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Lesson: The Differentiability of a Function

Worksheet • 25 Questions

Q1:

Given that is a continuous function, find the value of π‘Ž . What can be said of the differentiability of 𝑓 at π‘₯ = βˆ’ 2 ?

  • A π‘Ž = 3 , 𝑓 is not differentiable at π‘₯ = βˆ’ 2
  • B π‘Ž = 3 , 𝑓 is differentiable at π‘₯ = βˆ’ 2
  • C π‘Ž = βˆ’ 3 , 𝑓 is not differentiable at π‘₯ = βˆ’ 2
  • D π‘Ž = βˆ’ 3 , 𝑓 is differentiable at π‘₯ = βˆ’ 2

Q2:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = βˆ’ 1 ?

  • AThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 .
  • BThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 because 𝑓 ( βˆ’ 1 ) is undefined.
  • CThe function 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = βˆ’ 1 because 𝑓 β€² ( βˆ’ 1 ) β‰  𝑓 β€² ( βˆ’ 1 )   .
  • D The function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 1 as l i m l i m  β†’    β†’   οŽͺ  𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) but is not continuous.
  • EThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 because 𝑓 ( π‘₯ ) is continuous at 𝑓 ( βˆ’ 1 ) .

Q3:

Discuss the differentiability of the function 𝑓 at π‘₯ = 1 given

  • A 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 1 .
  • B 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) β‰  𝑓 β€² ( 1 )   .
  • C 𝑓 ( π‘₯ ) is discontinuous but differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) = 𝑓 β€² ( 1 )   .
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 ( π‘₯ ) is discontinuous at π‘₯ = 1 .

Q4:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = 4 ?

  • AThe function is continuous but not differentiable at π‘₯ = 4 because 𝑓 β€² ( 4 ) β‰  𝑓 β€² ( 4 ) βˆ’ + .
  • BThe function is continuous and differentiable at π‘₯ = 4 because l i m l i m π‘₯ β†’ 4 π‘₯ β†’ 4 βˆ’ + 𝑓 ( π‘₯ ) = 𝑓 ( π‘₯ ) = 𝑓 ( 4 ) .
  • CThe function is not continuous, so it is not differentiable at π‘₯ = 4 .
  • DThe function is not continuous but differentiable at π‘₯ = 4 because l i m l i m π‘₯ β†’ 4 π‘₯ β†’ 4 βˆ’ + 𝑓 ( π‘₯ ) = 𝑓 ( π‘₯ ) = 𝑓 ( 4 ) .

Q5:

Let Determine the values of π‘Ž and 𝑏 so that 𝑓 is continuous at π‘₯ = βˆ’ 2 . What can be said of the differentiability of 𝑓 at this point?

  • A π‘Ž = 5 , 𝑏 = βˆ’ 5 , not differentiable at π‘₯ = βˆ’ 2
  • B π‘Ž = 5 , 𝑏 = βˆ’ 5 , differentiable at π‘₯ = βˆ’ 2
  • C π‘Ž = βˆ’ 4 , 𝑏 = 1 , not differentiable at π‘₯ = βˆ’ 2
  • D π‘Ž = βˆ’ 4 , 𝑏 = 1 , differentiable at π‘₯ = βˆ’ 2

Q6:

Suppose Find the value of a such that the function 𝑓 is continuous at π‘₯ = 0 , and then discuss the differentiability of 𝑓 at π‘₯ = 0 .

  • A π‘Ž = 7 , differentiable at π‘₯ = 0
  • B π‘Ž = 7 , not differentiable at π‘₯ = 0
  • C π‘Ž = βˆ’ 7 , not differentiable at π‘₯ = 0
  • D π‘Ž = βˆ’ 7 , differentiable at π‘₯ = 0

Q7:

Find the values of π‘Ž and 𝑏 and discuss the differentiability of the function 𝑓 at π‘₯ = βˆ’ 1 given 𝑓 is continuous and

  • A π‘Ž = 2 , 𝑏 = βˆ’ 9 , and 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 .
  • B π‘Ž = 8 , 𝑏 = 4 , and 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 .
  • C π‘Ž = βˆ’ 2 , 𝑏 = βˆ’ 9 , and 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 1 .
  • D π‘Ž = βˆ’ 2 , 𝑏 = 9 , and 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 1 .
  • E π‘Ž = 2 , 𝑏 = 9 , and 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 1 .

Q8:

What can be said of the differentiability of 𝑓 ( π‘₯ ) = √ π‘₯ + 4 π‘₯ + 4 2 at π‘₯ = βˆ’ 2 ?

  • A 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = βˆ’ 2 because 𝑓 β€² ( βˆ’ 2 ) β‰  𝑓 β€² ( βˆ’ 2 ) + βˆ’ .
  • B 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 2 because 𝑓 is discontinuous at that point.
  • C 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 2 because 𝑓 ( βˆ’ 2 ) is undefined.
  • D 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 2 .

Q9:

What can be said of the differentiability of 𝑓 ( π‘₯ ) = 9 π‘₯ + 8 π‘₯ + 7  at π‘₯ = βˆ’ 2 ?

