Worksheet: Graphing Using Derivatives

In this worksheet, we will practice using derivatives to graph a function.

Q1:

Which of the following is the graph of 𝑓(𝑥)=(𝑥+1)(𝑥2)?

  • A
  • B
  • C
  • D
  • E

Q2:

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(𝑥5)(𝑥+1)(3𝑥7)
  • B164(𝑥5)(𝑥+1)(3𝑥7)
  • C(𝑥5)(𝑥+1)(3𝑥7)
  • D164(𝑥5)(𝑥+1)(3𝑥7)
  • E164(𝑥5)(𝑥+1)(3𝑥7)

Determine the intervals where 𝑃(𝑥)<1.

  • A4753,11,47+53
  • B,475347+53,
  • C4753,47+53
  • D4753,1(1,)
  • E,47531,47+53

Q3:

Which of the following is the graph of the function 𝑓(𝑥)=𝑥+6𝑥9𝑥+1?

  • A
  • B
  • C
  • D
  • E

Q4:

Which of the following could be the graph of the function 𝑓(𝑥)=(𝑥2)? Use derivatives to sketch the function.

  • A
  • B
  • C
  • D
  • E

Q5:

Which of the following is the graph of the function 𝑓(𝑥)=110𝑥+10𝑥?

  • A
  • B
  • C
  • D
  • E

Q6:

Which of the following is the graph of the function 𝑓(𝑥)=𝑥+𝑥6?

  • A
  • B
  • C
  • D
  • E

Q7:

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

  • A
  • B
  • C
  • D
  • E

Q8:

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

  • A
  • B
  • C
  • D
  • E

Q9:

Consider the function 𝑓(𝑥)=𝑥𝑥+25𝑥10.

Find all asymptotes of 𝑓.

  • A𝑓 has a horizontal asymptote at 𝑥=2 and an oblique asymptote at 𝑦=15𝑥.
  • B𝑓 has a horizontal asymptote at 𝑦=2 and an oblique asymptote at 𝑦=15𝑥+15.
  • C𝑓 has a vertical asymptote at 𝑦=2 and an oblique asymptote at 𝑦=15𝑥35𝑥.
  • D𝑓 has a vertical asymptote at 𝑥=2 and an oblique asymptote at 𝑦=15𝑥35𝑥.
  • E𝑓 has a vertical asymptote at 𝑥=2 and an oblique asymptote at 𝑦=15𝑥+15.

Find 𝑓(𝑥).

  • A5𝑥20𝑥(5𝑥10)
  • B20𝑥5𝑥(5𝑥10)
  • C10𝑥+35𝑥15(5𝑥10)
  • D10𝑥35𝑥+15(5𝑥10)
  • E2𝑥15

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 lim𝑓(𝑥).

  • A
  • B
  • C0
  • D15
  • E15

Which of the following could be the graph of 𝑓?

  • A
  • B
  • C
  • D
  • E

Q10:

Consider the function 𝑓(𝑥)=(𝑥1)(𝑥+2).

Find 𝑓(𝑥).

  • A4𝑥+1
  • B𝑥+2𝑥
  • C2𝑥+1
  • D2𝑥2
  • E3𝑥3

Find and classify the critical points of 𝑓.

  • A𝑓 has a local minimum at (1,0) and a local maximum at (1,4).
  • B𝑓 has a local maximum at (1,0) and a local minimum at (1,4).
  • C𝑓 has a local minimum at (12,278).
  • D𝑓 has a local maximum at (12,278).
  • E𝑓 has a local minimum at (0,2) and a local maximum at (2,0).

Find the intervals of increase and decrease for 𝑓.

  • A𝑓 is increasing on (,2) and (0,) and decreasing on (2,0).
  • B𝑓 is increasing on (1,1) and decreasing on (,1) and (1,).
  • C𝑓 is increasing on 12, and decreasing on ,12.
  • D𝑓 is increasing on (,1) and (1,) and decreasing on (1,1).
  • E𝑓 is increasing on ,12 and decreasing on 12,.

Find lim𝑓(𝑥).

  • A
  • B0
  • C1
  • D
  • E2

Which of the following is the graph of 𝑓?

  • A
  • B
  • C
  • D
  • E

Q11:

Consider a function 𝑓 whose first derivative 𝑓 is defined as 𝑓(𝑥)=4(𝑥+6)(5𝑥)(𝑥3)(𝑥+2).

Which of the following statements is correct?

  • A𝑓 is decreasing on (,6), (2,3), and (3,5).
  • B𝑓 is decreasing on (6,2) and (3,5).
  • C𝑓 is decreasing on (,6) only.
  • D𝑓 is decreasing on (,6) and (2,5).
  • E𝑓 is decreasing on (2,3) and (3,5) only.

Q12:

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

Q13:

Consider a function 𝑓 whose first derivative is 𝑓(𝑥)=(𝑥+11)(𝑥7)𝑥3. On which interval(s) is 𝑓 increasing?

  • A(11,7)
  • B(11,3) and (7,+)
  • C(11,+)
  • D(,+)
  • E(11,7) and [5,+)

Q14:

Consider the function 𝑓(𝑥)=4𝑥+5𝑥102𝑥+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

Q15:

Which of the following could be the graph of the function 𝑓(𝑥)=𝑥4𝑥?

  • A
  • B
  • C
  • D
  • E

Q16:

Sketch the graph of 𝑓(𝑥)=3𝑥𝑥4.

  • A
  • B
  • C
  • D
  • E

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