Problem
1
Points
812
2
16
3
32
4ab
18
4cde
27
5
9
6abc
27
6d
9
7
16
8
15
9
10
10
9
Total
200
M161, Final Exam, Spring 2006
Score
NAME:
SECTION:
INSTRUCTOR:
You may not use calculators.
Taylor series of the function f (x) about x = a.
∞
X
f ′′ (a)
f (n) (a)
f ′ (a)
(x − a) +
(x − a)2 + · · · =
(x − a)n
f (a) +
1!
2!
n!
n=0
Taylor polynomial plus error term
f (x) = Tn (x) + Rn (x) =
n
X
1
j=0
j!
f (j) (a)(x − a)j + (−1)n
1
n!
Z
x
(t − x)n f (n+1) (t) dt
a
Taylor Inequality: If M is a constant such that |f (n+1) (x)| ≤ M for a ≤ x ≤ b, then |Rn (x)| ≤
Length of a polar Curve: L =
lim
n→∞
ln(n)
= 0,
n
lim
n→∞
Z βr
√
n
r2 +
α
n = 1,
dr
dθ
2
dθ
lim x1/n = 1 any x, lim
n→∞
n→∞
Sum of a geometric series: a + ar + ar 2 + · · · + ar n = a
θ
sin =
2
r
M |x−a|n+1
.
(n+1)!
1 − cos θ
2
1+
x
n
n
= ex ,
lim
n→∞
xn
= 0, any x.
n!
n+1
1−r
1−r
θ
cos =
2
r
1 + cos θ
2
Error Bounds. Suppose |f ′′ (x)| ≤ K for a ≤ x ≤ b. If ET is the error in the Trapezoidal Rule, then
|ET | ≤
K(b − a)3
K∆x2 (b − a)
or it may be written as |ET | ≤
.
2
12n
12
Suppose |f (4) (x)| ≤ K for a ≤ x ≤ b. If ES is the error involved in using Simpson’s Rules, then
|ES | ≤
K∆x4 (b − a)
K(b − a)5
or it may be written as |ES | ≤
.
4
180n
180
cos2 θ =
1 + cos 2θ
2
sin2 θ =
1 − cos 2θ
2
1. Match the Taylor series with the functions. Place the number (Roman numeral) of
the appropriate series in the center column, to the right of the given function.
∞
X
A. ex/3
(i)
1 n
x
n
n=0 3 n!
B. 3x
(ii)
(−1)n+1 3n n
x
n
n=1
C. ln(1 + 3x)
(ln(3))n n
x
(iii)
n!
n=0
∞
X
∞
X
2. (a) Compute the 5th order Taylor polynomial, T5 , of f (x) = cos(2x) about x = π4 .
(b) Use the Taylor inequality to bound the error due to the approximation of f (x) = cos(2x)
i
h
.
by the 5th order Taylor polynomial of f (x) = cos(2x) about x = π4 , T5 , on the interval π8 , 3π
8
3. (a) Simplify 2i −
2 − 3i
. (Write as a + bi, a and b real.)
3 + 4i
(b) Find the cube roots of 1 − i.
3π
(c) Write 2e 4 i as a + bi, a and b real—evaluate any appropriate trig functions.
(d) Suppose w = c + id and z = e + if , c, d, e, f real. Simplify
and b real.)
w + w̄
. (Write as a + bi, a
z − z̄
4. Calculate
the following integrals. You must show your work.
Z
3/2
(a) x ln(3x) dx
(b)
Z
sin3 (2x) dx
1
dx
1 − 4x2
(c)
Z
√
(d)
Z
3x3 + 15x2 − 17x + 4
dx
x2 + 5x − 6
(e)
Z
x2 ex
3 +1
dx
5. Given the 1st order Taylor formula of the function f (x) centered at a = 0
′
f (x) = T1 (x) + R1 (x) = f (0) + xf (0) −
derive the 2nd order Taylor polynomial and error term.
Z
0
x
(t − x)f ′′ (t) dt,
6. Determine whether the series is absolutely convergent, conditionally convergent or divergent. In any case you must justify your answer and show complete work.
∞
X
3n2 + 2n + 3
(−1)n 2
(a)
4n + 3n + 4
n=1
(b)
(c)
∞
X
(−1)n
3n + 2
+ 3n + 4
n=1
4n2
∞
X
3
4n2 + 3n + 4
(−1)n
n=1
(d)
n4n
n=1 n!
∞
X
7. (a) Write a formula for the nth term of the sequence {2, −5, 10, −17, 26, · · ·}.
.
(b) Write the first 5 terms of the sequence derived by an =
n2
.
n3 + 1
8. Below are the plots of r = 2(1 + cos(θ)) and r = 2(1 − cos(θ)).
4
r=2(1+cos(θ))
3
r=2(1−cos(θ))
2
y
1
0
−1
−2
−3
−4
−4
−3
−2
−1
0
x
1
2
3
4
(a) Find the points of intersection of the two curves above.
(b) Find the area shared by the cardioids plotted above.
9.
How many subdivisions should be used in the Trapezoidal Rule to approximate ln 2 =
Z 2
1
dx with an error whose absolute value is less than 10−6 .
1 x
∞
X
1
and the function f (x) = √ , which is graphed below.
x3
n=1
Illustrate the proof of the integral test by drawing appropriate rectangles on the graph to
deduce that
Z n
n
X
1
1
1
1
√ dx.
ai = 3/2 + 3/2 + · · · + 3/2 ≤
2
3
n
1
x3
i=2
∞
X
1
Explain in your own words how this shows that the series
is convergent.
3/2
n=1 n
10. Consider the series
1
n3/2
y
1
y=
1
5
√1
x3
10
x