Midterm 3 with Solution - Mathematical Analysis for Business II | MATH 131, Exams of Mathematics

Download Midterm 3 with Solution - Mathematical Analysis for Business II | MATH 131 and more Exams Mathematics in PDF only on Docsity! — Salaction Math 131 Name: Spring 2030 Name.nn: Midterm 3 Lecturer: Form A Rec. Instructor: Rec. Time: Instructions: ¢ You have 48 minutes to complete this exam. It consists of 7 problems on 7 pages including this cover sheet. It is worth a total of 100 points. The value of each question is listed below. » You may not use any books or notes during this exam. Calculators are permitted EXCEPT those calculators that have symbolic algebra or calculus capabili- ties. In particular, the following calculators and their upgrades are not permitted: TI-89, TI-92, and HP-49. In addition, neither PDAs, laptops nor cell phones are permitted. Make sure to read each question carefully. * Please write clearly and make sure to justify your answers. Correct answers with no supporting work may receive no credit. * Please write your answers on the indicated lines. e A random sample of graded exams will be xeroxed before being returned. Problem | Point Value | Score 10 10 30 10 15 15 10 (1) Let f(a) = (6 — 2a)e”. Use derivatives and a sign graph to determine the interval(s) on which f(x) is increasing and on which f(z) is decreasing. (If there are none, please say so). fx) (G-2%) 6% + 2% (-2) (4 -2x)}e* = ~2(X~2)e% Sign chart for 4°: { tw) | YJ x) Answer (1.): increasing: (2.) Let f(z) = —2® + 185 — 122 +17, foo, 2) decreasing: C2, eo) Use derivatives and a sign graph to determine the interval(s) on which f(z) is concave up and on which f(x) is concave down. (If there are none, please say so). fx d= - b+ bor*- 12 F"x)= 20244 2hox? = ~20 %°(%-8), cum Choat Aor £" : (-) | (or (-) —> gy |b | o DB TU Answer (2.): concave up: 2 lo g) - 69 V8, 22 concave down: (5.) Let f(x) = 223 — 15x? — 1442. (a.) Use the Second Derivative Test to find where the local maximum(s) and the local minimum(s) of f(z) _ Fx) = ba? 20% - 144 2b (a $x~-24) 2 b(x-8) ea), oo Critical points are > Ke 8, Ke-z fx) = 12X-2o f'-2}2 = bh <0 (max) Lig)2 66 ?oO (wis ) Answer (5a.): local max(s). at 2 = -3 local min(s). at « = (b.) Pind where the absolute maximum and absolute minimum for f(x) occur over the interval (3, 10]. enol Ports = %=2, KaI0 critical pdats + OOS, 8 L(g) = -S13 — Gnax) £(g)2 -1082 (min) t lio)= — 94d Answer (5b.): absolute max(s}. at z = 3 absolute min(s). at c=_ & 5 (6.) A deli sells 480 sandwiches per day at a price of $4 each. A market survey shows that for every $0.10 reduction in price, 20 more sandwiches will be sold. (a.) How much should the deli charge in order to maximize revenue? Le+ 4 be the number at veoluction pree= A-O.104, sales = 480 +204 > Revenue z Rly) ~(4- 0-10 4) (480 204) Ri) ={4- 0.104) (2.0) + [480 4204) (0.10) = §0-24 - 48-24 = 22- 4y ce Cytkical port is Ye 3. Somce thine Is only ome critica point, usma, Secon ol devivetive test: Rlyz-4 => R'(S)= -4 <0 Se 42k 1s the absclite may, andl the price is 4 - C10l8\=3.2 Answer (6a.): price= 3. 2 (b.) What is the maximum revenue? The mow. vevenne is R(8) = 2948 Answer (6b.}: maximum revenne= 2048 6 (7.) Use the given information to sketch a graph of f(x): Domain: All real except « = —2 and x = 3. S(-1) =3 and f(1) = 0. F(z) < 0 on (—00, —2) and (—2, 3), f(z) > 0 on (3,00). f(x) <0 on (—00, —2), (1,3) and (3,00), f”(z) > 0 on (—2, 1). Vertical asymptotes: «= —2 and 2 = 3, Horizontal asymptotes: y = —1. 1X73.