Let f(x) = 1/(3-2x). (a) Find the 2nd order Taylor polynomial for f(x) centered at x = 1. (b) Use T²/(x; 1) to approximate 1/3. Round to 4 decimal places. (c) If |x-1] ≤ 0.1, find a "reasonable" upper bound on error when using T²(x; 1) to estimate f(x). Round to 4 decimal places past the leading 0s.
Let f(x) = 1/(3-2x). (a) Find the 2nd order Taylor polynomial for f(x) centered at x = 1. (b) Use T²/(x; 1) to approximate 1/3. Round to 4 decimal places. (c) If |x-1] ≤ 0.1, find a "reasonable" upper bound on error when using T²(x; 1) to estimate f(x). Round to 4 decimal places past the leading 0s.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Let f(x) = 1/(3-2x). (a) Find the 2nd order Taylor polynomial for f(x) centered at x = 1. (b) Use T²/(x; 1) to approximate 1/3. Round to 4 decimal places. (c) If |x-1] ≤ 0.1, find a "reasonable" upper bound on error when using T²(x; 1) to estimate f(x). Round to 4 decimal places past the leading 0s.
Expert Solution
Step 1
(a) To find the 2nd order Taylor polynomial for f(x) centered at x = 1, we first find the first and second derivatives of f(x):
f(x) = 1/(3-2x)
f'(x) = 2/(3-2x)^2
f''(x) = 8/(3-2x)^3
Then we plug these values into the formula for the 2nd order Taylor polynomial:
T²(x;1) = f(1) + f'(1)(x-1) + (1/2)f''(1)(x-1)^2
Plugging in the values for f(1), f'(1), and f''(1), we get:
T²(x;1) = 1/3 + (-2/9)(x-1) + (8/27)(x-1)^2
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