Problem
1abc
Points
15
1d
5
2
10
3
10
4
15
5
15
6
15
7
10
8
5
Total
100
Score
M161, Test 3, Fall 2004
NAME:
SECTION:
INSTRUCTOR:
You may use calculators.
stored in your calculators.
No formulas can be
1 + cos 2θ
1 − cos 2θ
sin2 θ =
2
2
′′
Error Bounds. Suppose |f (x)| ≤ K for a ≤ x ≤ b. If ET is the error in the Trapezoidal Rule, then
cos2 θ =
|ET | ≤
K∆x2 (b − a)
K(b − a)3
or
it
may
be
written
as
|E
|
≤
.
T
12n2
12
Suppose |f (4) (x)| ≤ K for a ≤ x ≤ b. If ES is the error involved in using Simpson’s Rules, then
|ES | ≤
K(b − a)5
K∆x4 (b − a)
or it may be written as |ES | ≤
.
4
180n
180
1. Determine whether the series is convergent or divergent. In either case explain why. You must justify
your answers.
∞
X
1
√
(a)
n+1
n=1
∞
X
(b)
1
√
3
n +n+1
n=1
(c)
∞
X
(−1)n
√
n+1
n=1
(d)
∞
X
32n
n5
n=4
2. Determine whether the series is absolutely convergent, conditionally convergent or divergent. In any case,
explain why.
∞
X
ln n
(a)
(−1)n
n
n=1
(b)
∞
X
(−100)n
n!
n=1
3. Find the interval of convergence of the series
∞
X
(x + 4)n
.
n3n
n=1
4. The parametric equations x = 4 sin(t − π), y = 2 cos(t − π), 0 ≤ t ≤ π describe the motion of a particle in
the x-y plane.
(a) Sketch the path of the particle indicating the direction of motion.
4
3
2
1
0
−1
−2
−3
−4
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
(b) What is the equation of the tangent to the curve at the point where t = 3π/4? Draw the tangent line on
the curve drawn above.
(c) What are the Cartesian coordinates of the point where t = π.
(d) Find the Cartesian equation for the parametric equations.
5. (a) Sketch the curve r = 1 + 2 cos θ.
4
3
2
1
0
−1
−2
−3
−4
−2
−1.5
−1
−0.5
0
x
0.5
(b) Find the area inside the large loop of the curve.
(c) Find the area inside the small loop of the curve.
1
1.5
2
6. (a) Find the Cartesian equation form of the polar curve r cos θ − y sin θ = 7.
(b) Find the polar equation for the Cartesian equation xy = 2.
(c) Find the Cartesian equation form of the polar curve r2 = 4r sin θ. Identify the curve.
7. Use power series to solve the initial value problem y ′ + 2y = 0, y(0) = 1.
8. Find the power series representation of f (x) =
1
d2 1
2
.
Hint:
.
=
(1 − x)3
dx2 1 − x
(1 − x)3