Math 155 Final Exam
- (13 points) Evaluate the following definite and indefinite If necessary, use substitution. Show all of your work.
- R 2t 1 — 5dt
- R 3 cos(x)esin(x)+5 dx
- Use integration by parts to evaluate R 2xe5x
- (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?)
- The population bt of a bacteria colony obeys the discrete-time dynamical system
—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?
- (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.
- (12 points) (a) Consider the function f(t) = t3 — ⇡t + 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) = B2e—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.
- (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).
- (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
- 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
cnext = ccurrent + dcOt, or cˆ(t + Ot) = cˆ(t) + c0(t)Ot).
- (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
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).
- 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:
- Find the exact area under the curve 1 + t2 between times t = 1 and t = What is the
average rate of starch production (that is, the average value of dS ) between times t = 1
and t = 3?
- (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.
- Let L(t) = the length (in cm) of a fish at time t (in years). Suppose that the fish grows at a rate dL = 0e—0.2t.
- Use a definite integral to determine the total change in length of the fish between times t = 5 and t =
- Determine L(t) if L(0) = (That is, find a solution to the di↵erential equation
dL = 5.0e—0.2t with initial condition L(0) = 2.)