## MAT 131 Calculus I ~~ Second Midterm Examination

April 12, 1994

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 (2x+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 cosx + e^x sinx) = 2e^x cosx

calculate \int_(pi/4)^(pi/3) 2e^x cosx 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:


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


Does the truck fall into the pothole, or does it stop in time? Explain your answer carefully.

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) = 3


Calculate d/dx f(g(x)) at x = 3.

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
/      cos x   Delta x
---         i
i=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:

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