Math 1220 (Underdown)
Exam 1
Name:
1. [5 points] Find the derivative of:
y = x ln(x) − x.
2. [10 points] Use logarithmic differentiation to find the derivative of
y = x(2x+3) .
Solving for y 0 is sufficient. No need to simplify your answer.
February 11, 2012
Math 1220 (Underdown)
Exam 1, Page 2 of 6
Z3 x2
3. [10 points] Evaluate x e − 1 dx.
1
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Math 1220 (Underdown)
Exam 1, Page 3 of 6
4. [10 points] Let f (x) = 4x + 5. Compute (f −1 )0 (x) at x = 10.
5. [10 points] Compute the derivative of y = 3ln x .
− ln 102
1
.
6. [10 points] Find the value of the expression
e
February 11, 2012
Math 1220 (Underdown)
Exam 1, Page 4 of 6
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7. [15 points] The Trouble with Tribbles
Lieutenant Uhura brought two small furry creatures called tribbles onto the starship
Enterprise. These adorable pets reproduced very quickly and three days later she counted
54 tribbles! How many days until there are one million tribbles? Assume that the tribble
population is governed by:
dP
= kP.
dt
You do not need to show how to get the solution to the above model. You
may jump right to using the general solution if you know it. If you don’t know it, you
can use separation of variables and integration to solve the differential equation, and
then use your solution and the data to answer the question.
Math 1220 (Underdown)
Exam 1, Page 5 of 6
February 11, 2012
8. [10 points] A brine tank initially contains 2 lbs of salt thoroughly mixed in 10 gallons
of water. The rate of change of salt in the tank is modeled by the equation:
1
dx
=4−
x.
dt
t + 10
Use the integrating factor method to solve for x(t), the amount of salt in the tank as a
function of time.
dx
+ p(t)x = q(t).
Hint: Remember to first put the equation into standard form:
dt
Math 1220 (Underdown)
Exam 1, Page 6 of 6
Question Points Score
1
5
2
10
3
10
4
10
5
10
6
10
7
15
8
10
Total:
80
February 11, 2012