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 0Does 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) = 3Calculate 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