Calculus 16 questions

Exam 3 Review

1. Find the second derivative.

a. 𝑓(𝑥) = 𝑥√36 − 𝑥2

b. 𝑓(𝑥) = 4√𝑥 − 3

c. 𝑓(𝑥) = 18𝑥3 − 32𝑥2

2. Find the critical numbers of the function. Then find the open intervals on which the function is

increasing or decreasing.

a. 𝑓(𝑥) = 1

4 𝑥4 −

1

3 𝑥3 − 6𝑥2

b. 𝑓(𝑥) = (𝑥 + 3)4

c. 𝑓(𝑥) = √25 − 𝑥2 Hint: Domain of 𝑓(𝑥) is [−5,5]

3. Use the first derivative test to find the relative extrema of the function.

a. 𝑓(𝑥) = 2𝑥3 − 3𝑥2 − 36𝑥 + 14

b. 𝑓(𝑥) = 𝑥4 − 𝑥3

4. Find the absolute extrema of the function on the closed interval.

a. 𝑓(𝑥) = 𝑥

𝑥−2 , [3,5]

b. 𝑓(𝑥) = 𝑥3 − 3𝑥, [0,4]

5. Use the second derivative test to find the relative extrema of the function.

a. 𝑓(𝑥) = 𝑥 + 16

𝑥

b. 𝑓(𝑥) = √4 − 𝑥2 Hint: Domain of 𝑓(𝑥) is [−2,2]

6. Determine the open intervals on which the function is concave upward or concave downward.

a. 𝑓(𝑥) = 𝑥(𝑥 − 4)3

b. 𝑓(𝑥) = −𝑥3 + 3𝑥2 − 5

7. Find the number of units 𝑥 that produces a maximum revenue 𝑅.

𝑅 = 33𝑥2 − 0.02𝑥3

8. Find the number of units 𝑥 that produces the minimum average cost per unit 𝐶 ̅.

𝐶 = 0.001𝑥3 + 2𝑥 + 54

9. A commodity has a demand function modeled by 𝑝 = 107 − 0.5𝑥 and a total cost function

modeled by 𝐶 = 30𝑥 + 32.75, where 𝑥 is the number of units.

a. What price yields a maximum profit?

b. When the profit is maximized, what is the average cost per unit?

10. Find the vertical and horizontal asymptotes.

𝑓(𝑥) = 𝑥2−3

2𝑥2−18

11. Find the horizontal asymptote of the graph of the function.

a. 𝑓(𝑥) = 𝑥3−6𝑥2+9𝑥+1

𝑥2−9𝑥+6

b. 𝑓(𝑥) = 2𝑥2−4𝑥−7

1−5𝑥−6𝑥2

12. Evaluate each limit.

a. lim 𝑥→∞

(−7𝑥2 + 2𝑥 − 8)

b. lim 𝑥→∞

(3 + 4

𝑥2 )

c. lim 𝑥→∞

4−5𝑥3

2𝑥+𝑥3

13. Compare the value of 𝑑𝑦 and 𝛥𝑦 for the function 𝑓(𝑥) = 0.6𝑥2 with 𝑥 = 6 and 𝛥𝑥 = 𝑑𝑥 =

−0.1

14. Use differential to approximate the change in cost corresponding to an increase in sales of one

unit. Then compare the actual change in cost.

𝐶 = 0.025𝑥2 + 9𝑥 + 4 , 𝑥 = 10

15. Use differential to approximate the change in profit corresponding to an increase in sales of one

unit. Then compare this with the actual change in profit.

𝑃 = −0.2𝑥3 + 500𝑥 − 50 , 𝑥 = 40

16. Find the differential 𝑑𝑦.

a. 𝑦 = 𝑥+2

9𝑥−1

b. 𝑦 = 5𝑥3 − 6𝑥