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 .
To assure a good approximation near x = a, we require that
f (a) = Tn (a)
f 0 (a) = Tn0 (a)
f 00 (a) = Tn00 (a)
..
.
(n)
f (a) = Tn(n) (a).
The higher derivatives of Tn (x) are
Tn (x)
Tn0 (x)
Tn00 (x)
Tn000 (x)
=
=
=
=
..
.
(n)
Tn (x) =
c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + c4 (x − a)4 + · · · + cn (x − a)n
c1 + 2c2 (x − a) + 3c3 (x − a)2 + 4c4 (x − a)3 + · · · + nan (x − a)n−1
2c2 + 3(2)c3 (x − a) + 4(3)(x − a)2 + · · · + n(n − 1)an (x − a)n−2
3(2)c3 + 4(3)(2)c4 (x − a) + · · · + n(n − 1)(n − 2)an (x − a)n−3
n!an .
Evaluating at x = a gives
Tn (a)
Tn0 (a)
Tn00 (a)
Tn000 (a)
=
=
=
=
..
.
(n)
Tn (a) =
It follows that the nth coefficient of Tn (x) is cn =
c0
c1
2!c2
3!c3
n!cn .
f (n) (a)
.
n!
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!
1
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.
2
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.
5
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].
6