Math 165 Section A
Professor Lieberman
November 8, 2004
PRACTICE FOURTH IN-CLASS EXAM
Carry out the solution of each problem: show steps of any required calculations and state
reasons that justify any conclusions. A short sentence is usually enough but answers without
any justification will receive no credit.
This test merely shows the number of questions and level of difficulty to expect on Wednesday’s exam. Questions on other topics (from Chapters 4,5, and 11) may appear.
1. (a) (15 points) Find the general solution for the differential equation
dy
x
= .
dx
y
(b) (10 points) Find the particular of the differential equation in part (a) that satisfies
the condition y = 1 at x = 1.
2. (20 points) Evaluate the integral
Z
t2 − 2 cos t dt.
3. (25 points) Sketch the graph of the function f (x) = x + 1 over the interval [−1, 2],
divide the interval into 6 equal subintervals, and calculate the sum of the areas of
the circumscribed rectangles.
4. (30 points) Sketch the graph of a function f that has the following properties:
(a) f is everywhere continuous;
(b) f (−4) = −3, f (0) = 0, f (3) = 2;
(c) f 0 (−4) = 0, f’(3)=0, f 0 (x) > 0 for x < −4, f 0 (x) > 0 for −4 < x < 3, f 0 (x) < 0
for x > 3;
(d) f 00 (−4) = 0, f 00 (0) = 0, f 00 (x) < 0 for x < −4, f 00 (x) > 0 for −4 < x < 0,
f 00 (x) < 0 for x > 0.