiable at 𝑥=1 because 𝑓(1)𝑓(1).
  • D 𝑓 ( 𝑥 ) is differentiable at 𝑥=1.
  • E 𝑓 ( 𝑥 ) is not differentiable at 𝑥=1 because 𝑓(1) is undefined.

Q6:

Suppose 𝑓(𝑥)=𝑥7𝑥+5𝑥8,3𝑥+4𝑥4𝑥>8.ifif What can be said of the differentiability of 𝑓 at 𝑥=8?

  • AThe function 𝑓(𝑥) is not continuous but differentiable at 𝑥=8 because 𝑓(8)=𝑓(8).
  • BThe function 𝑓(𝑥) is not differentiable at 𝑥=8 because 𝑓(8)𝑓(8).
  • CThe function 𝑓(𝑥) is differentiable at 𝑥=8 as limlim𝑓(𝑥)=𝑓(𝑥) but is not continuous.
  • DThe function 𝑓(𝑥) is not differentiable at 𝑥=8 because 𝑓 is discontinuous at 𝑓(8).
  • EThe function 𝑓(𝑥) is differentiable at 𝑥=8 because 𝑓(8) is undefined.

Q7:

What can be said of the differentiability of 𝑓(𝑥)=𝑥+4𝑥+4 at 𝑥=2?

  • A 𝑓 ( 𝑥 ) is continuous but not differentiable at 𝑥=2 because 𝑓(2)𝑓(2).
  • B 𝑓 ( 𝑥 ) is not differentiable at 𝑥=2 because 𝑓(2) is undefined.
  • C 𝑓 ( 𝑥 ) is differentiable at 𝑥=2.
  • D 𝑓 ( 𝑥 ) is not differentiable at 𝑥=2 because 𝑓 is discontinuous at that point.

Q8:

Discuss the differentiability of the function 𝑓(𝑥)=4𝑥+1𝑥 at 𝑥=7.

  • AThe function is differentiable at 𝑥=7 because 𝑓(7) exists.
  • BThe function is not differentiable at 𝑥=7 because 𝑓(𝑥) is not continuous at that point.
  • CThe function is differentiable at 𝑥=7 because 𝑓(7) exists.
  • DThe function is not differentiable at 𝑥=7 because 𝑓(7) does not exist.

Q9:

Let 𝑓(𝑥)=4𝑐+𝑚𝑥𝑥<1,𝑐𝑥4𝑚𝑥1.ifif If 𝑓(1)=12 and 𝑓 is continuous at 𝑥=1, determine the values of 𝑚 and 𝑐. What can be said of the differentiability of 𝑓 at this point?

  • A 𝑚 = 1 2 , 𝑐 = 6 , differentiable at 𝑥=1
  • B 𝑚 = 1 2 , 𝑐 = 6 , not differentiable at 𝑥=1
  • C 𝑚 = 4 , 𝑐 = 4 , not differentiable at 𝑥=1
  • D 𝑚 = 4 , 𝑐 = 4 , differentiable at 𝑥=1

Q10:

Find the values of 𝑎 and 𝑏 given the function 𝑓 is differentiable at 𝑥=1 where 𝑓(𝑥)=9𝑥+5𝑥<1,𝑎𝑥+𝑏𝑥4𝑥1.ifif

  • A 𝑎 = 9 , 𝑏 = 9
  • B 𝑎 = 4 , 𝑏 = 1
  • C 𝑎 = 1 0 , 𝑏 = 8
  • D 𝑎 = 1 8 , 𝑏 = 0

Q11:

Find the values of 𝑎 and 𝑏 given the function 𝑓 is differentiable at 𝑥=1 where 𝑓(𝑥)=𝑥+4𝑥1,2𝑎𝑥𝑏𝑥>1.ifif

  • A 𝑎 = 4 , 𝑏 = 5
  • B 𝑎 = 1 , 𝑏 = 5
  • C 𝑎 = 1 , 𝑏 = 5
  • D 𝑎 = 3 , 𝑏 = 6

Q12:

Let 𝑓(𝑥)=𝑎𝑥+8𝑥4𝑥<1,4𝑥=1,𝑎+𝑏𝑥𝑥>1.ififif Determine the values of 𝑎 and 𝑏 so that 𝑓 is continuous at 𝑥=1. What can be said of the differentiability of 𝑓 at this point?

