2011_Tutorial_4

# 2011_Tutorial_4 - (a x 1-x at x = 0(b 1 x 2 at x = 1(c x 1...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 4 1. Find the area of the following region. (a) The region bounded between y = 1 2 sec 2 x, y = - 4sin 2 x, x = - π 3 and x = π 3 . (b) The region in the ﬁrst quadrant bounded by y = x, y = 1 4 x 2 and below y = 1. (c) The region bounded by y = 4 - x 2 , y = 2 - x, x = - 2 and x = 3. Ans. (a) 4 3 π (b) 5 6 (c) 49 6 2. (a) Find the volume of the solid generated by revolving the region between the parabola x = y 2 + 1 and the line x = 3 about the line x = 3. (b) The region bounded by the parabola y = x 2 and the line y = 2 x in the ﬁrst quadrant is revolved about the y -axis to generate a solid. Find the volume of the solid. Ans. (a) 64 15 2 π (b) 8 3 π 3. Find the radius of convergence of the following series. (a) X 1 ( - 1) n ( x + 2) n n (b) X 1 (3 x - 2) n n (c) X 1 ( - 1) n (4 x + 1) n (d) X 0 3 n x n n ! (e) X 1 n n x n (f) X 1 (4 x - 5) 2 n +1 n 3 / 2 Ans . (a) 1 (b) 1/3 (c) 1/4 (d) (e) 0 (f) 1/4 4. Find the sum of the geometric series inside the interval of convergence 1 - 1 2 ( x - 3) + 1 4 ( x - 3) 2 - + ··· + ( - x - 3 2 ) n + ··· . Ans . 2 x - 1

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MA1505 Tutorial 4 5. Find the Taylor series for the following functions:
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Unformatted text preview: (a) x 1-x at x = 0; (b) 1 x 2 at x = 1; (c) x 1 + x at x =-2; Ans . (a) ∞ X n =0 x n +1 (b) ∞ X n =0 (-1) n ( n + 1)( x-1) n (c) 2 + ∞ X n =1 ( x + 2) n 6. Find a quadratic (2nd degree) polynomial to approximate each of the following functions near x = 0: (i) e sin x and (ii) ln(cos x ). Ans . (i) 1 2 x 2 + x + 1 (ii)-1 2 x 2 7. Let S = ∞ X n =0 1 n !( n + 2) . In this question, we will introduce two diﬀerent ways to ﬁnd the value of S , one by integration and the other by diﬀerentiation. (i) Integrate the Taylor series of xe x to show that S = 1. (ii) Diﬀerentiate the Taylor series of e x-1 x to show that S = 1. 8. Let f ( x ) = 1 x 2 + x + 1 . Suppose f ( x ) = ∞ X n =0 c n x n is the Taylor series representation for f ( x ) at x = 0. Find the value of c 18-c 19 + c 20 . Ans. 2 2...
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## This note was uploaded on 11/30/2011 for the course EEE 1001 taught by Professor Phoon during the Spring '11 term at National University of Singapore.

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2011_Tutorial_4 - (a x 1-x at x = 0(b 1 x 2 at x = 1(c x 1...

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