  • A 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 2 .
  • B 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 2 because 𝑓 ( βˆ’ 2 ) is undefined.
  • C 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 2 because 𝑓 ( π‘₯ ) is not continuous.
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 2 because 𝑓 ( βˆ’ 2 )  does not exist.

Q10:

Discuss the differentiability of the function 𝑓 at π‘₯ = βˆ’ 4 given

  • A The function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 4 .
  • B The function 𝑓 ( π‘₯ ) is continuous, but not differentiable at π‘₯ = βˆ’ 4 because 𝑓 β€² ( βˆ’ 4 ) β‰  𝑓 β€² ( βˆ’ 4 )   .
  • C The function 𝑓 ( π‘₯ ) is discontinuous, but differentiable at π‘₯ = βˆ’ 4 because 𝑓 β€² ( βˆ’ 4 ) = 𝑓 β€² ( βˆ’ 4 )   .
  • D The function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 4 because 𝑓 ( π‘₯ ) is discontinuous at π‘₯ = βˆ’ 4 .

Q11:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = 1 ?

  • A The function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 1 .
  • B The function 𝑓 ( π‘₯ ) is discontinuous but differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) = 𝑓 β€² ( 1 )   .
  • C The function 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) β‰  𝑓 β€² ( 1 )   .
  • D The function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 ( 1 ) is undefined.
  • E The function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 is discontinuous at π‘₯ = 1 .

Q12:

Let Determine the value of π‘Ž so that 𝑓 is continuous at π‘₯ = 2 . What can be said of the differentiability of 𝑓 at this point?

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

Q13:

Discuss the differentiability of the function 𝑓 ( π‘₯ ) at π‘₯ = 1 given 𝑓 ( π‘₯ ) = ( 6 π‘₯ βˆ’ 6 ) | 6 π‘₯ βˆ’ 6 | .

  • AThe function is differentiable at that point as 𝑓 β€² ( 1 ) = 𝑓 β€² ( 1 ) βˆ’ + .
  • BThe function is not differentiable at that point as 𝑓 β€² ( 1 ) β‰  𝑓 β€² ( 1 ) βˆ’ + .
  • CThe function is differentiable at π‘₯ = 1 as 𝑓 ( π‘₯ ) is continuous at that point.
  • DThe function is not differentiable at π‘₯ = 1 as 𝑓 ( π‘₯ ) is discontinuous at that point.

Q14:

Find the values of π‘Ž and 𝑏 given the function 𝑓 is differentiable at π‘₯ = 1 where

  • A π‘Ž = 1 , 𝑏 = βˆ’ 5
  • B π‘Ž = 3 , 𝑏 = 6
  • C π‘Ž = 4 , 𝑏 = 5
  • D π‘Ž = βˆ’ 1 , 𝑏 = 5

Q15:

Find the values of π‘Ž and 𝑏 given the function 𝑓 is differentiable at π‘₯ = βˆ’ 1 where

  • A π‘Ž = βˆ’ 9 , 𝑏 = βˆ’ 9
  • B π‘Ž = βˆ’ 1 8 , 𝑏 = 0
  • C π‘Ž = βˆ’ 4 , 𝑏 = 1
  • D π‘Ž = βˆ’ 1 0 , 𝑏 = βˆ’ 8

Q16:

Discuss the differentiability of the function 𝑓 at π‘₯ = βˆ’ 8 given

  • A 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 8 because 𝑓 ( π‘₯ ) is discontinuous at π‘₯ = βˆ’ 8 .
  • B 𝑓 ( π‘₯ ) is discontinuous but differentiable at π‘₯ = βˆ’ 8 because 𝑓 β€² ( βˆ’ 8 ) = 𝑓 β€² ( βˆ’ 8 ) + βˆ’ .
  • C 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = βˆ’ 8 because 𝑓 β€² ( βˆ’ 8 ) β‰  𝑓 β€² ( βˆ’ 8 ) + βˆ’ .
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 8 because 𝑓 ( βˆ’ 8 ) is undefined.
  • E 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 8 because 𝑓 ( π‘₯ ) is continuous at π‘₯ = βˆ’ 8 .

Q17:

Discuss the differentiability of the function 𝑓 at π‘₯ = 6 , given

  • A 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 6 .
  • B 𝑓 ( π‘₯ ) is discontinuous but differentiable at π‘₯ = 6 because 𝑓 β€² ( 6 ) = 𝑓 β€² ( 6 ) + βˆ’ .
  • C 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = 6 because 𝑓 β€² ( 6 ) β‰  𝑓 β€² ( 6 ) + βˆ’ .
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 6 because 𝑓 ( 6 ) is undefined.
  • E 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 6 because 𝑓 ( π‘₯ ) is discontinuous at π‘₯ = 6 .

Q18:

Discuss the differentiability of the function 𝑓 at π‘₯ = 1 given

  • A 𝑓 ( π‘₯ ) is differentiable at π‘₯ = 1 .
  • B 𝑓 ( π‘₯ ) is discontinuous but differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) = 𝑓 β€² ( 1 )   .
  • C 𝑓 ( π‘₯ ) is continuous but not differentiable at π‘₯ = 1 because 𝑓 β€² ( 1 ) β‰  𝑓 β€² ( 1 )   .
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 ( 1 ) is undefined.
  • E 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = 1 because 𝑓 is discontinuous at π‘₯ = 1 .

