Math 131 A exam 2 practice answers

1) Batman and the Penguin:
a) P(t) = [Graphics:ex2ansgr1.gif]*t
b) B'(t) = 8t - 40. Since acceleration is the derivative of velocity, velocity is an antiderivative of acceleration, and the Batmobile isn't moving until after t = 5.
c) B(t) = 4t2 - 40t + 100 (when t > 5, and B(t) = 0 if t < 5).
d) P(t) - B(t) is largest at t = 59 / 3 = 19.667. The distance between them is 1447.11 feet, Penguin is going 352/3 ft/sec then and the Batmobile is going 117.33 ft/ sec (the same speed as Penguin).
e) Yes, P(t) - B(t) = 0 at t = 38.6871 and B'(t) is 269.497 ft/ sec at this time.
f) The coolness of this chase is 354 / 352, just a tad cooler than any other chase ever.

2) You should draw F(t) increasing and concave up for a while, then increasing and concave down, leveling off after a few minutes. It should also be the case that F(0) is about 55 or so, and it should probably level off around 110. Possible graphs (with t in seconds) are shown below.
[Graphics:ex2ansgr4.gif][Graphics:ex2ansgr3.gif][Graphics:ex2ansgr4.gif][Graphics:ex2ansgr5.gif]

3) f '(3) = [Graphics:ex2ansgr7.gif][Graphics:ex2ansgr8.gif] = [Graphics:ex2ansgr9.gif][Graphics:ex2ansgr10.gif]= [Graphics:ex2ansgr11.gif][Graphics:ex2ansgr12.gif]=
[Graphics:ex2ansgr13.gif]13 + 2h = 13 + 0 = 13.

4) nderiv(2.3^X, X, -1) = 0.362134

5) given f '(x) graph.
a) Concave down as f '(x) is decreasing at x = 0.
b) For x = 0 to 1.9, f '(x) is negative, so f(x) is decreasing.
c) At x = 0, since f '(x) is positive for the entire time, f(x) just keeps getting bigger.
d) x = -2 is a chair seat. x = 0 is a hilltop and x = 1.9 is a valley bottom.
e) At x = -2, -0.75 and 1.25, the hills and valleys of f '(x) are inf pts of f(x).
f) The tangent line to f '(x) at x = 0 has a slope of about -6.4
g) y = -8(x - 1) + 4

6) f(x) = [Graphics:ex2ansgr14.gif]
a) This is just [Graphics:ex2ansgr15.gif]= 1.5.
b) Using nderiv, I get roughly f '(0) = 0.75.
c) The limits are:
i) -2/3
ii) -20 / 3
iii) -infinity

7) Calculus books and heights.
a) v(t) = 96 - 32t
b) 48 ft / sec
c) At t = 3 the book is 144 feet in the air.
d) s(t) = 6 (for the second time) at t = 5.93684 seconds.

8) Given a graph of f(x).
a) Negative as f(x) is decreasing at x = 1.
b) concave down.
c) Decreasing as f(x) is concave down there.
d) The limit is the defintion of f '(1), which, via a tangent line slope is about equal to -1.33.
e) Draw the line through (0.5, f(0.5) ) and (1.5, f(1.5) ).
f) Via a tangent line slope, I get f '(0.5) = 0.75.
g) Positive, since f(x) is concave up at x = 1.9.
h) The plot below is what you should get.
[Graphics:ex2ansgr4.gif][Graphics:ex2ansgr16.gif]