Download Calculus for Business and Social Science - Solved Final Exam | MATH 211 and more Exams Mathematics in PDF only on Docsity! MATH 211 FINAL EXAM SOLUTIONS FALL 2004 NAME (PRINT!): SIGNATURE: Z Number: SECTION: SHOW ALL WORK FOR CREDIT. NO GRAPHING CALCULATORS ARE ALLOWED TO BE USED IN CONJUNCTION WITH THIS EXAM. PLACE YOUR ANSWER TO EACH QUESTION IN THE BOX NEXT TO THE PROBLEM, THAT WILL BE YOUR FINAL ANSWER. page 1 /25 page 2 /30 page 3 /35 page 4 /30 page 5 /25 page 6 /30 page 7 /15 page 8 /10 Total /200 1. (25 pts.) Find the derivative of each of the following functions: (Do not simplify) (a) f(x) = 3 x5 −√3x − 1 Answer f ′(x) = −15x−6 − 1 2 3√ 3x − 1 (b) f(x) = ex (x5 + 2x3) 3 Answer f ′(x) = ex(x5 + 2x3)3 + ex · 3(x5 + 2x3)2 · (5x4 + 6x2) (c) f(x) = (2x3 + e−5x+2)5 Answer f ′(x) = 5(2x3 + e−5x+2)4 · (6x2 − 5e−5x+2) (d) f(x) = ( 1 x + x) · ln(5 − 2x2) Answer f ′(x) = ( − 1 x2 + 1 ) · ln(5 − 2x2) + ( 1 x + x ) −4x (5 − 2x2) (e) f(x) = 2x2 + 3x x3 + 2 Answer f ′(x) = (4x + 3)(x3 + 2) − (2x2 + 3x)(3x2) (x3 + 2)2 2 5. (15 pts.) Find the absolute maximum and minimum of the function f(x) = x + 16 x on the interval [1,5]. Answer f ′(x) = 1 − 16 x2 = 0 x = ±4 f(1) = 17 (1, 17) abs. max f(4) = 8 (4, 8) abs. min f(5) = 8 · 2 5 6. (15 pts.) A construction company is constructing a closed top, square -based, rectan- gular metal tank that will have a volume 64 cubic ft. What dimensions yield minimum surface area? x x h I. given: x2 · h = 64 h = 64 x2 II. Surface area: S = 2x2 + 4xh S(x) = 2x2 + 4x · 64 x2 = 2x2 + 256 x S ′(x) = 4x − 256 x2 = 0 → S ′′(x) = 4 + (256)(2) x3 > 0 for any x > 0 x = 4ft. ⇒ x = 4 is the min. for S(x) h = 64 16 = 4ft. 7. (15 pts.) A certain bacteria culture grows at a rate proportional to its size and it becomes 4 times its size in every 6 hours. Find the growth rate. Answer P (t) = P0e kt 4P∅ = P∅e6k ln 4 6 = k 6 8. (10 pts.) The decay rate of zirconium is 1.05% per day. What is its half life? Answer k = .0105 P (t) = P0e −.0105t 1 2 P0 = P0e −.0105t ln ( 1 2 ) = −.0105t t = ln 2 .0105 9. (15 pts.) An airconditioning company determines that the marginal cost of producing the xth airconditioner is given by C ′(x) = −.2x + 500, and that C(0) = 100. Find the total cost of producing 100 airconditioners. Answer C(x) = ∫ (−.2x + 500)dx = −.1x2 + 500x + C C(0) = 100 = 0 + 0 + C C(x) = −.1x2 + 500x + 100 C(100) = −.1(100)2 + 500(100) + 100 = 49, 100. 7