  • A 𝑎 = 8 , 𝑏 = 4 , 𝑓 is differentiable at 𝑥=1
  • B 𝑎 = 8 , 𝑏 = 4 , 𝑓 is not differentiable at 𝑥=1
  • C 𝑎 = 1 0 , 𝑏 = 6 , 𝑓 is differentiable at 𝑥=1
  • D 𝑎 = 1 0 , 𝑏 = 6 , 𝑓 is not differentiable at 𝑥=1

Q13:

Discuss the differentiability of the function 𝑓 at 𝑥=6, given 𝑓(𝑥)=7𝑥8𝑥2,8𝑥+5𝑥2<𝑥6.ifif

  • A 𝑓 ( 𝑥 ) is not differentiable at 𝑥=6 because 𝑓(𝑥) is discontinuous at 𝑥=6.
  • B 𝑓 ( 𝑥 ) is discontinuous but differentiable at 𝑥=6 because 𝑓(6)=𝑓(6).
  • C 𝑓 ( 𝑥 ) is not differentiable at 𝑥=6 because 𝑓(6) is undefined.
  • D 𝑓 ( 𝑥 ) is differentiable at 𝑥=6.
  • E 𝑓 ( 𝑥 ) is continuous but not differentiable at 𝑥=6 because 𝑓(6)𝑓(6).

Q14:

Given that 𝑓(𝑥)=8𝑥8𝑥<2,𝑎𝑥𝑥2.ifif is a continuous function, find the value of 𝑎. What can be said of the differentiability of 𝑓 at 𝑥=2?

  • A 𝑎 = 3 , 𝑓 is differentiable at 𝑥=2
  • B 𝑎 = 3 , 𝑓 is differentiable at 𝑥=2
  • C 𝑎 = 3 , 𝑓 is not differentiable at 𝑥=2
  • D 𝑎 = 3 , 𝑓 is not differentiable at 𝑥=2

Q15:

Suppose 𝑓(𝑥)=6𝑥4𝑥1,3𝑥𝑥>1.ifif What can be said of the differentiability of 𝑓 at 𝑥=1?

  • AThe function 𝑓(𝑥) is differentiable at 𝑥=1 as limlim𝑓(𝑥)𝑓(𝑥) but is not continuous.
  • BThe function 𝑓(𝑥) is not differentiable at 𝑥=1 because 𝑓(1) is undefined.
  • CThe function 𝑓(𝑥) is not differentiable at 𝑥=1.
  • DThe function 𝑓(𝑥) is not differentiable at 𝑥=1 because 𝑓(𝑥) is continuous at 𝑓(1).
  • EThe function 𝑓(𝑥) is continuous but not differentiable at 𝑥=1 because 𝑓(1)𝑓(1).

Q16:

Discuss the differentiability of the function 𝑓 at 𝑥=1 given 𝑓(𝑥)=8𝑥7𝑥+32𝑥<1,4𝑥1𝑥3.ifif

  • A 𝑓 ( 𝑥 ) is discontinuous but differentiable at 𝑥=1 because 𝑓(1)=𝑓(1).
  • B 𝑓 ( 𝑥 ) is continuous but not differentiable at 𝑥=1 because 𝑓(1)𝑓(1).
  • C 𝑓 ( 𝑥 ) is differentiable at 𝑥=1.
  • D 𝑓 ( 𝑥 ) is not differentiable at 𝑥=1 because 𝑓(𝑥) is discontinuous at 𝑥=1.

Q17:

Suppose 𝑓(𝑥)=𝑥9𝑥4,𝑥+3𝑥>4.ifif What can be said of the differentiability of 𝑓 at 𝑥=4?

  • AThe function is not continuous, so it is not differentiable at 𝑥=4.
  • BThe function is not continuous but differentiable at 𝑥=4 because limlim𝑓(𝑥)=𝑓(𝑥)=𝑓(4).
  • CThe function is continuous and differentiable at 𝑥=4 because limlim𝑓(𝑥)=𝑓(𝑥)=𝑓(4).
  • DThe function is continuous but not differentiable at 𝑥=4 because 𝑓(4)𝑓(4).

Q18:

Suppose 𝑓(𝑥)=8𝑥+7𝑥0,𝑎2𝑥𝑥>0.ifif Find the value of a such that the function 𝑓 is continuous at 𝑥=0, and then discuss the differentiability of 𝑓 at 𝑥=0.

  • A 𝑎 = 7 , not differentiable at 𝑥=0
  • B 𝑎 = 7 , differentiable at 𝑥=0
  • C 𝑎 = 7 , differentiable at 𝑥=0
  • D 𝑎 = 7 , not differentiable at 𝑥=0

Q19:

Find the values of 𝑎 and 𝑏 and discuss the differentiability of the function 𝑓 at 𝑥=1 given 𝑓 is continuous and 𝑓(𝑥)=9𝑥+𝑎𝑥+4𝑥<1,11𝑥=1,𝑎+𝑏𝑥𝑥>1.ififif

  • A 𝑎 = 2 , 𝑏 = 9 , and 𝑓(𝑥) is not differentiable at 𝑥=1.
  • B 𝑎 = 2 , 𝑏 = 9 , and 𝑓(𝑥) is not differentiable at 𝑥=1.
  • C 𝑎 = 2 , 𝑏 = 9 , and 𝑓(𝑥) is differentiable at 𝑥=1.
  • D 𝑎 = 2 , 𝑏 = 9 , and 𝑓(𝑥) is differentiable at 𝑥=1.
  • E 𝑎 = 8 , 𝑏 = 4 , and 𝑓(𝑥) is not differentiable at 𝑥=1.

