Lesson Worksheet: The Differentiability of a Function Mathematics • Higher Education

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 ๐‘“(๐‘ฅ)=๏ณ5๐‘Ž+๐‘๐‘ฅ,๐‘ฅ<โˆ’2,5,๐‘ฅ=โˆ’2,๐‘Ž๐‘ฅโˆ’3๐‘,๐‘ฅ>โˆ’2.๏Šจ 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?

  • A๐‘Ž=โˆ’4, ๐‘=1, not differentiable at ๐‘ฅ=โˆ’2
  • B๐‘Ž=5, ๐‘=โˆ’5, not differentiable at ๐‘ฅ=โˆ’2
  • C๐‘Ž=โˆ’4, ๐‘=1, differentiable at ๐‘ฅ=โˆ’2
  • D๐‘Ž=5, ๐‘=โˆ’5, differentiable at ๐‘ฅ=โˆ’2

Q2:

Discuss the continuity and differentiability of the function ๐‘“ at ๐‘ฅ=0 given ๐‘“(๐‘ฅ)=๏ฎโˆ’9๐‘ฅโˆ’6๐‘ฅ<0,๐‘ฅโˆ’9๐‘ฅโˆ’6๐‘ฅโ‰ฅ0.ifif๏Šจ

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

Q3:

Discuss the differentiability of a function ๐‘“ at ๐‘ฅ=โˆ’4 given ๐‘“(๐‘ฅ)=๏ญ8๐‘ฅ+7๐‘ฅ<โˆ’4,2๐‘ฅ+5๐‘ฅโ‰ฅโˆ’4.ifif

  • A๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’4 because ๐‘“โ€ฒ(โˆ’4)โ‰ ๐‘“โ€ฒ(โˆ’4)๏Šฐ๏Šฑ.
  • B๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’4 because ๐‘“ is continuous at ๐‘ฅ=โˆ’4 .
  • C๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’4 because ๐‘“(โˆ’4) is undefined.
  • D๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’4 because ๐‘“โ€ฒ(โˆ’4)=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)=(6๐‘ฅโˆ’6)|6๐‘ฅโˆ’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)=๏ฎ2๐‘ฅ+8๐‘ฅ<1,๐‘ฅ+9๐‘ฅโ‰ฅ1.ifif๏Šจ

  • A๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=1 because ๐‘“ is discontinuous at ๐‘ฅ=1.
  • B๐‘“(๐‘ฅ) is continuous but not differentiable at ๐‘ฅ=1 because ๐‘“โ€ฒ(1)โ‰ ๐‘“โ€ฒ(1)๏Šฐ๏Šฑ.
  • C๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=1.
  • D๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=1 because ๐‘“(1) is undefined.
  • E๐‘“(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)=๏ฎ๐‘ฅโˆ’7๐‘ฅ+5๐‘ฅโ‰คโˆ’8,3๐‘ฅ+4๐‘ฅโˆ’4๐‘ฅ>โˆ’8.๏Šจ๏Šจifif What can be said of the differentiability of ๐‘“ at ๐‘ฅ=โˆ’8?

  • AThe 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.
  • BThe 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.
  • CThe function ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’8 as limlim๏—โ†’๏Šฑ๏Šฎ๏—โ†’๏Šฑ๏Šฎ๏Žช๏Žฉ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ) but is not continuous.
  • DThe function ๐‘“(๐‘ฅ) is continuous at ๐‘ฅ=โˆ’8, but it is not differentiable.
  • EThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’8 because ๐‘“(โˆ’8) is undefined.

Q7:

What can be said of the differentiability of ๐‘“(๐‘ฅ)=โˆš๐‘ฅ+4๐‘ฅ+4๏Šจ at ๐‘ฅ=โˆ’2?

  • A๐‘“(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.
  • B๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’2 because ๐‘“(โˆ’2) is undefined.
  • C๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’2.
  • D๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’2 because ๐‘“ is discontinuous at that point.

Q8:

Discuss the differentiability of the function ๐‘“(๐‘ฅ)=โˆ’4๐‘ฅ+1๐‘ฅ at ๐‘ฅ=โˆ’7.

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

Q9:

Let ๐‘“(๐‘ฅ)=๏ญโˆ’4๐‘+๐‘š๐‘ฅ,๐‘ฅ<1,๐‘๐‘ฅโˆ’4๐‘š,๐‘ฅโ‰ฅ1.๏Šจ If ๐‘“(1)=12 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?

  • A๐‘š=โˆ’12, ๐‘=โˆ’6, differentiable at ๐‘ฅ=1
  • B๐‘š=โˆ’12, ๐‘=โˆ’6, not differentiable at ๐‘ฅ=1
  • C๐‘š=โˆ’4, ๐‘=โˆ’4, not differentiable at ๐‘ฅ=1
  • D๐‘š=โˆ’4, ๐‘=โˆ’4, 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)=๏ฎ9๐‘ฅ+5๐‘ฅ<โˆ’1,๐‘Ž๐‘ฅ+๐‘๐‘ฅโˆ’4๐‘ฅโ‰ฅโˆ’1.ifif๏Šจ

  • A๐‘Ž=โˆ’9, ๐‘=โˆ’9
  • B๐‘Ž=โˆ’4, ๐‘=1
  • C๐‘Ž=โˆ’10, ๐‘=โˆ’8
  • D๐‘Ž=โˆ’18, ๐‘=0

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