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 dd𝑦𝑥 and dd𝑦𝑥 both positive?

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

Q2:

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

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

Q3:

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

  • A𝑓 has a local minimum at 𝑥=4.
  • B𝑓 has a horizontal tangent at 𝑥=4.
  • C𝑓 has a vertical tangent at 𝑥=4.
  • D𝑓 has an inflection point at 𝑥=4.
  • E𝑓 has a local maximum at 𝑥=4.

Q4:

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

  • A𝑓 has inflection points at 𝑥=4 and 𝑥=6.
  • B𝑓 has inflection points at 𝑥=2 and 𝑥=6.
  • C𝑓 has inflection points at 𝑥=3 and 𝑥=5.
  • D𝑓 has inflection points at 𝑥=2, 𝑥=4, and 𝑥=6.
  • E𝑓 has inflection points at 𝑥=1 and 𝑥=7.

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.

  • A𝑓 has inflection points at 𝑥=2.5 and 𝑥=4.
  • B𝑓 has inflection points at 𝑥=2 and 𝑥=6.
  • C𝑓 has inflection points at 𝑥=0, 𝑥=1, 𝑥=6, and 𝑥=8.
  • D𝑓 has inflection points at 𝑥=1, 𝑥=6, and 𝑥=8.
  • E𝑓 has inflection points at 𝑥=2, 𝑥=3, 𝑥=5, and 𝑥=7.

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?

  • A𝑓 has inflection points at 𝑥=1, 𝑥=2, 𝑥=3, 𝑥=5, and 𝑥=7.
  • B𝑓 has inflection points at 𝑥=4, 𝑥=6, and 𝑥=8.
  • C𝑓 has inflection points at 𝑥=1.5, 𝑥=2.5, 𝑥=4, and 𝑥=6.
  • D𝑓 has inflection points at 𝑥=0 and 𝑥=9.
  • E𝑓 has inflection points at 𝑥=4 and 𝑥=6.

Q7:

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

  • A𝑓 has inflection points when 𝑥=4 and 𝑥=6.
  • B𝑓 has inflection points when 𝑥=3 and 𝑥=5.
  • C𝑓 has inflection points when 𝑥=1 and 𝑥=7.
  • D𝑓 has inflection points when 𝑥=2,𝑥=4 and𝑥=6.
  • E𝑓 has inflection points when 𝑥=2 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.

  • A𝑓 has inflection points at 𝑥=2 and 𝑥=4.
  • B𝑓 has inflection points at 𝑥=1 and 𝑥=5.
  • C𝑓 has an inflection point at 𝑥=3.
  • D𝑓 has inflection points at 𝑥=2, 𝑥=4, and 𝑥=8.
  • E𝑓 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?

  • A𝑓 has a local minimum at 𝑥=3.
  • B𝑓 has a local maximum at 𝑥=1 and a local minimum at 𝑥=5.
  • C𝑓 has a local maximum at 𝑥=0 and a local minimum at 𝑥=6.
  • D𝑓 has a local maximum at 𝑥=5 and a local minimum at 𝑥=1.
  • E𝑓 has a local maximum at 𝑥=3.

Q10:

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

  • A𝑓 is increasing on the interval (3,6) and decreasing on the interval (0,3).
  • B𝑓 is increasing on the intervals (0,1) and (5,6) and decreasing on the interval (1,5).
  • C𝑓 is increasing on the interval (1,5) and decreasing on the intervals (0,1) and (5,6).
  • D𝑓 is decreasing on the interval (0,6).
  • E𝑓 is increasing on the interval (0,3) and decreasing on the interval (3,6).

Q11:

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

  • A𝑓 is increasing on the intervals (0,2), (3,4), and (6,8) and decreasing on the intervals (2,3) and (4,6).
  • B𝑓 is increasing on the intervals (0,1) and (5,7) and decreasing on the intervals (1,5) and (7,8).
  • C𝑓 is increasing on the intervals (1,5) and (7,8) and decreasing on the intervals (0,1) and (5,7).
  • D𝑓 is increasing on the intervals (2,3) and (4,6) and decreasing on the intervals (0,2), (3,4), and (6,8).
  • E𝑓 is increasing on the intervals (1,2), (3,5), and (7,8) and decreasing on the intervals (0,1), (2,3), and (5,7).

Q12:

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

  • Apoint 𝐷
  • Bpoint 𝐵
  • Cpoint 𝐶
  • 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?

  • A𝑓 is concave upward on (1,2), (3,5), and (7,9) and concave downward on (0,1), (2,3), and (5,7).
  • B𝑓 is concave upward on (0,4) and (6,8) and concave downward on (4,6) and (8,9).
  • C𝑓 is concave upward on (4,6) and (8,9) and concave downward on (0,4) and (6,8).
  • D𝑓 is concave upward on (4,6) and (8,9) and concave downward on (1,4) and (6,8).
  • E𝑓 is concave upward on (0,1), (2,3), and (5,7) and concave downward on (1,2), (3,5), and (7,9).

Q14:

The graph of the function 𝑦=𝑓(𝑥) is shown. Determine over which intervals the function 𝑓(𝑥) is positive.

  • A[1,2] and [5,7]
  • B(,1), (2,5), and (7,)
  • CWe need more information about 𝑓(𝑥).
  • D(,1], [2,5], and [7,)
  • E(1,2) and (5,7)

Q15:

The graph of the function 𝑦=𝑓(𝑥) is shown. Determine over which intervals the function 𝑓(𝑥) is concave upward.

  • A(2,2)
  • B[2,1], [3,5], and [7,)
  • C(,2], [1,3], and [5,7]
  • D(,2), (1,3), and (5,7)
  • E(2,1), (3,5), and (7,)

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