Math 125
Carter
Test 2 Spring 2010
General Instructions: Do all your work and write your answers inside the blue book. Do
not write on the test. Write your name on only the outside of the blue book. There are
problems on both sides of this page. Good luck! Remember that pedestrians have the
right of way.
1. (5 points) Compute f 0 (x) for f (x) = sin (x) cos (x).
2. (5 points) Compute the equation of the line tangent to the function
f (x) = x tan (x) at the point x = π4 .
3. (5 points) Compute
dy
dx
where sin (x + y) = x cos (y).
4. (5 points) Compute the third derivative of y =
1
1+x .
5. (5 points) Compute f 0 (x) for f (x) = ln [(x + 1)(2x + 9)].
6. (5 points each) Compute the following definite and indefinite integrals:
(a)
Z
(x5 + 3x + 2) dt
(b)
8
+ 3ex
x
Z
!
dx
(c)
Z 3
−2
|x| dx
(d)
Z 27
1
(e)
π
4
Z
0
1
x1/3 dx
sec2 (t) dt
7. (10 points) Determine the maximal and minimal values for the function
f (θ) = cos (θ) + sin (θ) over the interval [0, 2π]. For which angles θ do
these optima occur?
8. (10 points) Determine the maximal and minimal values for the function
f (x) = −x2 + 10x + 43 over the interval [3, 8].
9. it (10 points) The radius of a sphere is increasing at a rate of 14 inches
per minute. Determine the rate at which the volume is increasing when
the radius is 8 inches.
10. (10 points) A tank in the shape of a right circular cone of radius 300 centimeters and height 500 centimeters leaks water from its vertex (which
points down) at a rate of 10 cubic centimeters per minute. Find the rate
at which the water level is falling when the level is 200 centimeters.
11. (10 points) Give a proof by induction that
N
X
j=1
j2 =
N (N + 1)(2n + 1)
.
6
2