Name:
Math 1090-007
Midterm 2
November 19, 2010
• The use of scientific calculators is permitted for this exam.
• No notes, books, other study materials or graphing calculators are allowed.
• Each problem is worth 20 points. Do every problem for full credit.
1. Consider the rational function
r(x) =
2x2 − 50
.
x2 − x − 12
(a) What is the domain of r(x)?
(b) Find the vertical asymptote(s), if there are any.
(c) Find the horizontal asymptote, if it exists.
1
(d) Find the y-intercept, if it exists.
(e) Find the x-intercept(s), if any exist.
(f) Sketch the graph of r(x) below
10
y
K10
5
K5
0
5
x
K5
K10
2
10
2. Evaluate the following expressions without a calculator. Show your work for full credit.
(a)
3 ln e − ln 1 + ln
1
√
e
+ ln e5
(b)
log6
3
1
27
− log6 8
3. The height, h, at time t, of an egg thrown upwards off the roof of a 128-foot building
with initial velocity 32 ft/sec can be written
h(t) = −16t2 + 32t + 128,
where h is measured in feet, and t is in seconds.
(a) Assuming this egg doesn’t have any special abilities, for how long after it is thrown
will it remain intact?
(b) What is the maximum height off the ground the egg will reach during its flight?
4
4. (a) Let g(x) =
q
3
x+7
.
2x−11
Find g −1 (x).
(b) Let h(x) = ex + 5. Find h−1 (x). What is the domain of h−1 ?
5
5. Suppose a colony of E. Coli with an initial population of 100 cells grows according to
the function
P (t) = 100e1.386 t
where t is measured in hours.
(a) What is the doubling time for this colony?
(b) How large will the population be in just 8 hours?
6