Question 1

(a) Let f(x) = x² + 1/(x+1). Say whether each of the following is defined. If it is, evaluate it.

f(2)

f(1)

f(0.9)

f(a1)

Use the graph to approximate the following.

h(0.5)

h(3+2) h(3)

h(3) h(3)

Answer

The number C(t) of coffee shops and related enterprises in the US can be approximated by the following function of time t in years since 1990.*

C(t) =
		500t + 800	if 0 t 4
		1,300t 2,400	if 4 < t 10

(a) Evaluate C(0), C(4) and C(5), and interpret the results.

(b) How fast was the number of coffee shops in the US growing in 1992 and in 1995?

(c) Use the model to estimate when there were 5,400 coffee shops in the US.

Answer

* Source for data: Specialty Coffee Association of America / The New York Times, August 13, 1995, p. F 10.

The following table shows the number of visitors to Hawaii during three different years.**

Year	1985	1991	1994
Number of Visitors (millions)	4.9	7.0	6.5

Which of the following kind of model would best fit the given data? Explain your choice of model. (a, b, and c are constants)

(A) n(t) = at² + bt + c

(B) n(t) = at + b

(C) n(t) = a + b/t

Answer

**Source: Hawaii Visitors Bureau/The New York Times, September 5, 1995, p. A12.

Find the equations of the following straight lines.

(a) through (1, 2) and (2, 1)

(b) through (1, 2) and at right angles to the line 2y + 3x = 6

Answer

Format (a) and (b) for entry into a graphing calcualtor:

(a)	(b)
Evaluate:	Simplify:
(c)	(d)

Answer

Carefully sketch the graph of y = x² + 10x + 96, showing the location of the vertex and intercepts.

Answer

The proprietor of Delta Nuttal Music Mania Store finds that, when the store offers old Slayer CDs at $10 per CD, it sells 10,000 discs per week. Dropping the price to $7.50 per disc has the effect of boosting sales to 17,500 per week. Use this data to set up a demand equation, and determine the unit price (to the nearest cent) the music store should charge to obtain the maximum revenue.

Answer

US imports of pasta increased from 290 million pounds in 1990 (t = 0) by an average of 52 million pounds per year.^‡

(a) Use this data to express annual US imports of pasta q (in millions of pounds) as a linear function of the number of years t since 1990.

(b) Use your model to predict US past imports in the year 2000.

Answer

‡ Data are rounded, and are for the period 1990 through 1994. Source: Department of Commerce/The New York Times, September 5, 1995, p. D4.

The following table shows math and science scores in the US for 17-year olds from 1977 to 1992. ^†

Year	1977	1982	1990	1992
Science Score	290	280	290	295
Math Score	304	296	304	308

(a) Use the data from 1977 and 1982 to model the math score m as a linear function of the science score s.

(b) Is your model consistent with the 1992 data?

(c) What does the slope in your model signify about science and math scores?

Answer

† Data are accurate to two significant digits, with the third digit "massaged" to simplify the calculation. Source: Department of Education's National Assessment of Educational Progress/The New York Times, August 18, 1994, p. A14.

Stefan Waner (matszw@hofstra.edu)

Steven R. Costenoble (matsrc@hofstra.edu)

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