Math 113 (Calculus Ii) Test 4 Form A With Answer Key - Brigham Young University Page 5
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1
2
3
4
5x
e
1
15. (8 points) Evaluate the indefinite integral
dx as an infinite series.
x
Solution:
n
n
x
x
x
1
e
1
n=0
n=1
n!
n!
=
=
x
x
x
n 1
x
=
.
n!
n=1
Thus
x
n
e
1
x
dx =
+ C.
n · n!
x
n=1
2
x
16. (10 points) Find the Taylor polynomial of degree 2 for f (x) = e
expanded about a = 0. Use
Taylor’s Inequality to estimate the accuracy of the approximation f (x)
T
(x) when x lies
2
in the interval [0, 0.1].
Solution: Note that
2
x
f (x) = 2xe
,
2
2
2
x
2
x
x
2
f (x) = 2e
+ 4x
e
= 2e
(1 + 2x
),
and
2
2
2
x
2
x
3
x
f (x) = 4xe
(1 + 2x
) + 2e
(4x) = 4(2x
+ 3x)e
.
Thus, f (0) = 1, f (0) = 0, and f (0) = 2. The Taylor polynomial of degree 2 is therefore
2
2
2
(x) = 1 + 0 · x +
T
x
= 1 + x
.
2
2!
x
2
(Note: You could also have found T
by taking the Maclaurin series for e
, substituting x
2
for x, and taking the terms up to the quadratic.)
Next, we bound the error. To do this, we need a bound of the third derivative (which is why
we found it above). Notice that for positive x, each of the terms are positive and increasing,
so the third derivative is increasing. A bound for f
on [0, 0.1] is therefore
.01
.01
M = f (0.1) = 4(2(.001) + .3)e
= (1.2008)e
.
3
0.1
M (0.1)
1.2008e
(.001)
Thus, |R
|
=
.
2
3!
6
END OF EXAM
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