December 16, 1992
This is a 3-hour test. Work all questions. You may use a
graphing calculator. Write your name and section number in
your ``bluebook''.
1. Sketch the following graphs.
2.
a. Calculate the derivative of
f(x) = x^2 cos(x).
3. The height h(t) of a marble falling through thick
syrup is given at various times t in the following table:
Estimate the velocity h' (in inches/sec) at t= 0.4 .
4.
a. Give the equation of the
line tangent to the graph of f(x) = cos(x^2)
when x = 2. (Note: x is in radians!)
5. If the sum of two non-negative
numbers is 12, what is the maximum value of one
times the square of the other?
6. A jet touches down on the deck of an aircraft carrier at t=0
and immediately starts decelerating. At touchdown it
has 200 feet to go before the end of the runway. The table below
gives the jet's velocity when it was
tested on a longer runway. Assuming these are the same velocities when it
attempts to land on the aircraft carrier, does the jet come to a stop before
it reaches the end of the runway, or does it go plunging off into the water?
Explain in detail.
7. You want to approximate
a. Plot the distance from the top of the
hill as a function of time. The sled started down the hill,
going faster and faster until it hit a wall.
b. Plot the distance from your dog to your
front door as a function of time. As you walked home with your
dog on a leash, he ran in circles around you, wrapping the
leash around your body (so the circles got smaller and smaller).
Finally you picked him up and carried him home.
b. Calculate the derivative of
f(x) = 1/[ e^{sin(2x)}].
c. Calculate the slope dy/dx
of the curve x^3 + xy + y^4 = 3 at the point
x= 1 , y= 1 .
d. Calculate an anti-derivative
for f(x) = 2x^2 - 5.
e. Calculate an anti-derivative
for f(x) =sin(2x).
t (sec) 0.0 0.2 0.4 0.6 0.8
h (inches) 12 11.5 10.8 9.9 8.8
b. Calculate the approximation
to f(2.1) given by the tangent line approximation
at x = 2.
t (sec) 0.0 0.5 1.0 1.5 2.0 2.5 3.0
v (feet/sec) 235 178 114 67 33 13 0
/1 | 2 | x | e dx | 0/using a left-hand sum. Suppose you do not know the true value. How many (equal) subdivisions do you need to get within .05 of the true value? Note that the function to be integrated is monotonic increasing on the interval given.
8. Given that f(0)=0, and that the derivative
f '(x) is given by
f '(x)= cos(x^2)- (sin x)^2
sketch the graph of f(x) for
0 < = x < = 3, showing on
your graph the coordinates of critical points and
inflection points. Hint: use your calculator
to produce the graph of f ', and use that information
to determine where f is increasing, where it is decreasing
and what the concavity of the graph of f will be.