2009_solutions

2009_solutions - Marks [3] 1. Short-Answer Questions. Put...

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1. Short-Answer Questions. Put your answers in the boxes provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. At most one mark will be given for an incorrect answer. Unless otherwise stated, simplify your answers as much as possible. Marks (a) Evaluate Z 3 + x 5 x dx [3] Z 3 + x 5 x dx = Z 3 x - 1 / 2 + x 9 / 2 dx = 6 x + 2 11 x 11 / 2 + C. (b) What integral is defined by the following expression? lim n →∞ n X i =1 π 4 n tan 4 n Do not evaluate the integral. [3] If one divides [0 ,π/ 4] into n equal subintervals, then Δ x = π 4 n and x i = i Δ x = 4 n . Therefore, lim n →∞ n X i =1 π 4 n tan 4 n = lim n →∞ n X i =1 tan( x i x = Z π/ 4 0 tan xdx. 1
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April 24, 2009 Math 101 Page 2 of 13 pages (c) Evaluate Z 1 0 ( y + 1) p 2 y + y 2 dy [3] Let u = 2 y + y 2 . Then du = (2 + 2 y ) dy = 2( y + 1) dy . When y = 0, u = 0 and when y = 1, u = 3. Thus, Z 1 0 ( y + 1) p 2 y + y 2 dy = 1 2 Z 3 0 udu = ± 1 3 u 3 / 2 ² 3 0 = 1 3 3 3 / 2 = 3 . (d) Evaluate Z x 2 ln x dx [3] Integrate by parts: let u = ln x and dv = x 2 dx so that du = dx/x and v = x 3 / 3. Therefore, Z x 2 ln xdx = x 3 3 ln x - Z x 2 3 dx = x 3 3 ln x - x 3 9 + C. 2
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April 24, 2009 Math 101 Page 3 of 13 pages [3] (e) Find the length of the curve x = 100 + 2 y 3 / 2 , 0 y 11. dx dy = 3 y so the length of the curve L is L = Z 11 0 p 1 + 9 y dy = 2 27 h (1 + 9 y ) 3 / 2 i 11 0 = 2 27 (1000 - 1) = 74 . (f) What is the average value of | sin θ - cos θ | over the interval 0 θ π/ 2? [3] Notice that cos θ sin θ for 0 θ π/ 4 and sin θ cos θ for π/ 4 θ π/ 2. Therefore the average value is 2 π Z π/ 2 0 | sin θ - cos θ | = 2 π ± Z π/ 4 0 (cos θ - sin θ ) + Z π/ 2 π/ 4 (sin θ - cos θ ) ! = 2 π ² [sin θ + cos θ ] π/ 4 0 + [ - cos θ - sin θ ] π/ 2 π/ 4 ³ = 2 π ² ( 2 - 1) + ( - 1 + 2) ³ = 4( 2 - 1) π . 3
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April 24, 2009 Math 101 Page 4 of 13 pages (g) Give the first three nonzero terms of the Maclaurin series (power series in x ) for [3] Z e - x 2 - 1 x dx Recall that e x = 1 + x + x 2 2 + x 3 6 + ... Therefore,
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2009_solutions - Marks [3] 1. Short-Answer Questions. Put...

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