Q20:

What can be said of the differentiability of 𝑓(𝑥)=9𝑥+8𝑥+7 at 𝑥=2?

  • A 𝑓 ( 𝑥 ) is not differentiable at 𝑥=2 because 𝑓(2) is undefined.
  • B 𝑓 ( 𝑥 ) is differentiable at 𝑥=2.
  • C 𝑓 ( 𝑥 ) is not differentiable at 𝑥=2 because 𝑓(2) does not exist.
  • D 𝑓 ( 𝑥 ) is not differentiable at 𝑥=2 because 𝑓(𝑥) is not continuous.

Q21:

Discuss the differentiability of the function 𝑓 at 𝑥=4 given 𝑓(𝑥)=6𝑥+7𝑥44𝑥<1,2𝑥1𝑥1.ifif

  • AThe function 𝑓(𝑥) is not differentiable at 𝑥=4 because 𝑓(𝑥) is discontinuous at 𝑥=4.
  • BThe function 𝑓(𝑥) is discontinuous, but differentiable at 𝑥=4 because 𝑓(4)=𝑓(4).
  • CThe function 𝑓(𝑥) is continuous, but not differentiable at 𝑥=4 because 𝑓(4)𝑓(4).
  • DThe function 𝑓(𝑥) is differentiable at 𝑥=4.

Q22:

Suppose 𝑓(𝑥)=4𝑥7𝑥<1,2𝑥5𝑥1.ifif What can be said of the differentiability of 𝑓 at 𝑥=1?

  • AThe function 𝑓(𝑥) is not differentiable at 𝑥=1 because 𝑓(1) is undefined.
  • BThe function 𝑓(𝑥) is differentiable at 𝑥=1.
  • CThe function 𝑓(𝑥) is continuous but not differentiable at 𝑥=1 because 𝑓(1)𝑓(1).
  • DThe function 𝑓(𝑥) is discontinuous but differentiable at 𝑥=1 because 𝑓(1)=𝑓(1).
  • EThe function 𝑓(𝑥) is not differentiable at 𝑥=1 because 𝑓 is discontinuous at 𝑥=1.

Q23:

Let 𝑓(𝑥)=12𝑥6𝑥<2,𝑎𝑥+6𝑥2.ifif Determine the value of 𝑎 so that 𝑓 is continuous at 𝑥=2. What can be said of the differentiability of 𝑓 at this point?

  • A 𝑎 = 3 , differentiable at 𝑥=2
  • B 𝑎 = 1 2 , differentiable at 𝑥=2
  • C 𝑎 = 3 , not differentiable at 𝑥=2
  • D 𝑎 = 1 2 , not differentiable at 𝑥=2

Q24:

Discuss the differentiability of the function 𝑓 at 𝑥=8 given 𝑓(𝑥)=6𝑥9𝑥8,6𝑥+6𝑥8<𝑥5.ifif

  • A 𝑓 ( 𝑥 ) is differentiable at 𝑥=8 because 𝑓(𝑥) is continuous at 𝑥=8 .
  • B 𝑓 ( 𝑥 ) is not differentiable at 𝑥=8 because 𝑓(𝑥) is discontinuous at 𝑥=8.
  • C 𝑓 ( 𝑥 ) is not differentiable at 𝑥=8 because 𝑓(8) is undefined.
  • D 𝑓 ( 𝑥 ) is discontinuous but differentiable at 𝑥=8 because 𝑓(8)=𝑓(8).
  • E 𝑓 ( 𝑥 ) is continuous but not differentiable at 𝑥=8 because 𝑓(8)𝑓(8).

Q25:

Suppose 𝑓(𝑥)=𝑥15𝑥1,2𝑥16𝑥>1.ifif What can be said of the differentiability of 𝑓 at 𝑥=1?

  • AThe function is not continuous, so it is not differentiable at 𝑥=1.
  • BThe function is not continuous but differentiable at 𝑥=1 because limlim𝑓(𝑥)=𝑓(𝑥)=𝑓(1).
  • CThe function is continuous and differentiable at 𝑥=1 because 𝑓(1)=𝑓(1).
  • DThe function is continuous but not differentiable at 𝑥=1 because limlim𝑓(𝑥)=𝑓(𝑥)=𝑓(1).

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