math021-final-fall2008-a-ans

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

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Unformatted text preview: HKUST MATH021 Concise Calculus Final Examination (Version 1) 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 e a b b d a c a d c 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) All of the above (e) None of the above Solution None of the above. 2. Let a , b , c be positive numbers. Evaluate lim x → (1- e- ax )cos 2 bx xe cx 2 . (a) a (b) b (c) c (d) 0 (e) 1 Solution The limit is a , by L’Hˆopital’s rule: lim x → (1- e- ax )cos 2 bx xe cx 2 = lim x → ae- ax cos 2 bx- 2 b (1- e- ax )cos bx sin bx e cx 2 + 2 cx 2 e cx 2 = a 3. How many local maximum should f ( x ) have if its derivative is f ( x ) = ( x + 1)( x- 2) 3 ( x + 4) 2 ( x- 5) ? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 Solution One local maximum f (2) at the critical point x = 2, by first derivative test. 1 4. Find Z 4 fl fl x 2 + 2 x- 3 fl fl dx . (a) 95 3 (b) 86 3 (c) 74 3 (d) 64 3 (e) 58 3 Solution x 2 + 2 x- 3 = ( x- 1)( x + 3) = 0 ⇐⇒ x = 1 or x =- 3. Z 4 fl fl x 2 + 2 x- 3 fl fl dx = Z 1- ( 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 + h 1 3 x 3 + x 2- 3 x i 4 1 = 86 3 5. The value of Z √ 2 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 = √ 2sin θ , dx = √ 2cos θdθ . Then Z √ 2 x 2 √ 2- x 2 dx = Z π 2 2sin 2 θ √ 2cos θdθ √ 2cos θ = Z π 2 2sin 2 θdθ = Z π 2 (1- cos2 θ ) dθ = h θ- 1 3 sin2 θ i π 2 = π 2 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)) is necessary to raise pro- duction level from t = 20 to t = 90 units? (a) 1500 (b) 1200 (c) 600 (d) 250 (e) 10 Solution The answer is 1500 thousands of dollars: c (90)- c (20) = Z 90 20 c ( t ) dt = Z 90 20 300(4 t + 1)- 1 / 2 dt = h 150(4 t + 1) 1 / 2 i 90 20 = 1500 2 7. Which of the following improper integrals is divergent? (a) Z ∞ 1 √ x x 2 + 4 dx (b) Z ∞ 1 ln x x 2 + 4 dx (c) Z ∞ 1 x ln x x 2 + 4 dx (d) Z ∞ 1 e sin x x 2 + 4 dx (e) Z ∞ 1 cos 2 x x 2 + 4 dx Solution The integarl 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 = + ∞ , 8. The definite integral Z 1 1 4 + x dx equals (a) lim n → + ∞ 1 n ˆ 1 4 + n + 1 4 +...
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math021-final-fall2008-a-ans - HKUST MATH021 Concise...

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