Worksheet: Interpreting Graphs of Derivatives

In this worksheet, we will practice connecting a function to the graphs of its first and second derivatives.

Q1:

The graph of a function 𝑦 = 𝑓 ( 𝑥 ) is shown. At which point are d d 𝑦 𝑥 and d d 𝑦 𝑥 both positive?

  • Apoint 𝐶
  • Bpoint 𝐴
  • Cpoint 𝐵
  • Dpoint 𝐷
  • Epoint 𝐸

Q2:

The graph of a function 𝑦 = 𝑓 ( 𝑥 ) is shown. At which point are d d 𝑦 𝑥 and d d 𝑦 𝑥 both negative?

  • Apoint 𝐴
  • Bpoint 𝐷
  • Cpoint 𝐵
  • Dpoint 𝐸
  • Epoint 𝐶

Q3:

Given that 𝑓 ( 4 ) = 0 and 𝑓 ( 4 ) = 0 , which of the following must be true?

  • A 𝑓 has a local maximum at 𝑥 = 4 .
  • B 𝑓 has a local minimum at 𝑥 = 4 .
  • C 𝑓 has an inflection point at 𝑥 = 4 .
  • D 𝑓 has a horizontal tangent at 𝑥 = 4 .
  • E 𝑓 has a vertical tangent at 𝑥 = 4 .

Q4:

Use the given graph of a function 𝑓 to find the 𝑥 -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at 𝑥 = 1 and 𝑥 = 7 .
  • B 𝑓 has inflection points at 𝑥 = 2 , 𝑥 = 4 , and 𝑥 = 6 .
  • C 𝑓 has inflection points at 𝑥 = 2 and 𝑥 = 6 .
  • D 𝑓 has inflection points at 𝑥 = 4 and 𝑥 = 6 .
  • E 𝑓 has inflection points at 𝑥 = 3 and 𝑥 = 5 .

Q5:

The graph of the first derivative 𝑓 of a continuous function 𝑓 is shown. State the 𝑥 -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at 𝑥 = 2 . 5 and 𝑥 = 4 .
  • B 𝑓 has inflection points at 𝑥 = 2 and 𝑥 = 6 .
  • C 𝑓 has inflection points at 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 6 , and 𝑥 = 8 .
  • D 𝑓 has inflection points at 𝑥 = 2 , 𝑥 = 3 , 𝑥 = 5 , and 𝑥 = 7 .
  • E 𝑓 has inflection points at 𝑥 = 1 , 𝑥 = 6 , and 𝑥 = 8 .

Q6:

The graph of the first derivative 𝑓 of a function 𝑓 is shown. What are the 𝑥 -coordinates of the inflection points of 𝑓 ?

  • A 𝑓 has inflection points at 𝑥 = 4 and 𝑥 = 6 .
  • B 𝑓 has inflection points at 𝑥 = 1 . 5 , 𝑥 = 2 . 5 , 𝑥 = 4 , and 𝑥 = 6 .
  • C 𝑓 has inflection points at 𝑥 = 4 , 𝑥 = 6 , and 𝑥 = 8 .
  • D 𝑓 has inflection points at 𝑥 = 0 and 𝑥 = 9 .
  • E 𝑓 has inflection points at 𝑥 = 1 , 𝑥 = 2 , 𝑥 = 3 , 𝑥 = 5 , and 𝑥 = 7 .

Q7:

Using the given graph of the function 𝑓 , at what values of 𝑥 does 𝑓 have inflection points?

  • A 𝑓 has inflection points when 𝑥 = 1 and 𝑥 = 7 .
  • B 𝑓 has inflection points when 𝑥 = 4 and 𝑥 = 6 .
  • C 𝑓 has inflection points when 𝑥 = 2 and 𝑥 = 6 .
  • D 𝑓 has inflection points when 𝑥 = 3 and 𝑥 = 5 .
  • E 𝑓 has inflection points when 𝑥 = 2 , 𝑥 = 4 and 𝑥 = 6 .

