## Review Exercises for Chapter 5 inApplied CalculusChapter 13 inFinite Mathematics & Applied CalculusTopic: Applications of the Derivative

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Question 1

Locate and classify all maxima and minima of the function

 f(x) = 1x - 1x2 (-5 x 5)

Include a sketch of its graph with your analysis.

Question 2

Here is the graph of f(x) = x3(x1/2 - 1), with domain [0, +¥).

Locate and classify all relative and absolute extrema.

Question 3

The number of cell phone subscribers in China for the period 2000-2005 was projected to follow the equation N(t) = 39t + 68 million subscribers in year t (t = 0 represents January 2000). The average annual revenue per cell phone user was \$350 in 2000.* Assuming that, due to competition, the revenue per cell phone user decreases continuously at an annual rate of 10%, we can model the annual revenue as

R(t) = 350(39t + 68)e-0.3t million dollars.

* Based on a regression of projected figures (coefficients are rounded). Source: Intrinsic Technology/New York Times, Nov. 24, 2000, p. C1.

(a) Determine when, to the nearest 0.1 year, the revenue is projected to peak, and also the revenue, to the nearest \$1 million, at that time.

(b) Determine when, to the nearest 0.1 year, the graph of R encounters a point of inflection. Interpret the result.

(c) If we were to project this model into the indefinite future, what does it predict? Is this prediction reasonable?

Question 4

Find the minimum value of C = x + 10y subject to xy = 10 (x and y both positive). To what values of x and y does this minimum correspond?

Question 5

Find x and y that minimize

C = 30,000x + y
subject to
40,000,000 = 4x0.1y     (x > 0, y > 0).
(Round x and y to the nearest whole number.) What is the corresponding value of C?

Question 6

The demand q (in weekly sales) for Hofstra lacrosse shorts at the HU Bookstore depends on the price p according to the demand equation

q = 360 / p1.5.
If shorts cost the HU Bookstore \$8 each, how much should the bookstore charge to maximize profit?

Question 7

A quadratic regression based on old sales data reveals the following demand equation for "E = mc2" T-shirts.

q = -2p2 + 33p (9 p 15)
Here, p is the price the club charges per T-shirt, and q is the number the Physics Club can sell each day at the flea market.

(a) How much should the Physics Club charge for the T-shirts in order to obtain the maximum revenue? What will this revenue be?

(b) Actually, the Physics club pays \$8 per T-shirt (see Question 1), as well as a daily \$100 fee for using the flea market, and the would like the profit to be as large as possible. How much should they be charging to accomplish this? (Round the answer to the nearest \$1.)

(c) Predict the daily profit the club will make from the sale of T-shirts at that price.

Question 8

Your Coffee House "After-Theater Chit-Chat" features two entertainers: André Rezig and fabulous Fiadora Fuffi. Costs can be broken down as follows:

 Rent: \$50 per day Salaries: \$300 per day André Rezig: \$200 per hour Fiadora Fuffi : \$250 per hour

Each artist draws varying numbers of guests, and you have calculated (using data-based regression on your graphing calculator) that, if Rezig performs for x hours and Fuffi for y hours, the number of guests they will draw is given by

g = 20x0.2y0.8.

Assuming you want to fill your establishment to its capacity of 100 customers at a minimum (daily) cost, how many hours should you have each artist perform? (Round answers to the nearest hour.)

† Actually, the function g is known as a "Cobb-Douglas Production Function," and it is indeed possible to use regression methods using your graphing calculutor -- See p. 661 in Calculus Applied to the Real World to find out how to do it.

Question 9

The demand equation for the Physics Club's "E=mc2" T-shirts is given by

q = -2p2 + 33p     (9 p 15)
where p is the price they charge per T-shirt, and q is the number they can sell each day at the local flea market.

(a) Obtain a formula for the price elasticity of demand, E for "E=mc2" T-shirts.

(b) Compute the elasticity of demand if the price is set at \$10 per shirt. Interpret the result.

(c) How much should the Physics Club charge for their T-shirts in order to obtain the maximum revenue? What will this revenue be?

Question 10

The demand for services from your newly created internet provider "Argentina Off-Line" is running into financial problems. The consumer demand for AOL has been modeled by the equation
 q = 1000e-p2+p,
where p is the monthly access charge, and q is the number of new subscribers per month.

(a) Obtain a formula for the price elasticity of demand, E for Argentina Off-Line services.

 E =

(b) Compute the elasticity of demand if the monthly access charge is set at \$2 per month. Interpret the result.

(c) How much should Argentina Off-Line charge in order to obtain the maximum revenue? What will this revenue be?