Math 125
Carter
Test 2 Fall 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. Please put
the test sheet inside the blue book as you leave. There are problems on both sides of
this page. On his last appearance on David Letterman, Warren Zevon said, “Enjoy every
sandwich.” Good luck!
1. Compute
dy
dx
for the following expressions (4 points each).
(a)
y = (x + 3)(x − 1)(x − 5)
(b)
y=
1 + sin (x)
1 − sin (x)
(c)
y = ecos (x)
(d)
sin (x + y) = x + cos (y)
(e)
x2 + y 2 = 5
(f)
y = x arctan (x)
(g)
y = ln (sin (x))
2. (10 points) A vertically rising hot air balloon is tracked by an observer
located 2 miles from the lift off point. At the moment at which the
observer’s line-of-sight and the horizontal is π6 , the angle is changing at
the rate of 0.2 radians/minute. How fast is the balloon rising at that
time?
3. (10 points) The volume of a sphere is increasing at the rate of 20 cubic centimeters per section. Determine the rate at which the radius is
increasing when the radius is 50 centimeters.
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4. (10 points) Determine the critical points and the regions of increase and
decrease for the function f (x) = sin (x) cos (x) on the interval [0, 2π].
5. (10 points) Determine the x-coordinates and the optimal values (maximum and minimum) for the function f (x) = −4x2 + 3x + 4 on the
interval [−1, 1].
6. Compute the definite and indefinite integrals (5 points each).
(a)
Z
(x5 + 3x − 8) dx
Z
(cos(x) − ex ) dx
(b)
(c)
Z π
0
sin(x) dx
(d)
Z 4
1
(x3 − x2 ) dx
(e)
Z 9
t−1/2
1
dt
7. (10 points) Use mathematical induction to show that
12 + 22 + · · · + n2 =
n
X
k=1
2
k2 =
n(n + 1)(2n + 1)
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