(4, 6)
(3, 2)
Lagrange Error Bound Worksheet
1. Let f be a function that has derivatives of all orders on the interval
1, 1
. Assume
0 1,f
4
1 1 3
0 , 0 , 0 , and 6
2 4 8
f f f f x
for all x in the interval (0, 1).
(a) Find the third-degree Taylor polynomial about x = 0 for the function f.
(b) Use your answer to part (a) to estimate the value of
0.5 .f
(c) What is the maximum possible error for the approximation made in part (b)?
2. Let f be the function defined by
.f x x
(a) Find the second-degree Taylor polynomial about x = 4 for the function f.
(b) Use your answer to part (a) to estimate the value of
4.2 .f
(c) Find a bound on the error for the approximation in part (b).
3. (Calc) Let f be a function that has derivatives of all orders. Assume
3 1,f
1 1 3
3 , 3 , 3 ,
2 4 8
f f f
and the graph of
4
fx
on [3, 4]
is shown on the right. The graph of
4
fx
is increasing on [3, 4].
(a) Find the third-degree Taylor polynomial about x = 3 for the function f.
(b) Use your answer to part (a) to estimate the value of
3.7 .f
(c) Use information from the graph of
4
y f x
to show
that
3.7 3.7 0.08.fP
Graph of
4
fx
(d) Could
3.7f
equal 1.283? Show why or why not.
4. Estimate the error that results when sin x is replaced by
3
1 for 0.2.
6
x x x
Show your reasoning.
5. Use series to find an estimate for
2
1
0
x
e dx
that is accurate to three decimal places. Justify your
answer.
