Math 135 Exam 3 Review
Note: Exam 3 also covers Optimization. Please see the Exam 2 Review Activity for practice problems on that topic!
1. An old rowboat has sprung a leak. Water is lowing into the boat at a rate, r(t), given in the table
|
t (minutes) |
0 |
5 |
10 |
15 |
|
r(t) (liters/min) |
12 |
20 |
24 |
16 |
Compute the right endpoint estimate and the left endpoint estimate for the total amount of water that has lowed into the boat during the 15 minutes.
2. The following table gives the US emissions, H(t), of hydroluorocarbons, or “super greenhouse gasses,”’ in teragrams per year, with t in years since 2000.
|
Year |
2000 |
2002 |
2004 |
2006 |
2008 |
2010 |
|
H(t) |
7104 |
7022 |
7163 |
7159 |
7048 |
6822 |
(a) What are the units and meaning of 10(10) H(t)dt?
(b) Estimate 10(10) H(t)dt.
3. Use RStudio to estimate the value of 127 (x + cos(x2 ))dx as indicated below.
(a) A right endpoint estimate using 4 rectangles.
(b) A left endpoint estimate using 8 rectangles.
(c) An estimate that is correct up to two decimal places.
4. Compute the following deinite and indeinite integrals.
(b)1 (cos(4x) + 5) dx
(c)125（e-x + x2 + 4x）dx
5. Bonnie and Clyde go for a bike ride. They start at the same location, and their velocities (in miles per hour) are shown in the two plots below.
(a) In addition to t = 0, when are Bonnie and Clyde at the same location?
(b) When is the distance between Bonnie and Clyde at its maximum value?
6. Your velocity after t seconds is given by v(t) = t2 + 1 in m/sec. Find the distance traveled between t = 0 and t = 5.
7. A news broadcast in early 1993 said the typical American’s annual income is changing at a rate of r(t) = 40(1.002)t dollars per month, where t is in months from January 1, 1993. How much did the typical American’s income change during the year 1993?
8. A pollutant spilled on the ground decays at a rate of 6% a day. In addition, cleanup crews remove the pollutant at a rate of 40 gallons a day.
(a) Write a diferential equation for the amount of pollutant P , in gallons, left after t days. (b) The initial amount of pollutant is 2000 gallons. Find the function P (t).
(c) How long does it take to clean up the pollutant? In other words, when is P (t) = 0?
9. A savings account earns interest at a continuous rate of 3% per year. In addition, deposits are made at a constant rate of $1200 per year.
(a) Write a diferential equation for the balance B(t) in the account where t is measured in years.
(b) The initial deposit amount was $5000. Find the function that is the solution to your diferential equation.
10. A rumor spreads through a college campus. The number of people who have heard the rumor after t days is a function R(t) where t is measured in days. The spread of the rumor satisies the diferential equation
= 0.12R - 0.000048R2
(a) This diferential equation is a constrained growth model. Rewrite the diferential equation so that it is of the form
= rR（ 1 - 
.
(b) In the long run, how many people hear the rumor?
(c) What is the size of the population when the rumor is spreading the fastest? How fast is the rumor spreading that that point?
(d) If R(0) = 1 then ind the formula for R(t).
11. The diferential equation
= 0.1P2 - 0.8P + 1.2
describes the behavior of an endangered popula- tion of sea turtles. Conservation ecologists are re- leasing turtles raised in capitivity into the wild at a constant rate.
(a) Find the populations where 
= 0. (Hint: multiply the derivative by 10 and then set it equal to zero.)
(b) Use the slope ield to the right to describe the population dynamics of these sea turtles for initial populations P = 1, P = 4 and P = 8. Comment on the e伍cacy of the ecologists’ actions for each of these initial conditions.
10
8
6
4
2
0
0 2 4 6 8 10
12. The acceleration of a large raindrop depends on gravity (32 ft/sec2 ), and the drag due to wind resis-
tance. This drag is proportional to the square of the velocity. We have
dt(dv) = 32 - 0.0015v2 .
(a) Find the terminal velocity (in ft/sec)
by solving dt(dv) = 0.
(b) The plot on the right is a trajectory
plot of raindrop velocity versus time.
What is the velocity of the raindrop
after 5 seconds? How long does it
take for the raindrop to achieve its
terminal velocity?
13. Use the fact that the derivative gives the slope of the curve to match each diferential equation to the plot that represents a possible solution.
(a) dx(dy) = x
(b) dx(dy) = y
(c) d(d)x(y) = x(1)
(d) d(d)x(y) = y(1)
(B)
y
x
(C)
y
x
x
14. A disease is spreading according to the SIR Model
dS dS dR
= -0.003SI = 0.003SI - 0.42I = 0.42I
dt dt dt
where t is measured in weeks.
(a) What is the threshold population for this model?
(b) Use traj plot to create an SI trajectory for this disease
. Use initial conditions S = 500 and I = 10. Run the trajectory for 20 weeks.
(c) What is the maximum size of the infected population? How many weeks does it take to achieve that maximum?
(d) There is a vaccine for this disease, which moves a susceptible person directly into the removed category. The updated model becomes
= -0.003SI - cS 
= 0.003SI - 0.42I 
= 0.42I + cS
where 0 c 1 represents the percentage of susceptibles who get vaccinated in a week. We want
to keep the maximum number of infecteds below 100. Use traj plot for various values of c to
determine the smallest c such that I < 100 for the entire outbreak, again starting with S = 500 and I = 10. Your answer should be correct up to two decimal places. Your answer should be correct up to two decimal places.
15. Consider the four diseases models below.
dS dS
= -0.005SI = -0.001SI
dt dt
(I) (III)
dI dI
= 0.005SI - 0.2I = 0.001SI
dt dt
= -0.002SI 
= -0.002SI + 0.3I
(II) (IV)
dI dI
= 0.002SI - 0.3I = 0.002SI - 0.3I
dt dt
For each description below, identify every model that matches the description. Note that each de- scription may match more than one model.
(a) The initial conditions S0 = 100 and I0 = 15 mean that the disease is in epidemic phase: the infected population will increase.
(b) Infecteds never recover.
(c) People do not develop immunity to the disease. Once you recover, you become susceptible again.
(d) Eventually, the outbreak MUST subside: the number of infected people must become 0 at some point.