Q8:

The graph of the first derivative 𝑓 of a continuous function 𝑓 is shown. State the 𝑥 -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at 𝑥 = 1 and 𝑥 = 5 .
  • B 𝑓 has inflection points at 𝑥 = 2 and 𝑥 = 4 .
  • C 𝑓 has inflection points at 𝑥 = 2 , 𝑥 = 4 , and 𝑥 = 8 .
  • D 𝑓 has an inflection point at 𝑥 = 3 .
  • E 𝑓 has an inflection point at 𝑥 = 6 .

Q9:

The graph of the derivative 𝑓 of a function 𝑓 is shown. At what values of 𝑥 does 𝑓 have a local maximum or minimum?

  • A 𝑓 has a local maximum at 𝑥 = 1 and a local minimum at 𝑥 = 5 .
  • B 𝑓 has a local maximum at 𝑥 = 3 .
  • C 𝑓 has a local maximum at 𝑥 = 5 and a local minimum at 𝑥 = 1 .
  • D 𝑓 has a local maximum at 𝑥 = 0 and a local minimum at 𝑥 = 6 .
  • E 𝑓 has a local minimum at 𝑥 = 3 .

Q10:

The graph of the derivative 𝑓 of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

  • A 𝑓 is increasing on the interval ( 0 , 3 ) and decreasing on the interval ( 3 , 6 ) .
  • B 𝑓 is increasing on the interval ( 3 , 6 ) and decreasing on the interval ( 0 , 3 ) .
  • C 𝑓 is increasing on the interval ( 1 , 5 ) and decreasing on the intervals ( 0 , 1 ) and ( 5 , 6 ) .
  • D 𝑓 is increasing on the intervals ( 0 , 1 ) and ( 5 , 6 ) and decreasing on the interval ( 1 , 5 ) .
  • E 𝑓 is decreasing on the interval ( 0 , 6 ) .

Q11:

The graph of the derivative 𝑓 of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

  • A 𝑓 is increasing on the intervals ( 0 , 1 ) and ( 5 , 7 ) and decreasing on the intervals ( 1 , 5 ) and ( 7 , 8 ) .
  • B 𝑓 is increasing on the intervals ( 2 , 3 ) and ( 4 , 6 ) and decreasing on the intervals ( 0 , 2 ) , ( 3 , 4 ) , and ( 6 , 8 ) .
  • C 𝑓 is increasing on the intervals ( 1 , 5 ) and ( 7 , 8 ) and decreasing on the intervals ( 0 , 1 ) and ( 5 , 7 ) .
  • D 𝑓 is increasing on the intervals ( 1 , 2 ) , ( 3 , 5 ) , and ( 7 , 8 ) and decreasing on the intervals ( 0 , 1 ) , ( 2 , 3 ) , and ( 5 , 7 ) .
  • E 𝑓 is increasing on the intervals ( 0 , 2 ) , ( 3 , 4 ) , and ( 6 , 8 ) and decreasing on the intervals ( 2 , 3 ) and ( 4 , 6 ) .

Q12:

The graph of a function 𝑦 = 𝑓 ( 𝑥 ) is shown. At which point is d d 𝑦 𝑥 negative but d d 𝑦 𝑥 positive?

  • Apoint 𝐶
  • Bpoint 𝐵
  • Cpoint 𝐷
  • Dpoint 𝐴
  • Epoint 𝐸

Q13:

The graph of the first derivative 𝑓 of a function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

  • A 𝑓 is concave upward on ( 4 , 6 ) and ( 8 , 9 ) and concave downward on ( 1 , 4 ) and ( 6 , 8 ) .
  • B 𝑓 is concave upward on ( 4 , 6 ) and ( 8 , 9 ) and concave downward on ( 0 , 4 ) and ( 6 , 8 ) .
  • C 𝑓 is concave upward on ( 0 , 1 ) , ( 2 , 3 ) , and ( 5 , 7 ) and concave downward on ( 1 , 2 ) , ( 3 , 5 ) , and ( 7 , 9 ) .
  • D 𝑓 is concave upward on ( 1 , 2 ) , ( 3 , 5 ) , and ( 7 , 9 ) and concave downward on ( 0 , 1 ) , ( 2 , 3 ) , and ( 5 , 7 ) .
  • E 𝑓 is concave upward on ( 0 , 4 ) and ( 6 , 8 ) and concave downward on ( 4 , 6 ) and ( 8 , 9 ) .

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