Section 7.6: The Taylor Approximation
Suppose that we want to approximate f (x) at x = a by a polynomial of degree n:
f (x) ≈ Tn (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + c4 (x − a)4 + · · · + cn (x − a)n .
How should the coefficients, ck , be defined to obtain a good approximation?
Definition: The nth degree Taylor polynomial for f (x) at x = a is given by
Tn (x) = f (a) + f 0 (a)(x − a) +
f 00 (a)
f 000 (a)
f (n) (a)
(x − a)2 +
(x − a)3 + · · · +
(x − a)n .
2!
3!
n!
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Example: Find the nth degree Taylor polynomial for f (x) = ex at x = 0.
Example: Find the nth degree Taylor polynomial for f (x) = sin x at x = 0.
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Example: Find the nth degree Taylor polynomial for f (x) = cos x at x = 0.
Example: Find the nth degree Taylor polynomial for f (x) =
3
1
at x = 0.
1−x
Example: Find the third degree Taylor polynomial for f (x) =
√
x at x = 4.
Definition: If the nth degree Taylor polynomial Tn (x) is used to approximate f (x) at x = a,
then the remainder of the approximation is
Rn (x) = f (x) − Tn (x).
How good is this approximation? How large should we take n to achieve a desired accuracy?
These questions can be answered using Taylor’s Inequality.
Theorem: (Taylor’s Inequality)
If |f (n+1) (x)| ≤ M for |x − a| < R, then the remainder Rn (x) of the Taylor series satisfies
|Rn (x)| ≤
M
|x − a|n+1 .
(n + 1)!
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Example: Consider the function f (x) = ln x.
(a) Find the third degree Taylor polynomial for f (x) at x = 3.
(b) Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ T3 (x)
for 2 ≤ x ≤ 4.
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Example: Determine the degree of the Taylor polynomial of f (x) = ln(x + 1) at x = 0 so
that the approximation f (x) ≈ Tn (x) has error less than 0.01 on the interval [0, 0.1].
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