Math 141
Name:
Exam 1
1. [12 points (6 pts each)] Evaluate each of the following limits.
(a) lim
x→∞
x2
x − 4x2
(3h + 2)2 − 4
(b) lim
h→0
h
2. [12 points (6 pts each)] In the theory of electrical circuits, Joule’s law describes a relationship between the power P dissipated by a resistor, the electrical current I passing through the resistor, and the
resistance R. The law can be written as follows:
P = I 2 R.
Usually power is measured in watts, current is measured in amps, and resistance is measured in ohms,
where 1 ohm = 1 watt/amp2 .
(a) Find an equation for
dI
dR
dP
in terms of I, R, , and
.
dt
dt
dt
(b) Suppose that the current through a resistor is increasing at a rate of 0.04 amps/sec, while the
resistance is increasing at a rate of 5 ohms/sec. How quickly is the power dissipated by the
resistor increasing when the current is 0.3 amps and the resistance is 60 ohms?
3. [26 points] A particularly fast-moving slug is crawling straight across an 8-foot-wide driveway. The
following graph shows the distance crawled by the slug as a function of time:
Slug Crawl
Distance Crawled HfeetL
5
4
3
2
asleep
1
1:00 pm
2:00 pm
3:00 pm
4:00 pm
Time
5:00 pm
6:00 pm
(a) [3 pts] What was the average speed of the slug between 1:00 pm and 6:00 pm?
(b) [3 pts] What was his average speed between 4:00 pm and 6:00 pm?
(c) [5 pts] At what time was the slug crawling the fastest? Explain.
Slug Crawl
Distance Crawled HfeetL
5
4
3
2
asleep
1
1:00 pm
2:00 pm
3:00 pm
4:00 pm
Time
5:00 pm
6:00 pm
(d) [4 pts] What was the slug’s crawl speed at 6:00 pm?
(e) [5 pts] Assuming his crawl speed remains the same after 6:00 pm, when will he reach the other end
of the 8-foot-wide driveway?
(f) [6 pts] Sketch a careful graph of the slug’s speed as a function of time:
PRACTICE SPACE
5
4
3
2
1
1 pm 2 pm 3 pm 4 pm 5 pm 6 pm
Time
6
Speed HfeethourL
Speed HfeethourL
6
FINAL ANSWER
5
4
3
2
1
1 pm 2 pm 3 pm 4 pm 5 pm 6 pm
Time
4. [18 points (6 pts each)]
(a) Find
dy
x2
10
if y =
+ √ + 3.
dx
5
x
(b) Find
Q2
dP
if P =
.
dt
Q+3
(c) Find f 0 (x) if f (x) =
p
x − 4x3 .
wall
5. [14 points] A 10-foot-long ladder rests against a vertical wall. The
ladder begins to fall, with the bottom end of the ladder moving away
from the wall at a rate of 2 feet/sec, and the top end of the ladder sliding
down. How quickly is the top end of the ladder moving towards the
ladder

ground when the bottom end is 6 feet from the wall?
ground

6. [10 points] Let f (x) = xx .
(a) [4 pts] Write a limit for f 0 (3). (Your answer to this question should be a limit formula.)
(b) [6 pts] Use a table of data to estimate the value of f 0 (3). To receive full credit, your final answer
must be accurate to within 0.01.
7. [8 points] Define a function f as follows:
f (n) = the area of a regular polygon with n sides and perimeter 6.
For example, f (3) is the area of an equilateral triangle whose perimeter is 6:
2
2
f (3) = the area of this triangle = 1.732,
2
and f (4) is the area of a square whose perimeter is 6:
1.5
1.5
1.5
1.5
What is lim f (n)? Explain your answer.
n→∞
f (4) = the area of this square = 2.25.