mid1s - th Taylor polynomial of f x based at a = 0 Simplify...

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Name Quiz Section MIDTERM I Math 126, Section C October 18, 2006 Problem Total Points Score 1 15 2 15 3 15 4 15 Total 60 - You may use a scientific calculator and one two-sided sheet of handwritten notes. No other notes, books or calculators are allowed. Please turn off your cell phone. - Show all your work to get full credit. - Read instructions for each problem CAREFULLY. - Leave all your answers in EXACT form. - Check your work! 1
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1. (15pts) Find the Taylor series for a given function f ( x ). Give your answer using summation notation. (a) f ( x ) = e x , based at a = 2 (b) f ( x ) = ln(1 - 2 x ), based at a = 0. 2
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2. (15pts) Let f ( x ) = 1 (1 - x )(1+ x ) . (a) Find the Taylor series for f ( x ) based at a = 0, and the interval of convergence. Give your answer using the summation notation. (b) Find the 6
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Unformatted text preview: th Taylor polynomial of f ( x ) based at a = 0. Simplify your answer as much as possible. (c) Find f (6) (0). 3 3. (15pts) Let f ( x ) = 2cos 2 x-1. (a) Find the quadratic approximation T 2 ( x ) of f ( x ) based at a = 0 (b) Use the quadratic approximation to estimate f ( π 8 ). (c) Using Taylor’s inequality, find the error bound for the estimate you computed in (b). 4 4. (15pts) Let A = (3 , , 0), B = (0 , 4 , 0), and C = (0 , , 1). (a) Find the area of the triangle ABC Hint. The following indentity may be useful: 3 2 + 4 2 + 12 2 = 13 2 . (b) Let CH be the height of the triangle from the vertex C to the base AB . Find the coordinates of the point H . 5...
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