Midterm Exam 1 with Answers - Analytic Geometry and Calculus III | MATH 126, Exams of Analytical Geometry and Calculus

Download Midterm Exam 1 with Answers - Analytic Geometry and Calculus III | MATH 126 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! MIDTERM I ANSWERS Math 126, Section C October 18, 2006 NOTE. Most problems had multiple solutions. Below we give only one possible solution for each problem. It is perfectly acceptable if you had a different (but correct!) solution. 1. Find the Taylor series for a given function f(x). Give your answer using summation notation. (a) (7pts.) f(x) = ex, based at a = 2 Answer. ex = ∞∑ 0 e2 n! (x− 2)n (b) (8pts.) f(x) = ln(1− 2x), based at a = 0. Answer. f ′(x) = − 2 1−2x = −2 ∞∑ 0 2nxn, hence, f(x) = − ∞∑ 0 2n+1 n+1 xn+1 = − ∞∑ 1 2n n xn 2. (15pts.) Let f(x) = 1 (1−x)(1+x) . (a) (7pts.) Find the Taylor series for f(x) based at a = 0, and the interval of convergence. Give your answer using the summation notation. Answer. f(x) = 1 (1−x)(1+x) = 1 1−x2 = ∞∑ 0 x2n. The interval of convergence is (−1, 1). (b) (4pts.) Find the 6th Taylor polynomial of f(x) based at a = 0. Answer. T6(x) = 1 + x 2 + x4 + x6 (c) (4pts.) Find f (6)(0). Answer. f (6)(0) = 6! 3. Let f(x) = 2 cos2 x− 1. (a) (6pts.) Find the quadratic approximation T2(x) of f(x) based at a = 0 Answer. Using the double angle formula, we get f(x) = cos(2x). Hence, f(x) = ∞∑ 0 (−1)n 22nx2n (2n)! . Cutting off the tail, we obtain T2(x) = 1− 2x2 (b) (3pts.) Use the quadratic approximation to estimate f(π 8 ). Answer. T2( π 8 ) = 1− 2(π 8 )2 = 1− π 2 32 . 1