Math 142 Week-in-Review #10 (Exam 3 Review: Sections 5.8 and )

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1 Math 142 WIR, copyright Angie Allen, Fall Math 142 Week-in-Review #10 (Exam 3 Review: Sections 5.8 and ) Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on sections 6.4 and 6.5). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your suggested and online homework. 1. Use the graph of f (x) with the indicated areas below to answer the following: y a A b B C c f(x) D d x area of A: 2.0 area of B: 1.5 area of C: 2.5 area of D: 9.5 c c a) Find f (x) dx 2 f (x) dx. a 0 c b b) Find 4 f (x) dx + f (x) dx. c d c) Find the area between f (x) and the x-axis from x = 0 to x = d. a d) Find f (x) dx. 0 e) Find the average value of the function f on [b,c]. f) When finding a left Riemann sum of f on the interval [a,0] using one rectangle, will we obtain an overestimate or underestimate?

2 Math 142 WIR, copyright Angie Allen, Fall Find dy for each of the following. You do not need to simplify your final answer. dx a) xlny + 2y = 2x 3 b) 4 3xy2 y + 3 x = 5

3 Math 142 WIR, copyright Angie Allen, Fall When finding a midpoint Riemann sum of a function f on the interval [a,b] using n rectangles, write the formula (using sigma notation) for the sum of the areas of the first, second, and third rectangles. 4. A 10-foot ladder is leaning against a wall. The bottom of the ladder slides away from the wall at a rate of 3 feet per second. How fast is the top of the ladder sliding down the wall when the bottom is 5 feet from the wall? 5. Write the following as a single integral: f (x) dx + f (x) dx f (x) dx 5 5

4 Math 142 WIR, copyright Angie Allen, Fall Acme Widget Company s marginal profit is given by w(x) = 35e 0.01x dollars per widget, where x is the number of widgets produced per day. The profit earned by selling 120 widgets is $300. (courtesy of Heather Ramsey) a) If the current production level is 250 widgets per day and the manager wishes to increase production to 275 widgets per day, how will this production increase affect profit? b) Find the profit earned by selling 200 widgets. 7. Suppose x 2 2xy y 2 = 7, where x and y are functions of t. If dy dt = 1 when x = 1 and y = 2, find dx dt.

5 Math 142 WIR, copyright Angie Allen, Fall Find bounds to estimate the value of (x 2 10x + 28) dx. 0 A t 2 + 4t 2 t 9. Evaluate 3 3 t 3 dt, where A is a constant and A > A rock thrown into a still pond causes a circular ripple. If the radius of the ripple is increasing by 2 feet per second, how fast is the area changing when the radius is 9 feet?

6 Math 142 WIR, copyright Angie Allen, Fall Use the graph of f (x) below to answer parts (a)-(f). (Courtesy of Joe Kahlig) 4 a) Find f (x) dx. 0 0 b) Find f (x) dx c) Find f (x) dx. 6 4 d) Find f (x) dx. 4 9 e) Find f (x) dx. 2 f) Find the area between the function and the x axis from x = 0 to x = 10.

7 Math 142 WIR, copyright Angie Allen, Fall Estimate 6 2 (3x x 2 ) dx using a Riemann sum with n = 4 and midpoints. What does your answer represent? What does the definite integral represent? B B A 13. If f (x) dx = 15 and [4 f (x) 3g(x)] dx = 62, find g(x) dx. A A B

8 Math 142 WIR, copyright Angie Allen, Fall Compute the following integrals: A 8 a) dx, where A > 2. 2 xlnx b) (2 +t) t 3 dt c) B 3 ( 3t t 5t 3 ) 4e t + π dt, where B > 3. d) 4 3x 6 + lnx dx

9 Math 142 WIR, copyright Angie Allen, Fall Let G(x) be an antiderivative of g(x). If g(x) dx = 15 and G(3) = 7, find G(0) Use the graph of f (t) below to answer the following. Note that the domain of f (t) is [0,7]. a) Approximate 5 rectangles of equal width. 2 f (t) dt using a left sum with 3 x b) If g(x) = f (t) dt, find g(6). 3 (Source: #3, pg. 372 of Single Variable Calculus: Concepts and Contexts, 4th ed., by Stewart) c) If f (t) gives the velocity (in ft/s) of an object after t seconds, determine the distance the object travels during the first 3 seconds.

10 Math 142 WIR, copyright Angie Allen, Fall Bootsie s Bandana Company s profit is given by P(x) = x x 300 dollars, where x is the number of bandanas produced and sold each month. a) Find the average profit when the number of bandanas sold each month is between 170 and 190. b) Find the average marginal profit when the number of bandanas sold each month is between 170 and 190. c) Find the change in profit when the number of bandanas sold each month increases from 170 to 190.

11 Math 142 WIR, copyright Angie Allen, Fall Given f (x) dx = 10 and f (x) dx = 10, find f (x) dx A x Find 2 x dx. 20. The research department of Acme, Inc. has determined the marginal cost function for one particular item to be m(x) = 0.12e 0.04x dollars per item, where x is the number of items produced. If Acme s fixed costs amount to $3,000, find the total cost when 150 items are produced. (courtesy of Heather Ramsey)

12 Math 142 WIR, copyright Angie Allen, Fall The price p (in dollars) and demand x for a product are related by 2x 2 + 5xp + 50p 2 = 80,000. (Source: #29, pg. 262 of Calculus for Business, Economics, Life Sciences, and Social Sciences, 11th ed., by Barnett, Ziegler, and Byleen) a) If the price is increasing at a rate of $2 per month when the price is $30, find the rate of change of the demand with respect to time. b) If the demand is decreasing at a rate of 6 units per month when the demand is 150 units, find the rate of change of the price with respect to time If f (7) = 12 and f (x) dx = 16, then find f (2). 2

13 Math 142 WIR, copyright Angie Allen, Fall The rate at which the concentration of a particular drug in the blood stream increases when taken daily can be modeled by r(t) = 2.2 t µg/ml per day, where t is the number of days since the daily regimen was started, 1 t 17. Five days after the regimen was started, the concentration of this drug in the blood stream was 4.5 µg/ml. (courtesy of Heather Ramsey) a) Find the average rate of change of the concentration of the drug in the blood stream from taking the second dose through taking the eighth dose. b) Find the average concentration of the drug in the bloodstream from taking the third dose through taking the tenth dose. c) Find the change in concentration of the drug in the bloodstream from taking the third dose through taking the tenth dose.

14 Math 142 WIR, copyright Angie Allen, Fall A boat is being pulled toward a dock by a person holding a rope 4 feet above the dock. If the rope is being pulled in at 3 feet per second, how fast is the boat moving toward the dock when it is 28 feet from the dock?