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M161, Final Exam, Spring 2004
NAME:
SECTION:
INSTRUCTOR:
You may use alulators. You are not allowed to have information stored in your
alulator. When you are told to do a
omputation analytially, you must show
all of your work. When you are told to
do a problem analytially, you will get no
points for alulator results.
sin 2 =
s1
Total
200
s
os os = 1 + os 2
2
2
Error Bounds. Suppose jf 00(x)j K for a x b. If ET is the error in the Trapezoidal
Rule, then
K x (b a)
b a)
or
it
may
be
written
as
j
ET j jET j K (12
n
12 :
2
3
2
Suppose jf (x)j K for a x b. If ES is the error involved in using Simpson's Rules,
then
(b a) or it may be written as jE j K x (b a) :
jES j K 180
S
n
180
(4)
4
5
4
1
1. (a) Find the Taylor series expansion of f (x) = ln x at x = 2. Write the result using
summation notation.
(b) For what values of x does the series found in part (a) onverge.
2
2. (a) Simplify i2 + 3i (1 + i).
p
p
(b) Write the omplex number 8 2 + 8 2i in polar form, (in the form rei , r
< ).
>
0 and
p p
() Find the fourth roots of 8 2 + 8 2i. (You may leave your answer in polar form.)
(d) Suppose that z = 2e
answer in polar form.)
1
i=4
and z = 3ei= . Compute z z and zz . (You may leave your
2
6
3
1 2
1
2
3. Calulate the following integrals.
Z
be integrated analytially.
You must show your work. These integrals must
If you just give the result from your alulator, you
will get zero redit.
(a)
Z
(b)
()
x1=3
x
x
Z
x
2
1
ln x dx
1 dx
1 dx
4
4. (a) Simplify sin(os
1 2y
3
).
(b) Determine whether the sequene onverge or diverge. If it onverges, nd the limit.
Support your answer with work or an explanation.
(i) an = 2n3n+ +n +n 1
2
2
(ii) an = 2 + ( 1)n
2 ln(x)
() Calulate the limit xlim
!1 ln(2x)
5
5. Determine whether the series is onvergent or divergent. In either ase you must justify
your1answer.
X
(a) n + nn ++1n + 1
n
=1
(b)
()
X1
n=1
X1
n=1
3
2
3n
(n + 1)2n
n
( 1)nn
(3n + 1)
2
6
6. Calulate the following derivatives (you do not have to simplify).
(a) dxd sin (2x)
1
(b) dxd sinh(2x)
7. Solve the following dierential equation:
satises y(0) = 4.
dy
dx
7
= 3y (solve for y). Find the solution that
8. (a) Sketh the urve of the parametri equations x = t(t 3), y = 3(t 3), 2 t 2
and indiate with an arrow the diretion in whih the urve is traed as the parameter t
inreases.
2
2
4
2
0
−2
−4
−6
−8
−10
−3
−2
−1
0
x
1
2
3
(b) Find the tangent to the urve at the point ( 2; 6). Draw the tangent line on your plot
of the urve.
() Find the tangent to the urve at the point (0; 0).
8
9. The viewing portion of the glass window in one of the sh tanks of the Fish-are-Us
Aquarium is the shape of the region between two parabolas y = x and x = y . See the
gure below. The water level is at 1 ft above the highest spot on the window (at y = 2). The
weight-density of seawater is 64 lb=f t . The problem is to nd the hydrostati fore on the
window.
(a) On the piture inluded below, draw an arbitary approximating retangle R (dierential
element).
2
2
3
2.5
y
water level
2
1.5
y
x=y2
1
0.5
y=x2
window
x
0
−0.5
−0.5
0
0.5
x
1
1.5
(b) Find the area of the dierential element R and the fore on the dierential element R.
() Approximate the hydrostati fore against the window by a Riemann sum. Then let n
approah innity to express the fore as an integral and evaluate it.
9