Math 106 Sections C and D
Final Exam (100 points)
Name:
Show all your work to receive full credit for a problem.
There are twelve questions. Questions are printed on both sides of a page.
To evaluate any integrals, do not use the calculator integral function. You may use
formulas 1-18 and 39-51 from the table of integrals.
You may use any of the following facts:
Z
b
Z
q
1 + (f 0 (x))2 dx
Arclength=
u dv = uv −
Z
v du
a
|I − Ln | ≤
K1 (b − a)2
2n
|I − Rn | ≤
K1 (b − a)2
2n
|I − Mn | ≤
K2 (b − a)3
24n2
|I − Sn | ≤
K4 (b − a)5
180n4
T (x) = f (x0) + f 0 (x0 )(x − x0 ) +
∞
1
Z
1
0
1
dx converges for p > 1 and diverges for p ≤ 1.
xp
1
dx converges for p < 1 and diverges for p ≥ 1.
xp
∞
X
1
dx converges for p > 1 and diverges for p ≤ 1.
p
n=1 n
sin x = x −
x3 x5 x7
+
− + · · · for x in (−∞, ∞).
3!
5!
7!
cos x = 1 −
x2 x4 x6
+
− + · · · for x in (−∞, ∞).
2!
4!
6!
ex = 1 + x +
S2n =
K2 (b − a)3
12n2
Tn + 2Mn
3
f 00(x0 )
f (n) (x0)
(x − x0 )2 + · · · +
(x − x0)n + · · ·
2!
n!
Kn+1
|x − x0|n+1
|f (x) − Pn (x)| ≤
(n + 1)!
Z
|I − Tn | ≤
x2 x3 x4
+
+ + · · · for x in (−∞, ∞).
2!
3!
4!
1
−(x − m)2
√
f (x) =
exp
2s2
2π s
sin(2x) = 2 sin x cos x
cos(2x) = cos2 x − sin2 x
!
1. (14 points) Evaluate the following integrals exactly.
(a)
(b)
Z
Z
6x − 20
dx
4x − x2
x2
√
dx
25 + x2
4. (6 points) Consider the IVP y 0 = x − y, y(2.5) = 2. Use Euler’s method with two steps to
estimate y(3.5). (Do not use a calculator program for this problem.)
5. (6 points) Consider the region bounded by the curve y = ex, and the lines x = 0, x = 1 and
the x-axis. Write (but do not evaluate) an integral to find the volume of the solid that is
formed when the region is rotated about the line y = e.
8. (15 points) Determine the convergence of the following series. If possible, find the sum of the
series if it converges.
(a)
7 7 7
7
+ + +
+ ···
2 4 8 16
(b)
41 401 4001 40001
+
+
+
···
10 100 1000 10000
(c)
∞
X
ln n
n=2 n
9. (8 points) Find the interval and radius of convergence of the series
∞
X
(x − 4)n
.
n3n
n=2
10. (10 points) Let f (x) = cos(2x2 ). Use this function to answer the following questions. In this
problem, you do not need to write any series using summation notation.
(a) Use a known power series to write the first four non-zero terms of the power series
representation for f .
(b) Use your answer in part (a) to write the power series representation of
Z
1
0
cos(2x2 ) − 1
.
x→0
x4
(c) Use your answer in part (a) to evaluate the limit: lim
cos(2x2 ) dx.
11. (8 points) Let f (x) = ln x.
(a) Find the first four non-zero terms of the Taylor series for f based at x0 = 1. Then write
the series using the summation notation.
(b) Let P3 (x) be the third-order Taylor polynomial for f based at x0 = 1. What does
Taylor’s theorem imply about the maximum approximation error committed by P3 over
the interval [0.5, 3]? (Find the best possible value for Kn+1 .)
12. (10 points) Suppose that the partial sums of the series
∞
X
ak are Sn =
k=1
n
X
k=1
ak =
2n + 5
.
3n − 1
(a) Does the sequence Sn converge? Justify your answer.
(b) Does the series
∞
X
ak converge? Justify your answer. If the series converges, what is its
k=1
sum?
(c) Find a1 and a2 , i.e., find the first two terms of the series.