Q19:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = βˆ’ 8 ?

  • AThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 8 because 𝑓 β€² ( βˆ’ 8 ) β‰  𝑓 β€² ( βˆ’ 8 ) βˆ’ + .
  • BThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 8 because 𝑓 ( βˆ’ 8 ) is undefined.
  • CThe function 𝑓 ( π‘₯ ) is not continuous but differentiable at π‘₯ = βˆ’ 8 because 𝑓 β€² ( βˆ’ 8 ) = 𝑓 β€² ( βˆ’ 8 ) βˆ’ + .
  • DThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 8 as l i m l i m π‘₯ β†’ βˆ’ 8 π‘₯ β†’ βˆ’ 8 βˆ’ + 𝑓 ( π‘₯ ) = 𝑓 ( π‘₯ ) but is not continuous.
  • EThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 8 because 𝑓 is discontinuous at 𝑓 ( βˆ’ 8 ) .

Q20:

Suppose What can be said of the differentiability of 𝑓 at π‘₯ = βˆ’ 6 ?

  • AThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 6 because 𝑓 β€² ( βˆ’ 6 ) β‰  𝑓 β€² ( βˆ’ 6 ) βˆ’ + .
  • BThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 6 because 𝑓 ( βˆ’ 6 ) is undefined.
  • CThe function 𝑓 ( π‘₯ ) is not continuous but differentiable at π‘₯ = βˆ’ 6 because 𝑓 β€² ( βˆ’ 6 ) = 𝑓 β€² ( βˆ’ 6 ) βˆ’ + .
  • DThe function 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 6 as l i m l i m π‘₯ β†’ βˆ’ 6 π‘₯ β†’ βˆ’ 6 βˆ’ + 𝑓 ( π‘₯ ) = 𝑓 ( π‘₯ ) but is not continuous.
  • EThe function 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 6 because 𝑓 is discontinuous at 𝑓 ( βˆ’ 6 ) .

Q21:

Let If 𝑓 ( 1 ) = 1 2 and 𝑓 is continuous at π‘₯ = 1 , determine the values of π‘š and 𝑐 . What can be said of the differentiability of 𝑓 at this point?

  • A π‘š = βˆ’ 4 , 𝑐 = βˆ’ 4 , not differentiable at π‘₯ = 1
  • B π‘š = βˆ’ 4 , 𝑐 = βˆ’ 4 , differentiable at π‘₯ = 1
  • C π‘š = βˆ’ 1 2 , 𝑐 = βˆ’ 6 , not differentiable at π‘₯ = 1
  • D π‘š = βˆ’ 1 2 , 𝑐 = βˆ’ 6 , differentiable at π‘₯ = 1

Q22:

Discuss the differentiability of a function 𝑓 at π‘₯ = βˆ’ 4 given

  • A 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 4 because 𝑓 β€² ( βˆ’ 4 ) β‰  𝑓 β€² ( βˆ’ 4 )   .
  • B 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 4 because 𝑓 is continuous at π‘₯ = βˆ’ 4 .
  • C 𝑓 ( π‘₯ ) is differentiable at π‘₯ = βˆ’ 4 because 𝑓 β€² ( βˆ’ 4 ) = 𝑓 β€² ( βˆ’ 4 )   .
  • D 𝑓 ( π‘₯ ) is not differentiable at π‘₯ = βˆ’ 4 because 𝑓 ( βˆ’ 4 ) is undefined.

Q23:

Discuss the differentiability of the function 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + 1 π‘₯ at π‘₯ = βˆ’ 7 .

  • AThe function is differentiable at π‘₯ = βˆ’ 7 because 𝑓 β€² ( βˆ’ 7 ) exists.
  • BThe function is not differentiable at π‘₯ = βˆ’ 7 because 𝑓 ( π‘₯ ) is not continuous at that point.
  • CThe function is differentiable at π‘₯ = βˆ’ 7 because 𝑓 ( βˆ’ 7 ) exists.
  • DThe function is not differentiable at π‘₯ = βˆ’ 7 because 𝑓 β€² ( βˆ’ 7 ) does not exist.

Q24:

Let Determine the values of π‘Ž and 𝑏 so that 𝑓 is continuous at π‘₯ = 1 . What can be said of the differentiability of 𝑓 at this point?

  • A π‘Ž = βˆ’ 8 , 𝑏 = 4 , 𝑓 is not differentiable at π‘₯ = 1
  • B π‘Ž = βˆ’ 8 , 𝑏 = 4 , 𝑓 is differentiable at π‘₯ = 1
  • C π‘Ž = βˆ’ 1 0 , 𝑏 = 6 , 𝑓 is not differentiable at π‘₯ = 1
  • D π‘Ž = βˆ’ 1 0 , 𝑏 = 6 , 𝑓 is differentiable at π‘₯ = 1

Q25:

Discuss the continuity and differentiability of the function 𝑓 at π‘₯ = 0 given

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