## Review Exercises for Chapter 4 inApplied CalculusChapter 11 inFinite Mathematics & Applied CalculusTopic: Techniques of Differentiation

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Note Use proper graphing calculator format to input your answers (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.

Evaluate dy/dx for each of the following.

1. y=
 x2(3x-1)
dy

dx
=
2. y=
 x-1(2x+1)
dy

dx
=
3. y=
 2x+12x-1
dy

dx
=
4. y=
 3x+2x2-1
dy

dx
=
5. y=
 (x2-2)ex
dy

dx
=
6. y=
 x2ln x
dy

dx
=
7. y=
 ex - e-xex
dy

dx
=
8. y=
 (2x2 - 2x)-4
dy

dx
=
9. y=
 (ln x - ex)3
dy

dx
=
10. y=
 ln (30x)
dy

dx
=
11. y=
 x+1(2x+1)2
dy

dx
=
12. y=
 x+12x-1 3
dy

dx
=
13. y=
 ln(x2-x)
dy

dx
=
14. y=
 ex2-x - x
dy

dx
=
15. y=
 2x - 1ex 2
dy

dx
=

Question 16

Daily sales of neon red Duffin Whallop Deluxe golfball gift-packs follows the formula

 s(t) = 20,000 - 20,0001 + 20,000e-0.2t ,

where t is time in days since January 1, and s(t) represents the number of Duffin Whallop Deluxe gift-packs sold per day at time t.

(a) Obtain a formula for s'(t), and hence estimate how fast sales were dropping at the end of March (t = 90).

(b) How fast were sales dropping 10 days later? Approximately how many gift-packs were sold that day?

Question 17

If \$5,000 is invested in a mutual fund whose value is growing at a rate of 15% per year (compounded once per year), how fast is the investment growing at the end of two years? (Round your answer to the nearest cent.)

Answer: The investment is increasing at a rate of \$ per year.

Question 18

The accompanying graph shows the number of transactions handled by Western Union for the years 1987-1994.*

* Source: Company Reports/The New York Times, September 24, 1994.

These data can be approximated by

Q(t) = 9.284e0.117t,

where t is time in years since 1987, and Q(t) is the number of Western Union transactions (in millions) each year.

(a) According to the model, how fast was Western Union's business (as measured by annual transactions) growing in 1990? (Round your answer to the nearest 0 01 million, as in 463.23 million)
Answer: growing at a rate of million transactions each year.

(b) What is the first year that the actual increase (from that year to the next) exceeded the rate of increase that year as predicted by the model?

Question 19 The weekly cost of manufacturing x pairs of Gauss-Jordan Sneakers is

C(x) = -0.00012x2 + 30x + 50,000 dollars.

(a) What is the marginal cost at a sales level of 1000 pairs per week?

 \$ per pair
(b) What is the average cost per pair at a sales level of 1000 pairs per week?
 \$ per pair

 (c) At the above sales level, the marginal cost is increasing decreasing at a rate of \$ per pair.

 (d) At the above sales level, the marginal cost is greater than Approximately equal to     less than the average cost.

 (e) In general, if the marginal cost is increasing, and there is no fixed cost, then the marginal cost is greater than Approximately equal to     less than the average cost.

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Last Updated:March, 2003
Copyright © 1999 StefanWaner and Steven R. Costenoble