Math 1A, Spring 2008, Wilkening
Sample Final Exam 2
You are allowed one 8.5×11 sheet of notes with writing on both sides. This sheet
must be turned in with your exam. Calculators are not allowed.
0. (1 point) write your name, section number, and GSI’s name on your exam.
1. (3 points) give precise definitions of the following statements or expressions:
(a) f(x) is neither even nor odd
(b) Rf(x)dx
(c) Rb
af(x)dx
2. (4 points) Show that the tangent lines to the curves y=x3and x2+ 3y2= 1 are
perpendicular where the curves intersect.
3. (3 points) Evaluate Z1
0
tan−1x
1 + x2dx.
4. If fis continuous and R4
0f(x)dx = 6, find R2
0f(2x)dx.
5. (5 points) A right circular cone of height hand base radius Rhas a hole of radius
rdrilled through its center (from the tip to the center of the base). Find the volume
of the solid that remains.
6. (5 points) Let f(x) = tanh−1(sin x) and g(x) = ln |sec x+ tan x|. Compute f0(x),
g0(x), f(nπ) and g(nπ) with nan integer. What do you conclude?
7. (5 points) A boat leaves a dock at 2:00 PM and travels due south at a speed of
20 km/h. Another boat has been heading due east at 10 km/h and reaches the same
dock at 3:00 PM. At what time were the two boats closest together? Verify that the
distance was minimized using one of the derivative tests.
8. (6 points) Let f(x) = x2(√x2+ 3 −x−1)
x2−1.
(a) find all vertical and horizontal asymptotes of f.
(b) show that y=−2x−1 is a slant asymptote, i.e. lim
x→−∞[f(x)−(−2x−1)] = 0.
Hint for (b): first show that limx→−∞[√x2+ 3+x] = 0, then manipulate [f(x)+2x+1]
to make use of this.
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