Math 125
Carter Test 2 Fall 2013
General Instructions: Do all of your work inside your blue book. Write your name on
only the outside of your blue book. Do not write on this test sheet since it may become
dislodged from the blue book. As you leave, please insert your test into the blue book. Give
neat, complete, and articulate solutions to each of the problems below. Do not perform extra
simplifications or arithmetic. There are 105 possible points on the test. Think clearly. I
hope that you do well.
When dicing vegetables, use a triangular bladed french knife. Keep the tip of the knife
on the cutting board, tilt the knife slightly away from the hand you use to guide it, and slide
the knife along the second segment of the fingers of your guiding hand. Do not look away
from the cutting surface even when a pretty waitress or waiter walks by.
1. Compute the derivatives for each of the following expressions. Do not
simplify your results!
(a)
y = 3x2 − 8x + 5
(b)
y = (2x + 3)(4x − 5)(3x + 8)
(c)
y=
(d)
(5x − 2)
(3x + 7)
√
y = cos ( x2 + 1)
(e)
y = earcsin (x)
(f)
y = ln (sin (x))
(g)
y = 83x+2
(h)
√
(x2 − 3x + 2) x2 + 1
y=
sin (x)
1
2. (10 points) Compute the equation of line tangent to the curve:
x2/3 + y 2/3 = 25
at the point (64, 27).
3. (10 points) A cartoon coyote plans to shoot an anvil upwards off the
edge of a 480 meter tall cliff at the rate of 20 meters per second. Assume
that the acceleration due to cartoon gravity is −10 meters per square
second. Then the height of the anvil as a function of time is given by
s(t) = −5t2 + 20t + 480.
(a) When does the anvil reach its acme?
(b) How high above the ground does the anvil travel?
(c) How many seconds does it take to reach the ground?
(d) What is the anvil’s terminal velocity?
4. (10 points) Determine the equation of the line tangent to the curve
y = cos (x) at the point x = π6 .
5. (5 points) Use the idea of local linearization to give an approximate value
for (126)1/3 .
6. (10 points) A spherical weather balloon is expanding at the rate of 20
cubic centimeters per second as the balloon rises in the atmosphere.
How fast is the radius increasing when the radius of the balloon is 100
centimeters?
7. (10 points) A hot air balloon rising straight up from a level field is tracked
by a range finder 200 meters from the liftoff point. At the moment that
the range finder’s elevation angle is π/3, the angle is increasing at the
rate of 0.14 rad/min. How fast is the balloon rising at that moment?
8. (10 points) Determine the Maximal and minimal values for the function
f (x) = x2 − 4x − 96 over the interval 0 ≤ x ≤ 12. At what x-coordinates
do these occur?
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