M161, Test 3, Spring 2006 Part 2
Problem
10
Points
5
11
5
12
5
13
5
Total
20
Score
NAME:
SECTION:
INSTRUCTOR:
You may not use calculators.
cos 2θ = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1
sin 2θ = 2 sin θ cos θ
Taylor series of the function f (x) about x = a.
∞
X
f (n) (a)
f ′′ (a)
f ′ (a)
(x − a) +
(x − a)2 + · · · =
(x − a)n
f (a) +
1!
2!
n!
n=0
Taylor Inequality: If M is a constant such that |f (n+1) (x)| ≤ M for a ≤ x ≤ b, then |Rn (x)| ≤
where
Z x
n 1
(t − x)n f (n+1) (t) dt
Rn (x) = (−1)
n! a
Length of a polar Curve: L =
Z
β
α
ln(n)
= 0,
n→∞
n
lim
lim
n→∞
s
r2
√
n
n = 1,
+
dr
dθ
2
(1)
dθ
x n
1+
= ex ,
n→∞
n
lim x1/n = 1 any x, lim
n→∞
M|x−a|n+1
(n+1)!
Partial sum of a geometric series: Sn = a + ar + ar2 + · · · + arn = a
1 − rn+1
1−r
xn
= 0, any x.
n→∞ n!
lim
10. On the axes given below plot the polar curve r = cos 3θ.
90
1
120
60
0.8
0.6
150
30
0.4
0.2
180
0
210
330
240
300
270
r = cos(3 t)
11. Plot the parametric curve x = cos t, y = t + sin t, 0 ≤ t ≤ 3π. Indicate the direction by arrows that the
curve is traversed as t goes from 0 to 3π.
x = cos(t), y = t+sin(t)
9
8
7
y
6
5
4
3
2
1
0
−5
0
x
5
12. Compute the integral
Z
π/2
sin4 (2θ)dθ.
0
Z
π/2
sin4 (2θ)dθ = .58904862254808623221174563436491
0
4
13. For the function f (x) = x the length of the curve for x between 0 and 2 is given by L =
0
Find L.
L=
Z
Z
2
0
p
1 + 16x6 dx = 16.646865484765381603158816799036
2
p
1 + 16x6 dx.