tut1 - ( x ) = (1 + x ) 1 / 3 . a) Find the Taylor...

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MATH 237 Assignment 1 Friday, September 15, 2006 1) Evaluate the following limits. a) lim x 0 x - sin x x 2 arctan x b) lim x →∞ x (e 1 /x - 1) c) lim x 0 x 4 sin ± 1 x ² d) lim x →∞ ln x x 2) Let g ( x ) = ³ 2 sin x - cos x when x 0 b e cx when x > 0 . a) Determine the value of b that makes g ( x ) continuous at x = 0. b) Find all b and c that make g ( x ) differentiable at x = 0, giving reasons for your answer. 3) Let h ( x ) = ³ x when x < 0 e x when x 0 . Is h continuous at x = 0? Is h differentiable at x = 0? Explain. 4) If f ( x ) = 1 1 + x , use the definition of the derivative to prove that f 0 ( a ) = - 1 (1 + a ) 2 for a 6 = - 1. 5) a) The displacement of a damped oscillator at time t is given by A ( t ) = e - t sin t where 0 t 2 π . Find the time t in the interval [0 , 2 π ] at which A ( t ) is maximum, and then compute the maximum value of A ( t ). b) A cylinder is inscribed in a cone of height H and radius R . Find the dimensions of the cylinder with the largest possible volume. What is the maximum volume? 6) Prove that if a function f is continuous on the closed interval [0 , 5] and f 0 ( x ) 2 on the open interval (0 , 5) and f (0) = 1, then f (5) 11.
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7) Let f
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Unformatted text preview: ( x ) = (1 + x ) 1 / 3 . a) Find the Taylor polynomial of degree 2, P 2 ,a ( x ), of f expanded about a = 0. b) For the given f use Taylors theorem to write an expression for the error term f ( x )-P 2 , ( x ). c) Show that when x &amp;gt; 0 the error f ( x )-P 2 ,a ( x ) is at most 5 81 x 3 . d) Write a fraction that estimates (1 . 2) 1 / 3 and show that the error in your estimate is at most 1 2025 . 8) Evaluate the following integrals: a) Z x 2 x 3 + 7 dx b) Z / 4 tan d c) Z 1 t ( t-1) dt d) Z x 2 ln x dx e) Z 1 arctan x 1 + x 2 dx f) Z 1 x e-2 x dx 9) Let f ( x ) = Z x cos( t 2 ) dt . a) Find f ( x ). b) Using the known power series for cos x , nd a power series for f (1). c) Use the rst 2 terms of the series to estimate the integral Z 1 cos( t 2 ) dt . Find an upper bound for the error of this approximation....
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tut1 - ( x ) = (1 + x ) 1 / 3 . a) Find the Taylor...

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