Math 155                 Final Exam

1. (13 points)   Evaluate   the   following   definite   and indefinite              If necessary, use substitution. Show all of your work.

 2
• R 2t 1 5dt

t

• R 3 cos(x)esin(x)+5 dx

 ⇡ x sin(x2) dx
 R

p

• 0

• Use integration by parts to evaluate R 2xe5x

1. (13 points)

• The population yt of a yeast colony obeys the discrete-time dynamical system

yt+1 = 0.3yt.

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

• Find the half-life for the yeast colony of part (a). (That is, at what time is the population    size equal to 1000?)

 ( 8 bt,                 if bt       5
• The population bt of a bacteria colony obeys the discrete-time dynamical system

bt+1 =

5

bt + 15,     if bt > 5

Accurately  graph  the  updating  function  for  0    bt    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?

1. (12 points)

• Suppose that the population mt of mooses satisfies the discrete-time dynamical system

mt+1 = mt(3 mt) hmt,

where h > 0 is a positive parameter, and mt is measured in thousands of mooses.

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

• 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.

1. (12 points) (a) Consider the function f(t) = t3 ⇡t + i) Find all critical points of f(t).
2. ii) Determine the global maximum and global minimum of f(t) on the interval [0, 3]. Justify

(b) Suppose that the production of a pharmaceutical agent, Q, depends  on  the  population  of some bacteria, B, in the following manner:

Q(B) = B2e0.002B.

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.

 3+x6
• (12 points) Consider the function f(x) = 2x2+3×4 .
• Find f0(x), the leading behavior of f(x) as x ! 0, and f1(x), the leading behavior of

f(x) as x ! 1.

• 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).

1. (12 points) Suppose that a bacterium is absorbing  a  certain  drug  from  its    At time t = 0, there is 0.2 mol of drug in the bacterium, and drug enters the bacterium at a rate

of     1 2 mol

1+t  min

• 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

• Apply Euler’s Method with Ot = 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
 dt

cnext  = ccurrent +  dcOt,  or  cˆ(t + Ot) = cˆ(t) + c0(t)Ot).

1. (13 points) A plant produces starch depending on the intensity of heat it receives during the Assume the rate of starch production of the plant is

dS            4t

dt = 1 + t2 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).

1. Estimate the total change in S(t) between 1pm and 3pm using right-hand Riemann sum with Ot = 5. Draw your rectangles or step functions on the figure below:

4t

1. Find the exact area under the curve 1 + t2 between times t = 1 and t = What is the
 dt

average rate of starch production (that is, the average value of dS ) between times t = 1

and t = 3?

1. (13 points)

• To celebrate completion of Math 155, you plant a tree on the   The tree is only 3m         tall when planted, but it grows 2m  per  year  after  being  planted.  Let  ht =  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.

 dt
• Let L(t) = the length (in cm) of a fish at time t (in years). Suppose that the fish grows at a rate dL = 0e0.2t.
1. Use a definite integral to determine the total change in length of  the  fish  between times t = 5 and t =

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

dL = 5.0e0.2t with initial condition L(0) = 2.)

Math 155 Final Exam Paper
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