Math 111 Exam 2, Oct. 26, 2011
Read each problem carefully and show ALL your work. No calculators.
1. (16 points) Evaluate the following limits. If the limit does not exist, explain why. (Include
as possible values of a limit).
a)
3
lim 64
x
x
x
b)
2
2
lim 24
x
xx
x
c)
2
0
1
lim cos
xxx
d)
2
lim 9 3
xx x x
2. (9 points) Find all horizontal, vertical and slant asymptotes of
2
2 3 4
() 2
xx
fx x
3. (12 points) Differentiate the following functions: (simplify answer in part b)
a)
2
3
3
( ) 4
xe
f x e x x x
b)
2
() x
f x x e
c)
23
() 1
x
xx
fx e
4. (15 points) Differentiate the following functions: (simplify answer in part c)
a)
3
( ) sin(4 )
x
f x e x
b)
2
( ) ln 3cos 4f x x
c)
12
( ) sin 1y x x x x
5. (14 points) Find
dy
dx
where: a)
3
2
y
xe xy y
b)
( ) 2 5 x
y x x
6. (12 points)
a. If
3
() 5
fx x
and
()gx
is the inverse of
()fx
, find
1
2
g
and
1
'2
g
.
b. Use linearization or differentials to approximate
325
.
7. (12 points) A rocket travels vertically upward at the constant speed of 4 km/minute. The
rocket is tracked by an observer using a telescope (at ground level) 9 km from the launching
pad. At what rate is the distance between the telescope and the rocket changing 3 minutes
after liftoff?
8. (10 points) We define a right sided slope to be the slope of a secant line connecting
( , ( ))x f x
and
( , ( ))x h f x h
. We define a centered slope to be the slope of a secant line
connecting
( , ( ))x h f x h
and
( , ( ))x h f x h
. Compute these two slopes (the right
sided slope and the centered slope) for the function
2
()f x x
at
3x
(your results may
depend on
h
). Which method gives a slope closest to the true slope of the function at
3x
when
h
is small? (Recall that you did something like this in your MATLAB assignment).
