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Math 152-copyright Joe Kahlig, 13A
Section 10.9: Application of Taylor Polynomials
Taylor Polynomials.
The Taylor series of a function, f (x), can be expressed: f (x) =
∞
X
f (k) (a)
k=0
k!
(x − a)k . The n-th
degree Taylor polynomial of f (x) at a, denoted Tn is given by
Tn (x) = f (a) + f 0 (a)(x − a) +
f 00 (a)
f (n) (a)
(x − a)2 + ... +
(x − a)n
2!
n!
Define Rn (x) to be the remainder of the series such that f (x) = Tn (x) + Rn (x)
Taylor’s Inequality: If |f (n+1) (x)| ≤ M for |x − a| < R, then the remainder Rn (x) of the Taylor
series satisfies the inequality
|Rn (x)| ≤
M
|x − a|n+1
(n + 1)!
for |x − a| < R
Example: Find the Taylor polynomials, T1 , T2 , and T3 , for f (x) = xex centered at a = 2.
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Math 152-copyright Joe Kahlig, 13A
Example: Find the Taylor polynomials, T1 , T3 , and T5 for f (x) =
Example:Estimate the accuracy of T3 on −2 ≤ x ≤ 2.
1
centered at a = 0
x2 + 25
Math 152-copyright Joe Kahlig, 13A
Example: Find T2 at a = 9 for f (x) =
7.5 ≤ x ≤ 11?
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√
x. How accurate is this Taylor polynomial on the interval
Math 152-copyright Joe Kahlig, 13A
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π
Example: Estimate |R3 | for the f (x) = cos(x) for the Taylor polynomial centered at on the interval
3
2π
0≤x≤
.
3
Example: Express f (x) = 2x3 + 4x2 + 7x + 6 as a Taylor polynomial
A) about a = 0.
B) about a = 1.