Math 165
Professor Lieberman
October 21, 2011
PRACTICE SECOND 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 exam is longer than the in-class exam, so that
you can see a fuller range of problems.
1. Suppose that f is a function satisfying f (3) = 8 and f ′ (3.05) = 14 . Use this information (and differentials) to approximate f (3.05).
2. Find the general antiderivative F (x) + C for
√
3
f (x) = 1/ x2 .
3. Determine where the function
√
f (x) = x x − 2
is increasing, decreasing, concave up, and concave down. Use this information to
sketch the graph.
4. Identify the critical points and find the maximum and minimum values of the function
s(t) = sin t − cos t on the interval I = [0, π].
5. Find the critical points for
r(s) = 3s + s2/5
and decide which critical points give a local maximum value and which give a local
minimum value.
6. Find the solution of the differential equation
du
= u3 (t3 − t)
dt
that satisfies the initial condition u = 4 at t = 0.
7. Find the equation of the line that is tangent to the ellipse
x2 y 2
+
=1
4
25
in the first quadrant so the triangle formed by this line and the coordinate axes has
the smallest possible area.