Section 11.6 – Taylor’s Formula with Remainder
The Lagrange Remainder of a Taylor Polynomial
n 1n 1
n
f zR x x
n 1 !
where z is some number between x and c
The Error of a Taylor Polynomial
n 1
nR x b cn 1 !
M
where M is the maximum value of n 1f x
on the interval [b, c] or [c, b]
Let f be a function that has derivatives of all orders on the Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and 4f x 6 for all x in the interval (0, 1).
a. Find the third-degree Taylor polynomial about x = 0 for f.
2 33
1 1/ 4 3 / 8p x 1 x x x
2 2! 3!
2 33
1 1 1p x 1 x x x
2 8 16
b. Use your answer to part a to estimate the value of f(0.5)
2 3
3
1 10.5 0.5 0.5
1p 1 0.5
2 8 16
3
157p 0.5 1.227
128
Let f be a function that has derivatives of all orders on the Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and 4f x 6 for all x in the interval (0, 1).
c. What is the maximum possible error for the approximation made in part b?
3 1
3R x b c3 1 !
M
4
3 1
3R x b c3 1 !
f x
4
3R x 01
40.016
6
6
!0
4.5
3
157p 0.5 1.227
128
Estimate the error that results when arctan x is replaced by
34x
x if x 0.2 and3
f x 4
3 1
3R x b c3 1 !
M
3 1
3R x 0.2 03 1 !
4
4
3
1R x 0.2 0.000267
6
Estimate the error that results when ln(x + 1) is replaced by 21
x x if x 0.12
2
3
f x ln x 1
1f ' x
x 11
f " x
2f "' x
x 1
x 1
F ‘’’ (x) has a maximum value at x = -0.1
2 1
2R x 0 0.12 1 !
f ''' 0.1
2R x 0.000457
Find an approximation of ln 1.1 that is accurate to three decimalplaces.
We just determined that the error using the second degreeexpansion is 0.000457.
22
1p x x x
2
2
2
1p 0.1 0.1 0.1
2
2p 0.1 0.095
Use a Taylor Polynomial to estimate cos(0.2) to 3 decimal places2 4 6x x x
cosx 1 ...2! 4! 6!
If x = 0.2, Alternating Series Test works for convergence
2n0.2
0.0052n !
20.21 0.98
2os
!c x
Use a Taylor Polynomial to estimate 1
0
sinxdx
x with three decimal
place accuracy.1 3 5
0
1 x xx ... dx
x 3! 5!
1 2 4
0
x x1 ... dx
3! 5!
3 5 71o
x x xx ... |
3 3! 5 5! 7 7!
1 1 11 ...
3 3! 5 5! 7 7!
Satisfies Alternating Series Test
1
0.0052n 1 2n 1 !
1 11 0.946
3 3! 5 5!
Suppose the function f is defined so that
2
32
2 3x 11 1f 1 , f ' 1 , f " x
2 2 x 1
a. Write a second degree Taylor polynomial for f about x = 1
2
2
1 1 1/ 2p x x 1
2!x 1
2 2
2
2
1 1 1p x x 1 x 1
2 2 4
b. Use the result from (a) to approximate f(1.5)
2
2 0.311 1 1
p 1.5 1.5 1 1.5 12 2 4
25
Suppose the function f is defined so that
2
32
2 3x 11 1f 1 , f ' 1 , f " x
2 2 x 1
c. 1If f " x
2 for all x in [1, 1.5], find an upper bound for the
approximation error in part b if
n 1
nR x b cn 1 !
M
2
42
24x 1 xf "' x
x 1
12
2R x1 !
M
21.5 1
3
2
0
6MR
.5x
3
2
0.R x
60.42 009 77
5.008
The first four derivatives of 1f x are
1 x
3 / 2
5 / 2
7 / 2
9 / 2
1f ' x
2 1 x
3f " x
4 1 x
15f "' x
8 1 x
105f "" x
16 1 x
a. Find the third-degree Taylor approximation to f at x = 0
b. Use your answer in (a) to find an approximation of f(0.5)
c. Estimate the error involved in the approximation in (b). Show your reasoning.
The first four derivatives of 1f x are
1 x
3 / 2
5 / 2
7 / 2
9 / 2
1f ' x
2 1 x
3f " x
4 1 x
15f "' x
8 1 x
105f "" x
16 1 x
a. Find the third-degree Taylor approximation to f at x = 0
1 3 15f 0 1 f ' 0 f " 0 f "' 0
2 4 8
2 33
1 3 / 4 15 / 8p x 1 x x x
2 2! 3!
2 33
1 3 5p x 1 x x x
2 8 16
b. Use your answer in (a) to find an approximation of f(0.5)
2 3
3
1 3 5p 0.5 1 0.5 0.5 0.5
2 8 16
3
103p 0.5
128
The first four derivatives of 1f x are
1 x
3 / 2
5 / 2
/ 2
9
7
/ 2
1f ' x
2 1 x
3f " x
4 1 x
15f "' x
105f "" x
16 1
x
x
8 1
c. Estimate the error involved in the approximation in (b). Show your reasoning.
13
3R x1 !
M
30.5 0
13
3 3R x
1 !0
105 /16.5 0
4
3
105 350.5 0.017
384 2048R x
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