University:
San Diego State University
Course:
MATH 151 | Calculus II
Academic year:

2023
Views:

472

Pages:

1
Author:

Leland Mcdonald

Workshop #27 1. Recall that the formula for the Maclaurin series is as follows. f (0) + f 0 (0) f 00 (0) 2 f 000 (0) 3 f (4) (0) 4 x+ x + x + x + ··· 1! 2! 3! 4! (a) Use this formula to find the Maclaurin series for f (x) = cos(2x) through the degree 4 term. (b) On a graphing calculator, graph both f (x) = cos(2x) and the degree four Maclaurin polynomial that you found above. On what interval would you say the polynomial is a good approximation of f (x)? (c) Recall that the Maclaurin series for cos x is as follows. cos x = 1 − 1 2 1 1 x + x4 − x6 + · · · 2! 4! 6! Use this to find the Maclaurin series for f (x) = cos(2x). Does this match what you found in part 1a? 2. Find the Maclaurin series for the function f (x) = (x + 2)2 − 3. √ 3. The power series representation of f (x) = 1 − x is given below, and it converges for −1 < x < 1. √ 1 1 1 5 4 1 · 3 · 5 · · · (2n − 3) n x − ··· − x − ··· 1 − x = 1 − x − x2 − x3 − 2 8 16 128 n!2n 1 1 (a) Using this series, find the first 4 terms in the power series for − (1 − x)− 2 . 2 1 (b) Use this to find the first four nonzero terms in the power series for √ . 1 − x2 (c) Use this to find the first four nonzero terms in the power series for arcsin x. arcsin x (d) Use this to find the first four nonzero terms in the power series for . x arcsin x . (e) Using what you have found thus far, find lim x→0 x (f) Check your answer to the above limit by using L’Hôpital’s Rule.