MATH 171 506, FINAL, FALL 2014
10 pts per question, show all work for credit
1. Find the derivatives of
sin−1 (sec x).
ex ln x ,
2. Find and classify the critical points of
1
.
(x − 1)(x − 5)
y=
3. Using the definition of derivative, find the derivative of y =
√
x + 1.
4. Initially man A is at (0, 0) walking west at 3 mph and man B is at (0, 1)
walking east at 4 mph.Find the rate at which the distance between them
is changing one hour later.
5. Give an ε, δ proof that y = x2 is continuous at x = 2.
6. Find
2ex − 2 − 2x − x2
.
x→0
x2 sin x
lim
7. A particle moves in the xy-plane with acceleration s00 (t) = 6ti + 4j. It
starts from the origin at time t = 0 with initial velocity 2i + j. Where
is it at time t = 1?
8. Find
Z
x2 (1 + x)100 dx.
9. Find
Z
0
1
x
dx.
1 + x4
10. I have available $54 to construct an open top box with a square base.
Material for the base costs $2 per square foot, $1 per square foot for
the sides. Find the dimensions that will maximize the volume.
11. x3 + y 3 + ln y +
√
y = 3. Find dy/dx at the point (1, 1).
12. Let
Z
x3
2
et dt.
F (x) =
x2
Find F 0 (x).
13. Let D be the square of the distance from the point (0, 1) on the y-axis
to points (x, x2 ) on the parabola y = x2 . Find the minimum value of
D.
14. Find the derivatives of
y=
tan x
,
ln x
15. Find
Z
0
1
2
y = cos(ex ).
x
dx.
x+1