Name
Student ID #
Math 1270-001
Fall 2011
EXAM I
Friday, September 23, 2011
Problem
1.
2.
3.
4.
5.
6.
Points
20
15
15
20
10
20
TOTAL
Score
(20 points) 1. A motorcycle waits at a stoplight next to a Porsche. When the light
turns green at t = 0 seconds, both vehicles proceed toward the next
intersection, 250 feet away. The motorcycle crosses the intersection at
t = 5 with instantaneous velocity 60 ft/s. The Porsche crosses the
intersection at the same time, with instantaneous velocity 70 ft/s.
(a) What is the average velocity of the motorcycle between t = 0 and
t = 5? What is the average velocity of the Porsche over the same
time interval?
(b) If f 0 (t) is the instantaneous velocity of the motorcycle, what is the
area under the graph of f 0 (t) from t = 0 to t = 5?
(c) Explain why the instantaneous velocity of the motorcycle must
have exceeded that of the Porsche at some time in the interval
[0, 5].
1
(15 points) 2. Suppose f is differentiable everywhere, and that f (0) = 2, f (1) = −1,
f (2) = 3, f 0 (0) = −4, f 0 (1) = 3, and f 00 (1) = −5.
(a) Write an equation for the secant line approximation to f (x) between the points x = 0 and x = 1.
(b) Write an equation for the tangent line approximation to the derivative function f 0 (x) at x = 1. Note this is not the same as the
tangent line for f (x).
f (h) − 2
.
h→0
h
(c) Determine lim
2
(15 points) 3. Let f (x) = 3x2 − 1. Use the definition of the derivative to find f 0 (x).
(20 points) 4. Calculate the following.
(a)
cos x
dy
where y =
+ 3x7 sin x.
2
dx
(x + 1)
x2 ,
if x < 1,
if x = 1,
(b) lim f (x), where f (x) = 0,

x→1

2 − x10 , if x > 1.



3
(10 points) 5. Suppose you have an electronic instrument with the property that once
every second n, it outputs a number an = 1/n, however, because of a
design flaw, it is not completely reliable. On average, once every 10
years, it outputs cos(1/n) instead of 1/n. Is it true that n→∞
lim an = 0?
Why or why not?
(20 points) 6. Find the maximum and minimum of the function f (x) = |x−1|−|x+1|
over the interval [−2, 3].
4