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Math 155 Final Exam

NAME:

Problem Points Score

1 13

2 13

3 12

4 12

5 12

6 12

7 13

8 13

Total 100

1. (13 points) Evaluate the following definite and indefinite integrals. If necessary, use

substitution. Show all of your work.

(a)

R 2t

1 2 � 5

t

dt

(b)

R 3 cos(x)e

sin(x)+5 dx

(c)

R p ⇡

0 x sin(x

2 ) dx

(d) Use integration by parts to evaluate

R 2xe

5x dx.

2

2. (13 points)

(a) The population y

t

of a yeast colony obeys the discrete-time dynamical system

y

t+1 = 0.3yt.

Find the solution of this discrete-time dynamical system if y0 = 2000.

(b) Find the half-life for the yeast colony of part (a). (That is, at what time is the population

size equal to 1000?)

(c) The population b

t

of a bacteria colony obeys the discrete-time dynamical system

b

t+1 =

( 8 5bt, if bt  5 �b

t

+ 15, if b

t

> 5

Accurately graph the updating function for 0  b t

 10, labeling your axes. Cobweb for at least five steps, starting at b0 = 7; clearly label all points (bt, bt+1) for the first three

steps of cobwebbing. What is the long-term behavior of this solution?

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3. (12 points)

(a) Suppose that the population m

t

of mooses satisfies the discrete-time dynamical system

m

t+1 = mt(3 � mt) � hmt,

where h > 0 is a positive parameter, and m

t

is measured in thousands of mooses.

Find all equilibria. For what values of h is there more than one equilibrium that makes

biological sense?

(b) For each equilibrium, use the Stability Theorem/Criterion to determine the values of

h for which that equilibrium is stable. Show clearly how you are using the Stability

Theorem/Criterion.

4

4. (12 points) (a) Consider the function f(t) = t

3 � ⇡t + 1. i) Find all critical points of f(t). ii) Determine the global maximum and global minimum of f(t) on the interval [0, 3]. Justify

your answer and show your work clearly for full credit.

(b) Suppose that the production of a pharmaceutical agent, Q, depends on the population of

some bacteria, B, in the following manner:

Q(B) = B

2 e

�0.002B .

The units of B are hundreds of bacteria. Find the positive critical point of the function

Q(B), and use either the first or second derivative test to determine if there is either a local

maximum or local minimum at that point. Show your work clearly for full credit.

5

5. (12 points) Consider the function f(x) =

2x2+3x4

3+x6 .

(a) Find f0(x), the leading behavior of f(x) as x ! 0, and f1(x), the leading behavior of f(x) as x ! 1.

(b) Use the method of matched leading behaviors to sketch a graph of f(x) on the interval

x � 0. Label your axes, and graph and label f0(x), f1(x), and f(x).

6

6. (12 points) Suppose that a bacterium is absorbing a certain drug from its environment. At

time t = 0, there is 0.2 mol of drug in the bacterium, and drug enters the bacterium at a rate

of

1 1+t2

mol

min

(a) Let c(t) represent the amount (mol) of drug in the bacterium at time t (minutes). Write a

pure-time di↵erential equation and an initial condition for the situation described above.

(b) Apply Euler’s Method with �t = 0.5 to estimate the amount of drug in the bacterium

at time t = 1.5. Show your work clearly using a table. (Recall the formula

c

next

= c

current

+

dc

dt

�t, or ĉ(t + �t) = ĉ(t) + c

0 (t)�t).

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7. (13 points) A plant produces starch depending on the intensity of heat it receives during the

day. Assume the rate of starch production of the plant is

dS

dt

=

4t

1 + t

2 grams per hour

where time t is measured in hours and S(t) is the amount of starch produced t hours after

noon each day (time t = 0 is noon, t = 1 is 1pm and so on).

a. Estimate the total change in S(t) between 1pm and 3pm using right-hand Riemann sum

with �t = 0.5. Draw your rectangles or step functions on the figure below:

b. Find the exact area under the curve

4t

1 + t

2 between times t = 1 and t = 3. What is the

average rate of starch production (that is, the average value of

dS

dt

) between times t = 1

and t = 3?

8

8. (13 points)

(a) To celebrate completion of Math 155, you plant a tree on the oval. The tree is only 3m

tall when planted, but it grows 2m per year after being planted. Let h

t

= the height of

the tree t years after being planted, and write down a discrete-time dynamical system,

together with an initial condition, that describes this situation.

(b) Let L(t) = the length (in cm) of a fish at time t (in years). Suppose that the fish grows

at a rate

dL

dt

= 5.0e

�0.2t .

i. Use a definite integral to determine the total change in length of the fish between

times t = 5 and t = 10.

ii. Determine L(t) if L(0) = 2. (That is, find a solution to the di↵erential equation

dL

dt

= 5.0e

�0.2t with initial condition L(0) = 2.)

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