APPM 1350 FINAL EXAM SUMMER 2006
On the front of your bluebook write: (1) your name, (2) your student ID number, and (3) a
grading table. You must work all of the problems on the exam. SHOW ALL YOUR WORK in
your bluebook and BOX in your final answers. A correct answer with no relevant work may
receive no credit, while an incorrect answer accompanied by some correct work may receive partial
credit. Text books, class notes, calculators and crib sheets are NOT permitted. Please start each
new problem on a new page of the bluebook.
1. (15 points) For each of the following unrelated questions, answer either ALWAYS TRUE,
ALWAYS FALSE or NEITHER. No justification is necessary.
(a) If f(x)>1 for all xand lim
x→0f(x) exists, then lim
x→0f(x)>1.
(b) All continuous functions have derivatives.
(c) The derivative of the function tan2xis the derivative of sec2x.
(d) Z5
−5
(ax2+bx +c)dx = 2 Z5
0
(ax2+c)dx
(e) If limx→6f(x)g(x) exists, then the limit is f(6)g(6).
2. (21 points) Find dy
dx in each case. No simplification is necessary.
(a) y=xln (arccos x)
(b) y=xex
(c) x ey= ln xy + arctan y
3. (28 points) Evaluate each of the following limits, if it exists. If the limit does not exist, state
this and state your justification. Show all your work.
(a) lim
t→0+tt2
(b) lim
r→∞ re1/r −r
(c) lim
x→0xarccot 1
x
(d) lim
h→0
1
hZ2+h
2p1 + t3dt
4. (21 points) Evaluate each of the following integrals.
(a) Ze−x
1 + e−2xdx
(b) Ztan xln (cos x)dx
(c) Z2
0
t
9−t2dt
5. (25 points) Do ONE of the following two problems. State clearly which problem you have
chosen in your bluebook. Only work for the chosen problem will be graded.
(a) What is the area of the largest rectangle in the first quadrant with two sides on the axes
and one vertex on the curve y=e−x?
(b) Plot the function f(x) = xln xon (0, e]. Find and label all critical points, local and
absolute extrema, and inflection points.
THE EXAM IS CONTINUED ON THE BACK
