Problem Points Sore
1
18
2
27
M161, Final Exam, Fall 2003
3
27
NAME:
4
29
SECTION:
5
27
6
18
7
18
8
18
9
9
10
9
Total
200
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
s
1
os 2
os =
2
1
s
1 + os 2
1
1. Determine the power series representation for the funtion f (x) =
and determine
1 + 9x2
the interval of onvergene.
2
2. (a) Is
false.
(b) Is
z
w
zw
=
= zw? If true, prove it. If false, nd omplex numbers
zw
z
and
w
for whih it is
jzj2 ? If true, prove it. If false, nd omplex numbers z and w for whih it is false.
10
() Without the use of your alulator, ompute 12 + 12 i .
3
3. Calulate the following integrals.
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.
Z2
x4 ln x dx
(a)
1
(b)
()
Z 2
3 1
1
t2
dt
Z2 1
0
px dx
4
4. (a) Simplify ot(os 1 x).
(b) Answer true or false.
(i) ln 1 = e.
(ii) The graph of the funtion y = ln x is dereasing.
(iii) The funtion f (x) = ln x is not one-to-one.
1
(iv) lim ln x =
x!0+
(v) The funtion f (x) = sin , 0 is one-to-one.
(vi) If f (x) =
p
(vii) If f (x) =
x
p
x
p
2, 2, then f 1 (x) = x2 + 2.
2 and g = f 1 , then g 0 (3) = 6.
(viii) The funtion f (x) = x2 is invertible beause it satises the vertial line test.
(ix)
1 1
X
n=3
2n
=2
(x) If nlim
!1 an = 0, then the series
1
X
n=3
an
onverges.
5
5. Determine whether the series is onvergent or divergent. In either ase you must justify
your answer.
1 sin(n=2)
X
(a)
n!
n=1
(b)
()
1
X
n
n
n=1 (n + 1)2
1
X
n
2
n=1 (3n + n + 1)
6
6. Calulate the following derivatives (you do not have to simplify).
d 2
(a) 3x 4
dx
(b)
()
d
dx
d
dx
sin 1 (2x)
sinh(2x)
7
7. (a) On the axes provided below, sketh the polar urve given by r = os 2.
0
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
(b) Find the area of the region enlosed by one loop of the urve given in part (a).
8
8. Find the entroid of the region R bounded by the urves y = x2 and y = 12.
(a) On the piture inluded below (next page), draw an arbitary approximating retangle R
(dierential element) and nd the entroid of the dierential element R.
(b) Find the moment of the dierential element R about the x-axis.
() Summing the moments over all of the approximating retangles and letting n ! 1, nd
the moment of the region R about the x-axis.
(d) Repeat steps (b) and () to nd the moment of the region R about the y -axis.
(e) Find the entroid of the region R.
9
14
y
12
y=12
y=x2
10
8
6
4
2
x
0
−4
−3
−2
−1
0
1
2
3
9. Solve the following dierential equation:
4
dy
dx
=
1+x
xy
10. Find the area under one arh of the yloid x = t
10
, x > 0, y (1) = 4. (Solve for y ).
sin t, y = 1
os t, 0 t 2 .