Fall 2010 Math 152
2. Given a power series for g(x) is
1
.
find g
2
Night Before Drill for Exam III
courtesy: David J. Manuel
3. ˆ
Write a series
1
ln(1 + x5 ) dx.
(covering 10.4-11.2)
which
is
∞
X
xn+1
,
n+1
n=0
equal
to
0
1
Section 10.4
4
∞
X
(−1)n
1. Determine whether the series
is
ln(n)
n=2
absolutely convergent, convergent (but not
absolutely), or divergent.
∞
X
1. Find the first 3 terms of the Maclaurin series
1
· ex .
for f (x) =
1−x
2. Find the Maclaurin series for f (x) = x2 cos x.
(−4)
is
n!
n=0
absolutely convergent, convergent (but not
absolutely), or divergent.
2. Determine whether the series
n
1
3. Find the Taylor Series for f (x) = centered
x
at x = 2.
∞
X
(−1)n
, what is the mini(n!)2
n=1
mum number of terms needed to estimate the
exact sum to within 10−6 ?
3. Given the series
2
5
Section 10.9
1.
(a) Find the third-degree
Taylor polynomial
√
for f (x) = 1 + x centered at a = 0.
Section 10.5
(b) Use Taylor’s Inequality to estimate the
error in
approximation on the in your
1
terval 0,
2
1
2. The formula I(α) = 4 α − sin(2α) gives
2
the moment of inertia of a circular sector of
radius 2 and central angle α about its bisector. Find the 3rd degree Taylor polynomial
of I about a = 0.
1. Find the radius and interval of convergence of
∞
X
(−1)n 3n (x − 1)n
f (x) =
.
(n + 2)2
n=0
2. Find the radius
∞
X
(−1)n (n!)2 xn
.
(2n)!
n=0
Section 10.7
of
convergence
of
∞
X
cn xn
3. Given the radius of convergence of
n=0
is 6, find the radius of convergence of
∞
X
2n cn xn
.
n
n=0
3
6
Section 11.1
1. Classify
the
triangle
with
vertices
P (4, 2, 1), Q(3, 3, 1), and R(3, 2, 3) (right,
isosceles, equilateral)
Section 10.6
2. Describe in words the region in R3 (threedimensional space) represented by the equation x2 − 4x + y 2 + z 2 − 3z = 0.
1
and
1 + 2x2
determine its radius of convergence.
1. Find a power series for f (x) =
1
3. Describe in words the region in R3 (threedimensional space) represented by the equation y + z = 2.
7
Section 11.2
1. Find the vector and scalar projection of
h1, −7, 2i onto the vector h3, −2, −1i.
2. Find the angle between the diagonal of a cube
of side length 1 and the diagonal of one of its
faces.
3. Given the vectors a = h1, 3, 2i and b = −3i +
2j + 4k, find the vector c shown in the figure
below:
b
c
a
2