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Question 1

The annual profit of Lotus Development Corp. for the years 1990-1994 are shown in the following chart. (Losses are shown as negative.)

During which one-year interval was the average rate of change of annual profits

(a) greatest
(b) least? Interpret your answers by referring to the rates of change.

† Figures are approximate. Source: Datastream: Company Reports/The New York Times, Jun 6, 1995, p. D8.

Question 2

Evaluate for each of the following.

(a) y = 4 4x1.2 + 4/x1.2 1/(2x)
(b) y = (2x2 4x)/(x + e2x)
(c) y = (2x ln x)-2
(d) y = e2x x2
(e) y = 2xe2x(x 2)
(f) y = [(2x1)/ex] - 2
(g) y = ln(2x x2)

Question 3

Annual sales of Eli Lilly Corp.'s human growth hormone Humatrope can be modeled by

S(t) = 0.4 / (0.0025 + 160e-2.7t) (0 t 10)
where S(t) represents annual sales in millions of dollars, and t represents the number of years since the drug's approval by the FDA in 1987.*

(a) Graph the function S on a graphing calculator, graphically estimate S(5) and S'(5), rounding your answers to the nearest whole number. Interpret your results.

(b) Without doing any calculation, mentally estimate S'(10). Explain how you obtained your answer.

(c) According to your graph, when, to the nearest year, were sales increasing most rapidly?

* The model is a very crude one, based on 1991 sales data, total sales data through May, 1992 and very rough estimates of the potential market and selling price. Source: Senate Judiciary Committee; Subcommittee on Antitrust and Monopoly/The New York Times, May 14, 1992, p. D1.

Question 4

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

s(t) = 20,000 20,000 / (1+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 5

The demand for Gauss Jordan sneakers is given by

 q(p) = 2,000,000p1.2 ,

where p is the price in dollars per pair and q is the monthly sales of said sneakers by Sammy Sporting Authority.

(a) Complete the following table

 h 1 0.5 0.01 0.001 q(120+h)q(120)h

(b) Estimate both q(120) and q'(120). Interpret your results. (Round all answers to the nearest whole number.)

Question 6

Calculate the derivatives of the following functions.

(a) f(x) = 3x2x0.5+4x 4 (b) g(x) = (x24x2)(3x1.2x)
(c) h(x) = (x2+2)ln x (d)  r(x) = x2x 2x1 4
(e) y = ln(x26/x) (f) y = ex24x
(g)  y = 3x 3x2 1x
(h)  g(x) = ex + exex ex