Exam 1
Tuesday, September 15
Remember to show an appropriate amount of work for each part.
1. [17 points] This problem has 4 parts, which are completely independent.
(a) [4 points]
f
(
x
) is an exponential function with
f
(0) = 2 and
f
(3) = 54. Find a formula
for
f
(
x
).
Show your work.
(b) [4 points]
g
(
x
) is a function with
g
(1) = 2,
g
(2) = 3 and
g
(3) = 5. Can
g
(
x
) be linear?
Justify
briefly
.
(c) [4 points] Let
h
(
x
) be the function
ln
x
x
-
5
. What is the domain of
h
(
x
)?
(d) [5 points] Suppose that
-
x
2
+ 4
x
≤
'
(
x
)
≤
x
2
-
4
x
+ 8, and
'
(2) = 4. Is
'
(
x
) continuous
at
x
= 2? Justify briefly, clearly stating any theorems you use.
2. [8 points] Let
h
(
x
) be the function:
h
(
x
) =
p
1 + ln(10
8
x
+ 7)
Find a formula for
h
-
1
(
x
).
Show your work.
3. [10 points] Solve for
x
:
log
(
x
1
log
e
)
+ ln(
x
) =
π
Show your work.
Hint
: You will likely want to use either log
x
=
ln
x
ln 10
or ln
x
=
log
x
log
e
.
4. [10 points] Let
f
(
x
) be the function:
f
(
x
) =
ax
+
b
if
x <
0
a
if
x
= 0
-
5
·
7
x
if
x >
0
For what value(s) of
a
and
b
is
f
(
x
) continuous?
Show your work.
5. [5 points] Let
f
(
x
) =
(
C
·
g
(
x
)
if
x <
0
ln(
x
2
+ 1)
if
x >
0
where
lim
x
→
0
-
g
(
x
) = 1
and
lim
x
→
0
+
g
(
x
) =
-
1
For what value(s) of
C
does lim
x
→
0
f
(
x
) exist?
Show your work.
