Math 125
Carter
Test 2 Fall 2009
General: 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 the Toxic Pets smash single “I’d
Rather be Killing Communist Babies,” appeared in the credits of the movie “Eight Heads
in a Duffle Bag,” but never was heard in the movie.
1. (10 points each) For each of the following, compute the derivative of y
dy
with respect to x. That is find y 0 , f 0 (x), or dx
(Same things — different
notation). Do not simplify your answer!
(a)
(b)
(c)
(d)
f (x) =
1+sin (x)
1−sin (x)
y = ecos (x
2
)
x2 + y 3 = 5
√
f (x) = t2 + 1 + arctan (t)
2. (10 points) Compute the equation of the line tangent to the curve at the
particular point, (a, f (a)) where y = 4 ln (9x + 2) at a = 2.
3. (10 points) Sand falls into the shape of a conical pile at the rate of 2 m3
per minute. The proportion of the radius to the height is always 1/3.
At what rate is the height increasing when the height is 5 meters? The
volume of a cone is V = π3 r2 h where r denotes the radius and h denotes
the height.
4. (10 points) A boat is pulled into a dock by a rope through a windlass
that is 3 feet above the boat. The rope is being hauled in at 1 foot per
second. At what rate is the boat approaching the dock when it is 4 feet
from the dock?
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5. (10 points each) Sketch the graphs of the functions given below. Indicate
critical behavior and when feasible the inflection points.
(a)
(b)
f (x) = (x − 1)(x − 2)(x − 5)
f (x) = e−x
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6. (10 points each) Determine the Maximum and minimum values of the
functions on the given intervals.
(a)
f (x) = −x2 + 10x + 8 on [3, 6].
(b)
f (θ) = cos (θ) + sin (θ) on [0, 2π].
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