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Homework_1 - (a f x = x 3 e x 2 x 2 1 2(b f x = x 2 ± x 1...

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MATH 230 HOMEWORK 1 - DUE IN CLASS ON 8/31/09 PAUL SIEGEL, INSTRUCTOR Instructions: Solve all problems completely. The use of calculators or computer aids is permitted but not required; full credit will be given only if all work is shown. You are encouraged to work with other students, but the solutions that you hand in must be written by you in your own words and they should be a re ection of your own thinking. If you do choose to work with other students, please hand in a list of all group members with your solutions. Problem 1. For each function f, nd the equation of the linear function which best approximates f near the given point a. (a) f ( x ) = x 2 sin 3 (5 x ), a = 10 (b) f ( x ) = 4 sin x 2 x +cos x , a = 0 (c) f ( x ) = x 2 e x x +1 , a = 1 2 Problem 2. Use calculus to solve each of the following optimization problems. (a) Find the maximum value of f ( x ) = log( x= ( x 2 + 1)) on the interval (0 ; 1 ). (b) Find the point on the graph of y = p x which is closest to the point (9 ; 0). (c) Find the dimensions of the rectangle bounded by the x -axis, the y -axis, and the graph of y = 8 x 3 whose area is maximal. Problem 3.
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Unformatted text preview: (a) f ( x ) = x 3 e x 2 ( x 2 +1) 2 (b) f ( x ) = x 2 ± x +1 ( x +1) 3 (c) f ( x ) = 1 x 2 p x 2 ± 36 Problem 4. Compute each of the following de±nite integrals. (a) Find the area under the curve y = sec 2 x between x = 0 and x = ± . Be careful! (b) Find the volume of the solid obtained by rotating the region in the plane bounded by y = x 2 and x = y 2 around the x axis. (c) Show that the surface area of a sphere of radius r is 4 ±r 2 by rotating the circle y = p r 2 ± x 2 around the x axis. 1 2 PAUL SIEGEL, INSTRUCTOR Problem 5. Challenge Problem (will not be graded): Where is the ±aw in the following \proof" that 0 = 1? Let us compute the inde±nite integral R 1 x dx using integration by parts. Set u = 1 =x and dv = dx , so that du = ± 1 =x 2 dx and v = x . We get: Z 1 x dx = uv ± Z v du = x x ± Z x ± 1 x 2 dx = 1 + Z 1 x dx Subtracting R 1 x dx from both sides, we are forced to conclude that 0 = 1!...
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