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.

** 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.

** 2.**
** a.** Calculate the derivative of
*f(x) = x^2 cos(x)*.

** 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)*.

** 3.** The height *h(t)* of a marble falling through thick
syrup is given at various times *t* in the following table:

t (sec) 0.0 0.2 0.4 0.6 0.8 h (inches) 12 11.5 10.8 9.9 8.8

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!)*

** b.** Calculate the approximation
to *f(2.1)* given by the tangent line approximation
at *x = 2*.

** 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.

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

** 7.** You want to approximate

/1 | 2 | x | e dx | 0/using a left-hand sum.

** 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.