Section 11.6 – Taylor’s Formula with Remainder
The Lagrange Remainder of a Taylor Polynomial
Rn x
z n 1
x
n 1 !
f
n 1
where z is some number between x and c
The Error of a Taylor Polynomial
Rn x
M
n 1 !
bc
n 1
where M is the maximum value of f
on the interval [b, c] or [c, b]
n 1
x
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 f 4 x 6 for all x in the interval (0, 1).
a. Find the third-degree Taylor polynomial about x = 0 for f.
p3 x 1
p3 x 1
1
2
1
2
x
1/ 4
x
1
2!
8
x
2
x
2
3/8
1
x
3
3!
x
3
16
b. Use your answer to part a to estimate the value of f(0.5)
p 3 0 .5 1
p 3 0 .5
1
2
157
128
1
0 .5 0 .5
8
1 .2 2 7
2
1
16
0 .5
3
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 f 4 x 6 for all x in the interval (0, 1).
c. What is the maximum possible error for the approximation
made in part b?
p 3 0 .5
R3 x
R3 x
R3 x
157
128
1 .2 2 7
M
3 1 !
4
f x
3 1 !
6
4 !
bc
bc
0 .5 0
4
3 1
3 1
1
64
0 .0 1 6
Estimate the error that results when arctan x is replaced by
x
x
3
3
if x 0 .2 a n d f
4
x
R3 x
R3 x
R3 x
4
M
3 1 !
4
3 1 !
1
6
0 .2
4
bc
3 1
0 .2 0
3 1
0 .0 0 0 2 6 7
Estimate the error that results when ln(x + 1) is replaced by
x
1
2
x if x 0 .1
2
f x ln x 1
f 'x
f "x
f "' x
1
R2 x
f ''' 0.1
2 1 !
R 2 x 0.000457
x 1
1
x 1
2
2
x 1
3
F ‘’’ (x) has a maximum value at x = -0.1
0 0.1
2 1
Find an approximation of ln 1.1 that is accurate to three decimal
places.
We just determined that the error using the second degree
expansion is 0.000457.
p2 x x
1
x
2
2
p 2 0 .1 0 .1
p 2 0.1 0.095
1
2
0 .1
2
Use a Taylor Polynomial to estimate cos(0.2) to 3 decimal places
co s x 1
x
2
2!
x
4
4!
x
6
6!
...
If x = 0.2, Alternating Series Test works for convergence
0.2
2n !
2n
0.005
cos x 1
0 .2
2!
2
0 .9 8
1
Use a Taylor Polynomial to estimate
place accuracy.
1
0
3
5
1
x
x
... dx
x
x
3!
5!
sin x
0
dx with three decimal
x
1
2n 1 2n 1 !
0 .0 0 5
2
4
x
x
1 3 ! 5 ! ... dx
0
1
3
5
7
1
x
x
x
... |o
x
3 3! 5 5! 7 7!
1
1
1
1
...
3 3! 5 5! 7 7!
Satisfies Alternating Series Test
1
1
3 3!
1
5 5!
0 .9 4 6
Suppose the function f is defined so that
f 1
1
2
, f ' 1
1
2
2 3x 1
, f " x
x
2
2
1
3
a. Write a second degree Taylor polynomial for f about x = 1
p2 x
p2 x
1
2
1
2
1
2
1
2
x 1
1/ 2
x 1
1
2!
4
x 1
x 1
2
2
b. Use the result from (a) to approximate f(1.5)
p 2 1 .5
1
2
1
2
1 .5 1
1
4
1 .5 1
2
0 .3 1 2 5
Suppose the function f is defined so that
f 1
c. If f " x
1
1
2
, f ' 1
1
2
, f " x
2 3x 1
x
R2 x
2
M
n 1 !
M
2 1 !
R2 x M
2
1
3
for all x in [1, 1.5], find an upper bound for the
approximation error in part b if f "' x
Rn x
2
0 .5
bc
24 x 1 x
x
2
4
0 .5
3
1
2
n 1
1 .5 1
2 1
3
6
R 2 x 0 .4 2 0 9
6
0 .0 0 8 7 7
The first four derivatives of f x
f 'x
1
1 x
are
1
2 1 x
f "x
f "' x
f "" x
a. Find the third-degree Taylor
approximation to f at x = 0
3/2
3
4 1 x
5/2
b. Use your answer in (a) to find
an approximation of f(0.5)
15
8 1 x
7/2
105
1 6 1 x
9/2
c. Estimate the error involved in the
approximation in (b). Show your
reasoning.
1
The first four derivatives of f x
f 'x
1
2 1 x
f "x
f "' x
f "" x
1 x
a. Find the third-degree Taylor
approximation to f at x = 0
3/2
f 0 1
3
4 1 x
f ' 0
5/2
p3 x 1
15
8 1 x
are
1
2
1
f " 0
2
x
3/4
x
3
2!
x
2
3
4
15 / 8
15
f "' 0
x
8
3
3!
7/2
p3 x 1
105
1 6 1 x
9/2
1
2
8
x
2
5
x
3
16
b. Use your answer in (a) to find
an approximation of f(0.5)
p 3 0 .5 1
p 3 0 .5
1
2
103
128
3
0 .5 0 .5
8
2
5
16
0 .5
3
The first four derivatives of f x
f 'x
1
2 1 x
f "x
f "' x
f "" x
1 x
3
5/2
R3 x
15
8 1 x
7/2
105
1 6 1 x
are
c. Estimate the error involved in the
approximation in (b). Show your
reasoning.
3/2
4 1 x
1
9/2
M
3 1 !
R3 x
105 / 16
R3 x
105
3 1 !
384
0 .5
0 .5 0
0 .5 0
4
3 1
3 1
35
2048
0 .0 1 7
0
