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math021-final-fall2008-b-ans

math021-final-fall2008-b-ans - HKUST MATH021 Concise...

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HKUST MATH021 Concise Calculus Final Examination (Version 2) Name: 13th Dec 2008 Student I.D.: 12:30-15:30 Lecture Section: Part I: Multiple choice questions. Question 1 2 3 4 5 6 7 8 9 10 Total Answer d b a d e d d b b a 1. Which of the following is the graph of a function y = f ( x ) in the xy plane? (a) The unit circle centered at the origin (b) The square with vertices (1 , 1), ( - 1 , 1), ( - 1 , - 1) and (1 , - 1) (c) The triangle with vertices ( - 1 , 0), (1 , 1), (0 , - 1) (d) None of the above (e) All of the above Solution None of the above. 2. Let a , b , c be positive numbers. Evaluate lim x 0 (1 - e - bx ) cos 2 ax xe cx 2 . (a) a (b) b (c) c (d) 0 (e) 1 Solution The limit is b , by L’Hˆopital’s rule: lim x 0 (1 - e - bx ) cos 2 ax xe cx 2 = lim x 0 be - bx cos 2 ax - 2 a (1 - e - bx ) cos ax sin ax e cx 2 + 2 cx 2 e cx 2 = b 3. How many local maximum should f ( x ) have if its derivative is f 0 ( x ) = ( x + 1)( x - 2) 3 ( x + 4) 2 ( x - 5) ? (a) 1 (b) 2 (c) 3 (d) 4 (e) 0 Solution One local maximum f (2) at the critical point x = 2, by first derivative test.
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1 4. The value of Z 2 0 x 2 2 - x 2 dx is (a) 1 2 (b) 2 (c) 1 (d) π 2 (e) π Solution The answer is π 2 . Let x = 2 sin θ , dx = 2 cos θdθ . Then Z 2 0 x 2 2 - x 2 dx = Z π 2 0 2 sin 2 θ 2 cos θdθ 2 cos θ = Z π 2 0 2 sin 2 θdθ = Z π 2 0 (1 - cos 2 θ ) = h θ - 1 3 sin 2 θ i π 2 0 = π 2 5. Which of the following improper integrals is divergent? (a) Z 1 e sin x x 2 + 4 dx (b) Z 1 cos 2 x x 2 + 4 dx (c) Z 1 x x 2 + 4 dx (d) Z 1 ln x x 2 + 4 dx (e) Z 1 x ln x x 2 + 4 dx Solution The integral Z 1 x ln x x 2 + 4 dx is divergent, since Z 3 x ln x x 2 + 4 dx > Z 3 x x 2 + 4 x 2 dx = 1 5 Z 3 1 x dx = + 6. If c = c ( t ) (unit in thousand dollars) is the cost to manufacture t units of certain product in a factory, its rate of change dc dt is called the marginal cost function . If the marginal cost is given by dc dt = 300 4 t + 1 ($1 , 000 / unit) , how many thousands of dollars of additional cost (i.e., c (90) - c (20)) are necessary to raise production level from t = 20 to t = 90 units? (a) 10 (b) 500 (c) 600 (d) 1500 (e) 5000 Solution The answer is 1500 thousands of dollars: c (90) - c (20) = Z 90 20 c 0 ( t ) dt = Z 90 20 300(4 t + 1) - 1 / 2 dt = h 150(4 t + 1) 1 / 2 i 90 20 = 1500
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2 7. Find Z 4 0 fl fl x 2 + 2 x - 3 fl fl dx . (a) 58 3 (b) 64 3 (c) 74 3 (d) 86 3 (e) 95 3 Solution x 2 + 2 x - 3 = ( x - 1)( x + 3) = 0 ⇐⇒ x = 1 or x = - 3. Z 4 0 fl fl x 2 + 2 x - 3 fl fl dx = Z 1 0 - ( x 2 + 2 x - 3) dx + Z 4 1 ( x 2 + 2 x - 3) dx = h - 1 3 x 3 - x 2 + 3 x i 1 0 + h 1 3 x 3 + x 2 - 3 x i 4 1 = 86 3 8. The definite integral Z 1 0 1 4 + x dx equals (a) lim n + ˆ 1 4 + 0 n + 1 4 + 1 n + 1 4 + 2 n + · · · + 1 4 + n - 1 n !
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