APPM 1345 Midterm 3 Spring 2009
On the front of your bluebook, please write: a grading key, your name, student ID, and instructor name. This
exam is worth 100 points and has 6 questions. There are 110 possible points available, so it is possible to earn up to 10
extra credit points. Show all work! Simplify answers unless otherwise indicated. Answers with no justification will
receive no points. Begin each problem on a new page. No Calculators. No Notes.
1. (10 points) Consider the functions y=−x3+ 6 and y= 3x2−1, defined for all real x.
(a) Which of the two functions has an inverse? Explain.
(b) Find the inverse.
2. (25 points) Find dy /dx for the following functions. Answers need not be simplified.
(a) y= ln(ln x)
(b) y=e2x
ln x
(c) y=x3x
√x2+ 1 (use logarithmic differentiation)
3. (30 points) Evaluate the following integrals.
(a) Zt2et3dt =
(b) Z2cos θsin θ dθ =
(c) Z3
e
log3x
xlog5xdx =
(d) Ze4
e
dt
tln t=
4. (15 points) Evaluate the following limits.
(a) lim
t→0
tet
1−et=
(b) lim
x→π
2
−
(sec x)1
sec x=
5. (15 points) We wish to estimate the value of ln 4 = Z4
1
1
tdt using numerical integration.
(a) Use the trapezoidal rule with n= 3 subintervals to approximate the value of ln 4.
(b) Find the minimum number of subintervals needed to ensure an error of magnitude less than 1/18.
(hint: |ET| ≤ b−a
12 h2M.)
6. (15 points) In early January Malcolm receives a collection of rabbits as a surprise gift. His initial delight soon turns
to dismay as the rabbits start multiplying. In just one month, the number of rabbits increases by 50%. If the rate
of increase is always proportional to the number of rabbits, in which month can Malcolm expect the size of his
collection to reach 8times the original size?
(You may use the approximations ln 1.5≈.4and ln 8 ≈2.1.)