Question 1

Evaluate each of the following integrals.

(a)	2x2 + x0.1 + 4 x + 1 2x2 dx
(b)	(4x6)3 dx
(c)	x2 (3x2 12x + 5)3 dx
(d)	3x ex22 dx

Answer

Evaluate each of the following integrals analytically.

(a) 4 4x1.2 + 4/x1.2 e2x+1dx	(b) 032x(x2+16)1/2 dx
(c) (3x+2)e(3x2+4x) dx	(d) (x + (1/x) ln x) dx
(e) - 22 x(1+x2)2.3 dx	(f) (3x+1) / (x+2) dx

Answer

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

S(t) = 0.4 / (0.0025 + 160e^-2.7t)

(b) Referring to the model in part (a), calculate both S'(6) and ₀⁶S(t) dt (round both answers to three significant digits) and carefully explain what is measured by each.

Ý 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

The number of number of firms registered as investment advisers with the Securities and Exchange Commission increased significantly in the early 1990s, at a rate of approximately 1,400 per year.^ý Further, a total of 21,600 firms were registered by September 1994. Use a definite integral to estimate the number registered in September 1991.

ý The model is based on linearized data. Source for data: Cerulli Associates/The New York Times, June 25, 1995, p. 6F.

Answer

A 1995 article in the New York Times gave projections of the total cost of a four-year college education from 1996 to 2011, based on a continuous growth rate of 6.8% per year. These costs can be modeled as follows.

Public college: C(t) = 45,000e^0.068t;
Private college: D(t) = 90,000e^0.068t,

(a) Represent the area enclosed by the two graphs and the vertical lines t = 0 and t = 10 as a definite integral.
(b) Estimate that integral from the graph.
(c) Suppose that the state decided to subsidize the cost of a four-year college education for 100 students per year. What does the answer to part (b) tell you about that cost?

* Models are based on published data. Source: Dreyfus Service Corporation/The New York Times, August 6, 1995, Section 3, p. 1.

Answer

Stefan Waner (matszw@hofstra.edu)

Steven R. Costenoble (matsrc@hofstra.edu)

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