Math 171 — Calculus I Spring, 2016
Practice Questions for Exam I
1
Problem 1. Determine the one-sided limits at c = 1, 2 and 4 for the function g shown
in the figure and state whether the limit exists at these points.
(a) lim− g(t) =
(e) lim− g(t) =
(b) lim+ g(t) =
(f ) lim+ g(t) =
t→1
t→4
t→1
(h) Does the limit
exist at c = 2?
Yes/No
t→4
(c) lim− g(t) =
t→2
(g) Does the limit
exist at c = 1?
Yes/No
(d) lim+ g(t) =
(i) Does the limit
exist at c = 4?
Yes/No
t→2
Problem 2.
In the theory of relativity, the mass of a particle with velocity v is:
m= r
m0
1−
v2
c2
where m0 is the mass of the particle at rest and c is the speed of light. What happens
to m as v → c− ?
The same theory tells us that the length observed by an observer in relative motion with
respect to the object is given by
L = L0
p
1 − v 2 /c2
where L0 is the length of the object in its rest frame. What happens to L as v → c− ?
(
cx if x < 1
Problem 3. Let c be a number and consider the function f (x) =
. Is
v if x ≥ 1
there a number c such that f (x) is continuous at x = 1? Justify your answer.
Problem 4.
For each limit, evaluate the limit or or explain why it does not exist.
x+2
(a) lim 2
x→2 x − 4
x−2
(b) lim 2
x→2 x − 4
1
1
(c) lim+ √ − √
2
x→0
x
x +x
22t + 2t − 6
(d) lim
t→1
2t − 2
x−9
(e) lim √
x→9
x−3
(2 + x)2 − 4
x→0
x
(f) lim
2
Problem 5.
What does the Squeeze Theorem say about lim f (x) if lim l(x) =
x→7
x→7
lim u(x) = 6 and f (x), u(x), and l(x) are related as in Figure below? The inequality
x→7
f (x) ≤ u(x) is not satisfied for all x. Does this affect the validity of your conclusion?
The following identity may be useful for the next problems.
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
Problem 6.
Use equation (1), to simplify and compute the limit
cos(x + h) − cos(x)
h→0
h
lim
Problem 7.
Evaluate the following limits
sin(t)
t→0
t
sin2 (t)
(b) lim
t→π/4
t2
sin10 (t)
(c) lim
t→0
t9
tan(x)
(d) lim
x→0
x
(a) lim
Problem 8.
1 − cos(2x)
x→0
sin(2x)
(e) lim
sin(2h)(1 − cos(h))
h→0
h2
(f) lim
cos x − cos(4x)
Hint: Rewrite
x2
cos(4x) as cos(x + 3x) and use (1).
(g) lim
x→0
Consider the graph of f below
What is lim f (x)?
x→∞
What is lim f (x)?
x→−∞
(1)
Math 171 — Calculus I Spring, 2016
Problem 9.
Practice Questions for Exam I
Evaluate the following limits
t−2
(d) lim √
t→−∞
t2 + 4
t3 + 9t
(e) lim
t→−∞ 10t + 3
t4/3 + t1/3
(f) lim
t→∞ (2t2/3 + 2)2
x+2
x→∞ x2 − 4
(a) lim
x−2
x→−∞ x2 − 4
(b) lim
x−2
(c) lim √
x→∞
x2 + 4
Problem 10. The Monod Growth function is given by
G(S) = µmax
S
K +S
where µmax and K are constant. Find lim G(S).
S→∞
Problem 11.
The Lennard-Jones potential has the following form,
U (r) =
A
B
− 6
12
r
r
where A and B are constants and r is the distance between the particle in question.
Compute lim U (r).
r→∞
ex
. Show that f (0) < 1 < f (ln(4)).
ex − 2
Can you use the intermediate value theorem to conclude that there is a solution of
f (x) = 1?
Problem 12.
Let f (x) =
Can you find a solution to f (x) = 1?
Problem 13. An epidemiologist finds that the percentage N (t) of susceptible children
who were infected on day t during the first three weeks of a measles outbreak is given,
to a reasonable approximation, by the formula
N (t) =
100t2
t3 + 5t2 − 100t + 380
a. Is the rate of decline greater at t = 8 or t = 16?
b. Draw the secant line whose slope is the average rate of change in infected children
over the interval [0, 4].
c. Estimate ∆N/∆t on the interval [0, 6]. What are the units of ∆N/∆t.
3
4
Problem 14. The graphs in the figure represent the positions s of moving particles
as functions of time t. Match each graph with a description:
a. Speeding up
b. Speeding up and then slowing down
c. Slowing down
d. Slowing down and then speeding up
Problem 15.
questions
Use the tuna swimming data in figure below to answer the following
a. Determine the average velocity of each of these two fish over the 35 hours shown
in the figure.
b. What is the fastest average velocity shown in this figure, and over what time interval
and for which fish did it occur?