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Worksheet: Graphing Using Derivative

Q1:

Which of the following is the graph of 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ + 1 ) ( π‘₯ βˆ’ 2 ) 3 ?

  • A
  • B
  • C
  • D
  • E

Q2:

Given that the curve π‘₯ 𝑦 + π‘Ž π‘₯ + 𝑏 𝑦 = 0 2 has an inflection point at ( 3 , βˆ’ 2 ) , what are the values of constants π‘Ž and 𝑏 ?

  • A π‘Ž = βˆ’ 4 , 𝑏 = 3
  • B π‘Ž = βˆ’ 8 , 𝑏 = 3
  • C π‘Ž = 2 4 , 𝑏 = 3
  • D π‘Ž = 8 , 𝑏 = 3

Q3:

Use the given graph of a function 𝑓 β€² β€² to find the π‘₯ -coordinates of the inflection points of 𝑓 .

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

Q4:

The graph of the derivative 𝑓 β€² 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 intervals ( 0 , 1 ) and ( 5 , 6 ) and decreasing on the interval ( 1 , 5 ) .
  • C 𝑓 is increasing on the interval ( 3 , 6 ) and decreasing on the interval ( 0 , 3 ) .
  • D 𝑓 is increasing on the interval ( 1 , 5 ) and decreasing on the intervals ( 0 , 1 ) and ( 5 , 6 ) .
  • E 𝑓 is decreasing on the interval ( 0 , 6 ) .

Q5:

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

  • A 𝑓 is increasing on [ 0 , 1 ] and [ 5 , 7 ] .
  • B 𝑓 is increasing on [ 0 , 1 ] and [ 3 , 5 ] .
  • C 𝑓 is increasing on [ 1 , 3 ] only.
  • D 𝑓 is increasing on [ 0 , 1 ] and [ 3 , 7 ] .
  • E 𝑓 is increasing on [ 3 , 7 ] only.

Q6:

The graph of the derivative 𝑓 β€² of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

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

Q7:

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

  • A 𝑓 is decreasing on ( 5 , 7 ) .
  • B 𝑓 is decreasing on ( 0 , 1 ) and ( 3 , 5 ) .
  • C 𝑓 is decreasing on ( 0 , 1 ) , ( 3 , 5 ) , and ( 5 , 7 ) .
  • D 𝑓 is decreasing on ( 1 , 3 ) .
  • E 𝑓 is decreasing on ( 1 , 3 ) and ( 5 , 7 ) .

Q8:

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

  • A 𝑓 is increasing on [ 1 , 3 ] .
  • B 𝑓 is increasing on [ 0 , 1 ] and [ 3 , 4 ] .
  • C 𝑓 is increasing on [ 4 , 6 ] .
  • D 𝑓 is increasing on [ 1 , 3 ] and [ 4 , 6 ] .
  • E 𝑓 is increasing on [ 3 , 4 ] .

Q9:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. State the π‘₯ -coordinates of the inflection points of 𝑓 .

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

Q10:

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

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

Q11:

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

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

Q12:

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

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

Q13:

Consider the polynomial function whose graph 𝑦 = 𝑃 ( π‘₯ ) is given below.

Use the given points and the fact that π‘₯ = 1 is a critical point of the function 𝑃 to determine 𝑃 ( π‘₯ ) .

  • A 1 6 4 ( π‘₯ βˆ’ 5 ) ( π‘₯ + 1 ) ( 3 π‘₯ βˆ’ 7 ) 2
  • B 1 6 4 ( π‘₯ βˆ’ 5 ) ( π‘₯ + 1 ) ( 3 π‘₯ βˆ’ 7 )
  • C 1 6 4 ( π‘₯ βˆ’ 5 ) ( π‘₯ + 1 ) ( 3 π‘₯ βˆ’ 7 ) 2
  • D 1 6 4 ( π‘₯ βˆ’ 5 ) ( π‘₯ + 1 ) ( 3 π‘₯ βˆ’ 7 ) 2
  • E ( π‘₯ βˆ’ 5 ) ( π‘₯ + 1 ) ( 3 π‘₯ βˆ’ 7 ) 2

Determine the intervals where 𝑃 ( π‘₯ ) < 1 .

  • A ο€Ώ βˆ’ 4 √ 7 βˆ’ 5 3 , 1  βˆͺ ο€Ώ 1 , 4 √ 7 + 5 3 
  • B ο€Ώ βˆ’ 4 √ 7 βˆ’ 5 3 , 1  βˆͺ ( 1 , ∞ )
  • C ο€Ώ βˆ’ ∞ , βˆ’ 4 √ 7 βˆ’ 5 3  βˆͺ ο€Ώ 1 , 4 √ 7 + 5 3 
  • D ο€Ώ βˆ’ 4 √ 7 βˆ’ 5 3 , 4 √ 7 + 5 3 
  • E ο€Ώ βˆ’ ∞ , βˆ’ 4 √ 7 βˆ’ 5 3  βˆͺ ο€Ώ 4 √ 7 + 5 3 , ∞ 

Q14:

The graph of the derivative 𝑓 β€² of a function 𝑓 is shown. At what values of π‘₯ does 𝑓 have a local maximum or minimum?

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

Q15:

The graph of the first derivative 𝑓 β€² of a function 𝑓 is shown. What are the π‘₯ -coordinates of the inflection points of 𝑓 ?

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

Q16:

Using the graph, discuss the monotony of the function.

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

Q17:

Using the given graph of the function 𝑓 , at what values of π‘₯ does 𝑓 have inflection points?

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

Q18:

Using the given graph of the function 𝑓 β€² , at what values of π‘₯ does 𝑓 have inflection points?

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

Q19:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. State the π‘₯ -coordinates of the inflection points of 𝑓 .

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