* This is an 90-minute test. Work all questions. You may use a
programmable, graphing
calculator except when instructed otherwise. *

** As usual, no credit for
unexplained work!**

Total score = 100.

** 1.** (20 points) Calculate the following derivatives:

** a.**
*d*/*dx* *x*^(1/3)

** b.**
*d*/*dx* (2*x*+1) / (*x*^2+1)

** c.**
*d*/*dx* *x*^2 e^*x*

** d.**
*d*/*dx* e^(*x*^2+2)

** e.**
*d*/*dx* (e^*x*+3e^(-*x*))^3

** 2.** (20 points) Given that

*d*/*dx* (e^*x* cos*x* + e^*x* sin*x*) =
2e^*x* cos*x*

calculate \int_(pi/4)^(pi/3) 2e^*x* cos*x* *dx*

/ pi/3 | x | 2 e cos x dx | pi/4 /

*using the Fundamental Theorem of Calculus*. Show all
your work! A numerical calculation is **not** acceptable and
will score 0.

** 3.** (20 points)
A truck is traveling at 88 ft/sec (= 60 m.p.h.) on the BQE
when the driver, spotting a giant pothole 140 ft. ahead, hits the brakes.
The truck's speed *v*(*t*)
is then recorded every half second from *t*=0
until the truck comes to a stop.
The measurements appear in the following table:

Does the truck fall into the pothole, or does it stop in time? Explain your answer carefully.t0.0 0.5 1.0 1.5 2.0 2.5 3.0v(t) 88 82 73 60 43 23 0

** 4.** (20 points) Suppose *f* and *g* are two functions with the
following properties.

g(3) = 5 g'(3) = -2 g(5) = 2 g'(5) = 0 f(3) = 7 f'(3) = -4 f(5) = 0 f'(5) = 3Calculate

** 5.** (20 points) You are going to calculate the area under the
graph of the function

*f*(*x*) = cos(*x*^2) between *x* = 0 and
*x* = sqrt(pi/4).
You only have a scientific calculator that *does not calculate
definite integrals*, and you can't use the Fundamental Theorem of Calculus
because cos(*x*^2)
is not the derivative of any elementary function. So you are going to
* use a left-hand sum*. You divide the interval 0 to sqrt(pi/4) into
*n* equal subintervals and calculate

Left hand sum = \sum_{*i*=0}^{*n*-1} cos(*x*_*i*^2)
Delta *x*

n-1 --- \ 2 / cosxDeltax---ii=0

as usual, where Delta *x* = (1/*n*)sqrt(pi/4) and *x*_*i* =
*i* Delta *x* as usual.

How large should *n* be to guarantee that your left-hand sum will be
within .01 of the exact value?
(Note that this function is positive and decreasing for 0 < *x*
< sqrt(pi/2).)

An alternative set of numbers for Problem 3:

tv(t) 0.0 88 0.5 82 1.0 73 1.5 60 2.0 43 2.5 23 3.0 0