## Review Exercises for Applied Calculus Finite Mathematics & Applied Calculus Topic: Introduction to the Derivative

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True/False Quiz
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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) Average rate of change of f over the interval [0, 2] is (b) Average rate of change of f over the interval [-1, 1] is (c) f has the greatest average rate of change on the interval Select one [-2, -1] [-1, 0] [0, 1] [1, 2]

Question 2 The number of network managers in the US over the years 1989-2001 followed the curve

N(t) = 1.8t2 - 40t + 320     thousand managers*       (9 t 21)

where t is time in years (t = 0 represents 1980).

* Source: Bureau of Labor Statistics via Economy.com, Tech Target/New York Times, August 8, 2001, p. G1.

(a) Compute the average rate of change of N over the intervals [10, 15] and [15, 20].

 Answser: Interval [10, 15] Interval [15, 20] Select one thousand managers years thousand managers per year years per 1000 managers dollars per 1000 managers thousand managers per month miles per hour managers

 (b) Interpret the results of part (a).

 (c) The rate of change of N is Select one increasing decreasing increasing, and then decreasing decreasing, and then increasing approximately unchanged over the given period 1989-2001.

Question 3 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? Select one 1990-1991 1991-1992 1992-1993 1993-1994 (b) least? Select one 1990-1991 1991-1992 1992-1993 1993-1994 (c) 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 4 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, rounding all anwers to two decimal places.

 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 5 Look carefully at each of the graphs below, and select the corect answers for each graph.

 The slope of the tangent is 0 at the point Select one P Q R None of the above The slope of the tangent is 2 at the point Select one P Q R None of the above The slope of the tangent is -1 at the point Select one P Q R None of the above The slope of the tangent is 0 at the points Select one P & Q P & R Q & R None of the above The slope of the tangent is 1 at the point Select one P Q R None of the above The slope of the tangent is -1 at the point Select one P Q R None of the above

Question 6   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 7 Calculate the derivative of the following functions algebraically.
Note You should enter the answers using computer formula notation -- e.g. enter 3x-2 as 3*x^(-2)
 (a) f(x) = 3x2 - 4x-1 f'(x) = (b) f(x) = 2x4 + 3x-2 f'(x) = (c) g(x) = x1.2 + x2.3 g'(x) = (d) g(x) = x/5 - 5/x g'(x) = (e) h(x) = x^2/5 - 5/x^2 h'(x) =

Question 8 The weekly cost of manufacturing x pairs of Gauss-Jordan Sneakers is

C(x) = -0.00012x2 + 30x + 50,000 dollars.

(a) What is the marginal cost at a sales level of 1000 pairs per week?

 \$ per pair
(b) What is the average cost per pair at a sales level of 1000 pairs per week?
 \$ per pair

 (c) At the above sales level, the marginal cost is increasing decreasing at a rate of \$ per pair.

 (d) At the above sales level, the marginal cost is greater than Approximately equal to     less than the average cost.

 (e) In general, if the marginal cost is increasing, and there is no fixed cost, then the marginal cost is greater than Approximately equal to     less than the average cost.