## Review Exercises for Applied Calculus Finite Mathematics & Applied Calculus Topic: Functions and Linear Models

Chapter 1 Summary
True/False Quiz
Index of Review Exercises
Everything for Calculus
Everything for Finite Math
Everything for Finite Math & Calculus
Utilities: Function Evaluator & Grapher | Excel Grapher | On-Line Regression Utilty | All Utilities

Question 1 Let f be the function specified by following table.

 x -2 -1 0 1 2 f(x) 5 2 1 2 5

 (a) f(1) = (b) f(1 - 2) = (c) f(1)-f(2) =
 (d) What kind of model best fits the data? Select one linear quadratic exponential logistic
 (e) Why does no linear function exactly fit the data?
 (f) The linear function which passes through the points where x = 0 and 2 is: (Note Use proper input syntax: e.g., use "3*x" instead of "3x".) g(x) =

 Question 2 Given f(x) = x - 1x , find the following. (Note Use proper input syntax: e.g., use "3*x" instead of "3x".)

 (a) f(1) = (b) f(-1) = (c) f(x+h) = (d) f(x)+h = (e) f(x)+x =

Question 3 Let g be the function with the graph shown.

 g(-1) = g(2) - g(-1)3 = -g(1)g(-1) =

 Question 4 Find linear functions for the following straight lines. (Note Use proper input syntax: e.g., use "3*x" instead of "3x".)
 (a) Through the origin with slope 3 y = (b) Through the point (1,-1) with slope 3 y = (c) Through the points (-3, -6) and (1, -1) y = (d) Through the point (1, -2) and parallel to the line x + 3y = 1 y = (e) Through the point (1, -1) and parallel to the line 2x - 3y = 11 y =

Question 5 Select a function to match each of the following graphs.

 Select one f(x) = -|x| f(x) = |x| - 1 f(x) = 1/x f(x) = 1/x^2 f(x) = x + 1/x^2 f(x) = x^2 - 1 f(x) = |x| + 2x^2 f(x) = 1 - x^2 f(x) = x + 1/x Select one f(x) = -|x| f(x) = |x| - 1 f(x) = 1/x f(x) = 1/x^2 f(x) = x + 1/x^2 f(x) = x^2 - 1 f(x) = |x| + 2x^2 f(x) = 1 - x^2 f(x) = x + 1/x Select one f(x) = -|x| f(x) = |x| - 1 f(x) = 1/x f(x) = 1/x^2 f(x) = x + 1/x^2 f(x) = x^2 - 1 f(x) = |x| + 2x^2 f(x) = 1 - x^2 f(x) = x + 1/x Select one f(x) = -|x| f(x) = |x| - 1 f(x) = 1/x f(x) = 1/x^2 f(x) = x + 1/x^2 f(x) = x^2 - 1 f(x) = |x| + 2x^2 f(x) = 1 - x^2 f(x) = x + 1/x

Question 6 US imports of pasta increased from 290 million pounds in 1990 (t = 0) by an average of 52 million pounds per year.* Use this to express annual US imports of pasta q (in millions of pounds) as a linear function of the number of years t since 1990, and use your model to predict US past imports in the year 2000.

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

Question 7 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.

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

Question 8 The 50-member rugby team is planning to buy new gear for their next road trip to California. They are told by the suppliers that, if they order x game shirts, the cost per shirt is given by the following data:

 x (Number of shirts) 5 25 40 100 125 A(x)(Cost per shirt \$) 25.05 21.25 21.025 21.25 21.45

(a) Which of the following functions best models the data?

 0.005x + 20.75
 0.01x + 20
 0.01x + 20 - 25/x

 0.01x + 20 + 25/x
 0.0005x2 - 0.07x + 23.25

(b) Graph A(x) in the window 10 x 100, 20 y 23 and use your graph to estimate the lowest cost per shirt, and the number of shirts they should buy to obtain the lowest price per shirt.

Question 9 The following table shows the average daily circulation of all the newspapers of two major newspaper companies.

 Number of Newspapers 82 26 Daily Circulation (Millions) 5.8 3

† Circulation figures are rounded to the nearest 0.1 million, and reflect average daily circulation as of September 30, 1994. Source: Newspaper Association of America/The New York Times, Juy 30, 1995, p. E6

(a) Use these data to express a company's daily circulation, c, as a linear function of the number, n, of newspapers it publishes.

(b) What information does the slope give to newspaper publishers?

(c) Your company plans to add 11 newspapers to its current 10. According to your model, how would this effect daily circulation?

Question 10 The two fraternities Sigma Mu and Epsilon Alpha plan to raise money in a joint effort to benefit homeless people on Long Island. They will sell Jurassic Park T-shirts in the student center, but are not sure how much to charge. Sigma Mu treasurer Solo recalls that they once sold 400 shirts in a week at \$8 each, while Epsilon Alpha treasurer Justino claims that, based on past experience, they can sell 600 per week if they charge \$5 each.

(a) Based on the above anecdotal information, construct a linear demand equation for Jurassic Park T-shirts, giving weekly sales q as a linear function of unit price p?

(b) Express the weekly revenue as a function of p. At what price should Sigma Mu and Epsilon Alpha sell the T-shirts in order to obtain the largest possible revenue? [Hint: Revenue = price quantity. You could graph revenue as a function of p and locate the highest point.]

(c) Approximately how many T-shirts would they sell at that price?

Question 11 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. [See the hint for the last question]

Question 12 Continuing Question 11, suppose that old Slayer CDs cost the Delta Nuttal store \$2.00 each, with fixed costs of \$1,000 per week for advertising.
(a) Use this data with that of Question 11 to give the weekly cost as a function of the unit price x they charge per CD.
(b) Use the results of part (a) and Question 11 to give the weekly profit in terms of the unit price x, and hence determine how much they should charge for maximum weekly profit.

Question 13 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?

† 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.