InsallW98T4 - i. Write a dierential equation that is...

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Insall - WS98 Calculus II Exam #4 April 20, 1998 1. (20 pts.) Power Series: (a) Find the radius of convergence and the interval of convergence of the series X n =0 x n n +2 . (b) Use a power series to compute the integral Z 0 . 2 0 1 1+ x 4 . Express your answer as a convergent series. Then approximate its value using 101 terms. 2. (20 pts.) Taylor Series and Taylor Polynomials: (a) Find the Taylor Series for f , centered at ln(3), if f ( x )= e x . (b) A car is moving with a speed of 20 m s and acceleration 2 m s 2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute? 3. (20 pts.) Differential Equations: (a) Solve the differential equation yy 0 = x . (b) One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor.
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Unformatted text preview: i. Write a dierential equation that is satised by y . ii. Solve the dierential equation obtained above. Leave your answer in implicit form. 4. (20 pts.) Arc Length and Surface Areas of Revolution: (a) Find the length of the curve described by y = ln(cos( x )); 0 x 4 . (b) Let f ( x ) = 1 3 ( x 2 + 2) 3 2 for 1 x 2. Approximate to 4 decimal places the area of the surface obtained by rotating the graph of f about the x-axis. 5. (20 pts.) Moments and Centers of Mass: (a) Masses m 1 ,...m 4 are located at the points P 1 ,...,P 4 , respectively, where m 1 = 3 , m 2 = 3 , m 3 = 8 and m 4 = 6, and where P 1 = (-1 ,-2) , P 2 = (-2 , 4) , P 3 = (3 ,-4) and P 4 = (-6 ,-5). Find their moments about the x- and y-axes, and nd the center of mass of the system. (b) Find the centroid of the region bounded by the curves with equations y = x , y = 0 and x = 4....
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This note was uploaded on 01/24/2011 for the course MATH 21 taught by Professor Mathdep during the Winter '98 term at Missouri S&T.

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