Legibly Printed Family/Last Name: Legibly Printed Given/FirstName: Signature: M119 Seat number: EXAM 3 14th April 2015 READ THIS: For each of the problems that begin on the next page, find the one correct answer choice from the options listed. Indicate your answer choice by placing the appropriate CAPITAL letter in the space next to the question numbers below AND by circling the correct answer in your test itself (because this answer sheet may not be returned to you). Only your answers on this answer sheet will be graded. You may write on the test as much as needed when working out the problems, but no credit is given for anything written in the test itself, and no partial credit is given. Good luck. You MUST fill in your answers before the allotted time is up. Once the time is up, you are NOT allowed to write on your test! Please do NOT remove the cover sheet from your exam. Indicate your answer choices below in CAPITAL LETTERS. (1) (11) (2) (12) (3) (13) (4) (14) (5) (15) (6) (16) (7) (17) (8) (18) (9) (19) (10) (20) 1. If p is a critical point of a function f , which one of the following statements is TRUE? (A) If f 0 < 0 on the left of p and f 0 > 0 on the right of p, then f has a local maximum at p. (B) If f 0 > 0 on the left of p and f 0 < 0 on the right of p, then f has a local maximum at p. (C) If f 0 < 0 on the left of p and f 0 > 0 on the right of p, then f has an infliction point at p. (D) If f 0 < 0 on the left of p and f 0 < 0 on the right of p, then f has local minimum at p. (E) If f 0 > 0 on the left of p and f 0 > 0 on the right of p, then f has local maximum at p. (F) None of these 2. Consider the function f (x) = 5x 2 4x + c where c is a constant. Find the critical point of f (x). 4 5 5 (B) 4 (A) (C) 4 5 5 2 2 (E) 5 (F) None of these (D) 3. If f (x) = 2x 3 + 18x 2 + ax + b, where a and b are constants, find the x-coordinate of the inflection point of f (x)? (A) 3 (B) 9 (C) 9 (D) 3 (E) There is no inflection point for f (x) (F) None of these 4. Find the x-coordinate of the local maximum point of f (x) = 100x x 2 + 10000. (A) 1000 (B) 10000 (C) 500 (D) 100 (E) 50 (F) None of these 5. A product sells for $250 each. The cost for producing this product is given by C(q) = 100 + 5q 2 . Find the production level that yields maximum profit. (A) 75 (B) 5 (C) 25 (D) 200 (E) 100 (F) None of these 6. At a price of $8 per ticket, a drama theater group can fill every seat in the theater, which has a capacity of 1200. For every additional dollar increase in the price, the number of people buying tickets decreases by 75. Find the ticket price that maximizes the revenue. (A) $9 (B) $10 (C) $6 (D) $16 (E) $12 (F) None of these 7. Consider the logistic curve function 400 1 + Ce 0.4t that represents the number of people heard a rumor, where time t is in hours. Suppose that two people originally started this rumor. Find the value of C. P = (A) 201 (B) 402 (C) 199 (D) 399 (E) 299 (F) None of these 8. Find the time (up to 2 decimal places) in previous problem when the rumor is spreading at the fastest rate. (A) 14.07 hrs (B) 15.09 hrs (C) 13.23 hrs (D) 14.97 hrs (E) 15.59 hrs (F) None of these 9. The following table gives the velocity of a car every 5 seconds. Estimate the distance travelled by the car during this 20 seconds by calculating overestimate and underestimate. Time (Sec) 0 5 10 15 20 Velocity (ft/sec) 22 21 23 24 26 (A) 425.5 ft (B) 462.5 ft (C) 460 ft (D) 445 ft (E) 402.5 ft (F) None of these 10. Using the following graph of r (t), find the exact value of the integral R 12 4 r (t) dt. rHtL 7 6 5 4 3 2 1 0 2 4 6 8 10 12 14 16 t (A) 22.5 (B) 20 (C) 28.5 (D) 26 (E) 32.5 (F) None of these 11. If R8 2 f (x) dx = 15 and R3 2 f (x) dx = 7, find the value of R8 3 f (x) dx. (A) 14 (B) 3 (C) 12 (D) 8 (E) 22 (F) None of these 12. Let f (x) = 10 x 2 and g(x) = x 2 8. Find the integral that represents the area between these two graphs. R3 (A) 3 2x 2 18 dx R 10 (B) 8 18 2x 2 dx R 10 (C) 8 2x 2 18 dx R3 (D) 3 (10 x 2 )(x 2 8) dx R3 (E) 3 18 2x 2 dx (F) None of these 13. If the graph of the derivative P 0 (t) of a function P (t) is given below, find the value of t where P has the maximum value. PHtL 1 2 3 4 5 6 7 8 9 10 t (A) 2 (B) 8 (C) 10 (D) 4 (E) 9 (F) None of these 14. The marginal cost function of a company is C 0 (q) = q 2 10q + 50 dollars per unit and the fixed cost is $1600. Find the expression that represents the total cost for producing 200 units. R 200 (A) 0 q 2 10q + 50 dq R 1600 2 (B) 200 0 q 10q + 50 dq R 200 2 (C) 1600 + 0 q 10q + 50 dq R 200 (D) 1600 0 q 2 10q + 50 dq R 1600 2 (E) 200 + 0 q 10q + 50 dq (F) None of these 15. Suppose that a bicyclist pedals along a straight road at the rate of r (t) = 10 2t mph. Find the exact distance travelled by the bicyclist in the first 2 hrs. (A) 25 miles (B) 20 miles (C) 6 miles (D) 16 miles (E) 8 miles (F) None of these Problems 16-20 are short answer questions. Pay attention to the required format. There will be no partial credit. 16. Find the x-coordinate of the global maximum of f (x) = x 3 12x + 100 on the interval [6, 3]. Give exact answer (NO approximation) and leave your answer as an integer or simplified fraction. 17. Find the x-coordinate of the global minimum of f (x) = x 3 12x + 100 on the interval [6, 3]. Give exact answer (NO approximation) and leave your answer as an integer or simplified fraction. 18. Find the area enclosed by the graphs of the function f (x) = 6x + 3 and g(x) = 3x 2 + 3. Leave your answer as an integer, simplified fraction or a terminating decimal number. 19. A company that makes mountain bikes has the marginal cost function: C 0 (q) = 500 0.5q + 6 where q is the quantity produced. If the bikes are sold $300 each, find the profit or loss on the first 30 bikes. (Write your answer in dollars and cents, i.e., up to two decimal places in dollars.) 20. A virus spreads in a community of 5000 people according to a logistic function P (t) = 5000 1 + Ce 0.05t where C is a constant. It was known that the rate of spread of the virus reaches the maximum rate on 20th day. Find the value of C up to four decimal places. This page is left blank intentionally This page is left blank intentionally