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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:

True or False: If a function is continuous at a point, then it must be differentiable at that point.

  • ATrue
  • BFalse

Q2:

Suppose ๐‘“(๐‘ฅ)=๏ญโˆ’6๐‘ฅโˆ’4,๐‘ฅโ‰คโˆ’1,3๐‘ฅ,๐‘ฅ>โˆ’1.๏Šจ What can be said of the differentiability of ๐‘“ at ๐‘ฅ=โˆ’1?

  • AThe function ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’1 as limlim๏—โ†’๏Šฑ๏Šง๏—โ†’๏Šฑ๏Šง๏Žช๏Žฉ๐‘“(๐‘ฅ)โ‰ ๐‘“(๐‘ฅ) but is not continuous.
  • BThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1 because ๐‘“(โˆ’1) is undefined.
  • CThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1.
  • DThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1 because ๐‘“(๐‘ฅ) is continuous at ๐‘“(โˆ’1).
  • EThe 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.

Q3:

The figure shows the graph of ๐‘“. What can be said of the differentiability of ๐‘“ at ๐‘ฅ=โˆ’4?

  • AThe function is not differentiable at ๐‘ฅ=โˆ’4 because it is undefined at that point.
  • BThe function is differentiable at ๐‘ฅ=โˆ’4 because the functionโ€™s rate of change is the same on both sides of that point.
  • CThe function is not differentiable at ๐‘ฅ=โˆ’4 because the functionโ€™s rate of change is different on both sides of that point.
  • DThe function is differentiable at ๐‘ฅ=โˆ’4 because it is continuous at that point.

Q4:

Consider the function ๐‘“(๐‘ฅ)=โˆš๐‘ฅ๏Žข.

What is the domain of ๐‘“?

  • A๐‘ฅโ‰ฅ0
  • B๐‘ฅ<0
  • Cโ„
  • D๐‘ฅโ‰ค0
  • E๐‘ฅ>0

Find an expression for the derivative of ๐‘“.

  • A13โˆš๐‘ฅ๏Žข๏Šจ
  • B13โˆš๐‘ฅ๏Žข๏Šจ
  • C3โˆš๐‘ฅ๏Žข๏Šจ
  • D1โˆš๐‘ฅ๏Žข๏Šจ
  • E3โˆš๐‘ฅ๏Žข๏Šจ

What is the domain for the derivative ๐‘“โ€ฒ?

  • A๐‘ฅโ‰ฅ0
  • B๐‘ฅ>0
  • Cโ„โˆ’{0}
  • D๐‘ฅโ‰ค0
  • Eโ„

Q5:

The figure shows the graph of ๐‘“. At which points is the derivative of the function not defined?

  • A๐‘ฅ=โˆ’1, ๐‘ฅ=1, and ๐‘ฅ=0
  • B๐‘ฅ=โˆ’1 and ๐‘ฅ=1
  • C๐‘ฅ=1
  • D๐‘ฅ=โˆ’1
  • E๐‘ฅ=0

Q6:

Suppose ๐‘“(๐‘ฅ)=๏ฒโˆ’1+3๐‘ฅ,๐‘ฅโ‰ค1,โˆ’๐‘ฅ+3,๐‘ฅ>1.๏Šฉ What can be said of the differentiability of ๐‘“ at ๐‘ฅ=1?

  • AThe function ๐‘“(๐‘ฅ) is continuous but not differentiable at ๐‘ฅ=1 because ๐‘“โ€ฒ(1)โ‰ ๐‘“โ€ฒ(1)๏Šฑ๏Šฐ.
  • BThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=1 because ๐‘“(๐‘ฅ) is discontinuous at ๐‘“(1).
  • CThe function ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=1.
  • DThe function ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=1 as limlim๏—โ†’๏Šง๏—โ†’๏Šง๏Žช๏Žฉ๐‘“(๐‘ฅ)โ‰ ๐‘“(๐‘ฅ) but is not continuous.
  • EThe function ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=1 because ๐‘“(1) is undefined.

Q7:

Consider a function with ๐‘“(โˆ’8)=3 and ๐‘“โ€ฒ(โˆ’8)=7. What is lim๏—โ†’๏Šฑ๏Šฎ๐‘“(๐‘ฅ)?

Q8:

Given that ๐‘“(๐‘ฅ)=๏ญ8๐‘ฅโˆ’8,๐‘ฅ<โˆ’2,๐‘Ž๐‘ฅ,๐‘ฅโ‰ฅโˆ’2.๏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?

  • A๐‘Ž=โˆ’3, ๐‘“ is differentiable at ๐‘ฅ=โˆ’2
  • B๐‘Ž=3, ๐‘“ is differentiable at ๐‘ฅ=โˆ’2
  • C๐‘Ž=โˆ’3, ๐‘“ is not differentiable at ๐‘ฅ=โˆ’2
  • D๐‘Ž=3, ๐‘“ is not differentiable at ๐‘ฅ=โˆ’2

Q9:

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 ๐‘“(๐‘ฅ)=๏ณ9๐‘ฅ+๐‘Ž๐‘ฅ+4๐‘ฅ<โˆ’1,11๐‘ฅ=โˆ’1,๐‘Ž+๐‘๐‘ฅ๐‘ฅ>โˆ’1.๏Šจififif

  • A๐‘Ž=2, ๐‘=โˆ’9, and ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1.
  • B๐‘Ž=2, ๐‘=9, and ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1.
  • C๐‘Ž=โˆ’2, ๐‘=9, and ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’1.
  • D๐‘Ž=โˆ’2, ๐‘=โˆ’9, and ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=โˆ’1.
  • E๐‘Ž=8, ๐‘=4, and ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=โˆ’1.

Q10:

Is the function ๐‘“(๐‘ฅ)=๏ฑ๐‘ฅ๏€ผ1๐‘ฅ๏ˆ,๐‘ฅโ‰ 0,0,๐‘ฅ=0sin differentiable at ๐‘ฅ=0?

  • AYes
  • BNo

This lesson includes 107 additional questions and 610 additional question variations for subscribers.

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