University of California, Davis
Department of Applied Science
Spring 2004
David M. Rocke
Numerical Methods
EAD 115
April 27, 2004
Midterm Examination
NAME
For all problems on this midterm examination, we will use the function
f
(
x
) = 2
x
3

3
x
2

11
x
+ 6
.
1. Compute the 0 through third order Taylor series approximations of
f
(1) at
x
= 0 (i.e., the Maclaurin series). Compute the true absolute error
E
t
and the
approximate relative error
E
a
in each case, where the latter is obtained from
the change in the approximation from one iteration to the next.
Solution
We have
f
(
x
)
=
2
x
3

3
x
2

11
x
+ 6
f
(
x
)
=
6
x
2

6
x

11
f
(
x
)
=
12
x

6
f
(
x
)
=
12
f
(0)
=
6
f
(0)
=

11
f
(0)
=

6
f
(0)
=
12
The first four approximations of
f
(1) using the Maclaurin series
f
(
x
) =
f
(0) +
f
(0)
x
+
1
2
f
(0)
x
2
+
1
6
f
(0)
x
2
+
· · ·
are
ˆ
f
0
(1) = 6
ˆ
f
1
(1) = 6

11(1) =

5
ˆ
f
2
(1) = 6

11(1)

(1
/
2)(6)(1) =

8
ˆ
f
3
(1) = 6

11(1)

(1
/
2)(6)(1) + (1
/
6)(12)(1) =

6
The respective true absolute errors are 12, 1, 2, and 0, while the estimated
absolute errors of the last three are 11, 3, and 2 (the first one has no estimated
error since there is no previous approximation).
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 Spring '10
 ROCKE
 Numerical Analysis, absolute errors, true absolute error, David M. Rocke

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