8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
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for all x in [0, c]. This enables us to conclude that
|E n (x)| ≤ M
|x| n+1
(n + 1)!
for all x in [0, c], which shows an important bound on the approximation’s error, E n .
Our work above was based on the approximation centered at a = 0; the argument may
be generalized to hold for any value of a, which results in the following theorem.
The Lagrange Error Bound for P n (x). Let f be a continuous function with n +
1 continuous derivatives. Suppose that M is a positive real number such that
f (n+1) (x)
≤ M on the interval [a, c]. If P n (x) is the nth order Taylor polynomial for
f (x) centered at x = a, then
|P n (c) − f (c)| ≤ M
|c − a| n+1
(n + 1)!
.
This error bound may now be used to tell us important information about Taylor
polynomials and Taylor series, as we see in the following examples and activities.
Example 8.4. Determine how well the 10th order Taylor polynomial P 10 (x) for sin(x),
centered at 0, approximates sin(2).
Solution. To answer this question we use f (x) = sin(x), c = 2, a = 0, and n = 10 in the
Lagrange error bound formula. To use the bound, we also need to find an appropriate
value for M. Note that the derivatives of f (x) = sin(x) are all equal to ± sin(x) or ± cos(x).
Thus,
f
(n+1) (x) ≤ 1
for any n and x. Therefore, we can choose M to be 1. Then
|P 10 (2) − f (2)| ≤ (1)
|2 − 0| 11
(11)!
=
2 11
(11)!
≈ 0.00005130671797.
So P 10 (2) approximates sin(2) to within at most 0.00005130671797. A computer algebra
system tells us that
P 10 (2) ≈ 0.9093474427 and sin(2) ≈ 0.9092974268
with an actual difference of about 0.0000500159.
513
for all x in [0, c]. This enables us to conclude that
|E n (x)| ≤ M
|x| n+1
(n + 1)!
for all x in [0, c], which shows an important bound on the approximation’s error, E n .
Our work above was based on the approximation centered at a = 0; the argument may
be generalized to hold for any value of a, which results in the following theorem.
The Lagrange Error Bound for P n (x). Let f be a continuous function with n +
1 continuous derivatives. Suppose that M is a positive real number such that
f (n+1) (x)
≤ M on the interval [a, c]. If P n (x) is the nth order Taylor polynomial for
f (x) centered at x = a, then
|P n (c) − f (c)| ≤ M
|c − a| n+1
(n + 1)!
.
This error bound may now be used to tell us important information about Taylor
polynomials and Taylor series, as we see in the following examples and activities.
Example 8.4. Determine how well the 10th order Taylor polynomial P 10 (x) for sin(x),
centered at 0, approximates sin(2).
Solution. To answer this question we use f (x) = sin(x), c = 2, a = 0, and n = 10 in the
Lagrange error bound formula. To use the bound, we also need to find an appropriate
value for M. Note that the derivatives of f (x) = sin(x) are all equal to ± sin(x) or ± cos(x).
Thus,
f
(n+1) (x) ≤ 1
for any n and x. Therefore, we can choose M to be 1. Then
|P 10 (2) − f (2)| ≤ (1)
|2 − 0| 11
(11)!
=
2 11
(11)!
≈ 0.00005130671797.
So P 10 (2) approximates sin(2) to within at most 0.00005130671797. A computer algebra
system tells us that
P 10 (2) ≈ 0.9093474427 and sin(2) ≈ 0.9092974268
with an actual difference of about 0.0000500159.