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Lesson: Graphing Using Derivative

Sample Question Videos

Worksheet • 19 Questions • 1 Video

Q1:

Which of the following is the graph of 𝑓 ( 𝑥 ) = ( 𝑥 + 1 ) ( 𝑥 2 ) 3 ?

  • A
  • B
  • C
  • D
  • E

Q2:

Given that the curve 𝑥 𝑦 + 𝑎 𝑥 + 𝑏 𝑦 = 0 2 has an inflection point at ( 3 , 2 ) , what are the values of constants 𝑎 and 𝑏 ?

  • A 𝑎 = 8 , 𝑏 = 3
  • B 𝑎 = 8 , 𝑏 = 3
  • C 𝑎 = 2 4 , 𝑏 = 3
  • D 𝑎 = 4 , 𝑏 = 3

Q3:

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

Q4:

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

  • A is increasing on the interval and decreasing on the intervals and .
  • B is decreasing on the interval .
  • C is increasing on the intervals and and decreasing on the interval .
  • D is increasing on the interval and decreasing on the interval .
  • E is increasing on the interval and decreasing on the interval .

Q5:

Use the given graph of to find all possible intervals on which is increasing.

  • A is increasing on and .
  • B is increasing on only.
  • C is increasing on and .
  • D is increasing on only.
  • E is increasing on and .

Q6:

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

  • A is increasing on the intervals and and decreasing on the intervals and .
  • B is increasing on the intervals , , and and decreasing on the intervals , , and .
  • C is increasing on the intervals and and decreasing on the intervals and .
  • D is increasing on the intervals , , and and decreasing on the intervals and .
  • E is increasing on the intervals and and decreasing on the intervals , , and .

Q7:

Use the given graph of to find all possible intervals on which is decreasing.

  • A is decreasing on .
  • B is decreasing on and .
  • C is decreasing on and .
  • D is decreasing on , , and .
  • E is decreasing on .

Q8:

Use the given graph of to find all possible intervals on which is increasing.

  • A is increasing on and .
  • B is increasing on .
  • C is increasing on and .
  • D is increasing on .
  • E is increasing on .

Q9:

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

Q10:

Using the graph, determine the intervals of increase and decrease of the function.

  • A 𝑓 ( 𝑥 ) is increasing on the interval ] , 6 [ and decreasing on the interval ] 6 , [ .
  • B 𝑓 ( 𝑥 ) is increasing on .
  • C 𝑓 ( 𝑥 ) is increasing on the interval ] 6 , [ and decreasing on the interval ] , 6 [ .
  • D 𝑓 ( 𝑥 ) is increasing on the interval ] , 7 [ and decreasing on the interval ] 7 , [ .
  • E 𝑓 ( 𝑥 ) is increasing on the interval ] 7 , [ and decreasing on the interval ] , 7 [ .

Q11:

Using the graph, determine the intervals of increase and decrease of the function.

  • A 𝑓 ( 𝑥 ) is increasing on the interval ] , 1 [ and decreasing on the interval ] 1 , [ .
  • B 𝑓 ( 𝑥 ) is increasing on .
  • C 𝑓 ( 𝑥 ) is increasing on the interval ] 1 , [ and decreasing on the interval ] , 1 [ .
  • D 𝑓 ( 𝑥 ) is increasing on the interval ] , 6 [ and decreasing on the interval ] 6 , [ .
  • E 𝑓 ( 𝑥 ) is increasing on the interval ] 6 , [ and decreasing on the interval ] , 6 [ .

Q12:

Using the graph, determine the intervals of increase and decrease of the function.

  • A 𝑓 ( 𝑥 ) is increasing on the interval ] 4 , [ and decreasing on the interval ] , 4 [ .
  • B 𝑓 ( 𝑥 ) is increasing on .
  • C 𝑓 ( 𝑥 ) is increasing on the interval ] , 4 [ and decreasing on the interval ] 4 , [ .
  • D 𝑓 ( 𝑥 ) is increasing on the interval ] , 1 [ and decreasing on the interval ] 1 , [ .
  • E 𝑓 ( 𝑥 ) is increasing on the interval ] 1 , [ and decreasing on the interval ] , 1 [ .

Q13:

Consider the polynomial function whose graph is given below.

Use the given points and the fact that is a critical point of the function to determine .

  • A
  • B
  • C
  • D
  • E

Determine the intervals where .

  • A
  • B
  • C
  • D
  • E

Q14:

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

Q15:

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

Q16:

Using the graph, discuss the monotony of the function.