Lesson Worksheet: Curve Sketching using Derivatives Mathematics

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In this worksheet, we will practice using derivatives to graph different functions.

Q1:

Consider the polynomial function whose graph 𝑦=𝑃(𝑥) is given below.

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

  • A164(𝑥−5)(𝑥+1)(3𝑥−7)
  • B164(𝑥−5)(𝑥+1)(3𝑥−7)
  • C(𝑥−5)(𝑥+1)(3𝑥−7)
  • D164(𝑥−5)(𝑥+1)(3𝑥−7)
  • E164(𝑥−5)(𝑥+1)(3𝑥−7)

Determine the intervals where 𝑃(𝑥)<1.

  • A−4√7−53,1∪1,4√7+53
  • B−∞,−4√7−53∪4√7+53,∞
  • C−4√7−53,4√7+53
  • D−4√7−53,1∪(1,∞)
  • E−∞,−4√7−53∪1,4√7+53

Q2:

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

  • A
  • B
  • C
  • D
  • E

Q3:

Which of the following is the graph of the function 𝑓(𝑥)=√𝑥+𝑥−6?

  • A
  • B
  • C
  • D
  • E

Q4:

Which of the following is the graph of the function 𝑓(𝑥)=2𝑥2𝜋𝑥sin on the interval [−1,1]?

  • A
  • B
  • C
  • D
  • E

Q5:

Which of the following could be the graph of the function 𝑓(𝑥)=(𝑥−2)? Use derivatives to sketch the function.

  • A
  • B
  • C
  • D
  • E

Q6:

Consider the function 𝑓(𝑥)=𝑥−𝑥+25𝑥−10.

Find 𝑓(𝑥).

  • A5𝑥−20𝑥(5𝑥−10)
  • B20𝑥−5𝑥(5𝑥−10)
  • C−10𝑥+35𝑥−15(5𝑥−10)
  • D10𝑥−35𝑥+15(5𝑥−10)
  • E2𝑥−15

Find and classify all critical points of 𝑓.

  • A𝑓 has a local minimum at 𝑥=3 and a local maximum at 𝑥=4.
  • B𝑓 has a local minimum at 𝑥=12 and a local maximum at 𝑥=3.
  • C𝑓 has neither a local maximum nor a local minimum at 𝑥=2.
  • D𝑓 has a local maximum at 𝑥=0 and a local minimum at 𝑥=4.
  • E𝑓 has a local maximum at 𝑥=12 and a local minimum at 𝑥=3.

Find the intervals of increase and decrease for 𝑓.

  • A𝑓 is increasing on (0,2) and (2,4) and decreasing on (−∞,0) and (4,∞).
  • B𝑓 is increasing on (−∞,0) and (4,∞) and decreasing on (0,2) and (2,4).
  • C𝑓 is increasing on −∞,12 and (3,∞) and decreasing on 12,2 and (2,3).
  • D𝑓 is increasing on (−∞,−4) and (0,∞) and decreasing on (−4,0).
  • E𝑓 is increasing on 12,2 and (2,3) and decreasing on −∞,12 and (3,∞).

Find lim→∞𝑓(𝑥).

  • A∞
  • B−∞
  • C0
  • D−15
  • E15

Which of the following could be the graph of 𝑓?

  • A
  • B
  • C
  • D
  • E

Q7:

Consider the function 𝑓(𝑥)=162𝑥+𝑥sin.

Use the second derivative of 𝑓 to determine the 𝑥-values for which the concavity of the graph of 𝑓 changes. Which of the following graphs is the graph of 𝑓?

  • A
  • B
  • C
  • D

Q8:

Consider the function 𝑓(𝑥)=4𝑥+5𝑥−102𝑥+5.

Find the equation of the oblique asymptote of the graph of 𝑓 to decide which of the following graphs is the graph of 𝑓.

  • A
  • B
  • C
  • D

Q9:

Which of the following could be the graph of the function 𝑓(𝑥)=𝑥√4−𝑥?

  • A
  • B
  • C
  • D
  • E

Q10:

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

  • A
  • B
  • C
  • D
  • E

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