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.

Answer

Evaluate for each of the following.

(a) y = 4 4x^1.2 + 4/x^1.2 1/(2x)

(b) y = (2x² 4x)/(x + e^2x)

(c) y = (2x ln x)^-2

(d) y = e^{2x x2}

(e) y = 2xe^2x(x 2)

(f) y = [(2x1)/e^x] ^{- 2}

(g) y = ln(2x x²)

Answer

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)

(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.

Answer

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)

(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?

Answer

The demand for Gauss Jordan sneakers is given by

q(p)

=

2,000,000 p1.2

,

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.)

Answer

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 3 x2 1 x	(h)	g(x) = ex + ex ex ex

Answer

Stefan Waner (matszw@hofstra.edu)

Steven R. Costenoble (matsrc@hofstra.edu)

Last Updated:May, 1997 Copyright © 1996 StefanWaner and Steven R. Costenoble