Spring /30/2013

Size: px

Start display at page:

Download "Spring /30/2013"

Bertram Whitehead
6 years ago
Views:

1 MA Calculus 2 for the Life Sciences FINAL EXAM Spring /30/2013 Name: Sect. #: Answer all of the following questions. Use the backs of the question papers for scratch paper. No books or notes may be used. You may use a calculator. You may not use a calculator that has symbolic manipulation capabilities. When answering these questions, please be sure to: check answers when possible, clearly indicate your answer and the reasoning used to arrive at that answer (unsupported answers may receive NO credit). QUESTION SCORE TOTAL Bonus. 10 TOTAL 100

2 Please make sure to list the correct section number on the front page of your exam. In case you forgot your section number, consult the following table: Sections # Lecturer Time/Location Kate Ponto MWF 10:00 am - 10:50 am, FPAT Alberto Corso MWF 01:00 pm - 01:50 pm, SRB 303 Section # Recitation Instructor Time/Location 002 Laura Graham TR 12:30 pm - 01:20 pm, CB Laura Graham TR 02:00 pm - 02:50 pm, CP Jonathan Thompson TR 12:30 pm - 01:20 pm, CP Jonathan Thompson TR 02:00 pm - 02:50 pm, BS 109

3 1. Find the area of the region bounded by the graphs of the functions y = 2x x 2 and y = x 2. y x

4 2. (a) Use the substitution method to evaluate the definite integral ln 7 ln 4 e x (e x 3) 2 dx (b) Let f be a twice differentiable function of x. Use the integration by parts method and the Fundamental Theorem of Calculus to evaluate the integral 3 1 xf (x) dx, given that f(1) = 2 f(3) = 5 f (1) = 2 f (3) = 4.

5 3. The logistic ( model for growth of a population is described by the differential equation dn dt = rn 1 N ), where r is the growth rate and K is the carrying capacity. Both r K and K are positive constants. For some species a very small population limits reproduction since there is a lack of suitable mates. We can model this using the differential equation where 0 < a < K. dn dt ( = rn (N a) 1 N ), K (a) Find the equilibria N for the latter differential equation. (b) Use the analytic (eingenvalues) method to classify these points. ( Hint: dn dt = g(n), where g(n) = r ( K N 3 + r 1 + a ) ) N 2 ran. K

6 4. Suppose we have two strains of bacteria with population sizes denoted by a = a(t) and b = b(t), where t represents time. If the growth rates of the bacteria are proportional to the size of the respective population, we have differential equations da dt = µ a and db dt = λ b, where µ and λ are positive constants. We also assume that µ λ. The fraction of the population of type a is given by the expression p = a a + b. It can be shown that this fraction p = p(t) satisfies the following differential equation dp dt = (µ λ) p (1 p). (i) Find the equilibria for this differential equation and use the graphical method in order to classify these points. case µ > λ: dp dt p

7 case µ < λ: dp dt p (ii) What is the biological meaning of your findings in part (i) in terms of the two strains of bacteria? In the case µ > λ we have: In the case µ < λ we have:

8 5. Suppose a scientist has four colonies of the same bacteria. She wants to estimate the growth rate of this species of bacteria. To do this, she measures the population of each colony at time t = 0 and one day later at time t = 1. The data from her measurements are listed below: Colony P 0 P 1 A 9 17 B C 4 9 D 5 7 Find the least squares approximation of the form P 1 = m P 0 + b that expresses the population P 1 at time t = 1 as a function of the population P 0 at t = 0.

reducing., How many solutions does the system have?

9 6. Solve the following system of linear equations x + 2y z = 2 x + 4y + 3z = 8 3x + 8y + z = 12 by writing the corresponding augmented matrix and then by row reducing., How many solutions does the system have? Which of the two pictures below illustrates the geometric situation described by the given system of linear equations?

10 7. Match each of the following functions f(x, y) = 3 x+y g(x, y) = xy(1 x y) h(x, y) = xye x2 y 2 k(x, y) = 4 x 2 y 2 with its graph (labeled A.-D.) and its level curves (labeled I.-IV.). f(x, y) corresponds to graph and level curves A. I. g(x, y) corresponds to graph and level curves B. II. h(x, y) corresponds to graph and level curves C. III. k(x, y) corresponds to graph and level curves D. IV.

8. Consider the function f(x, y) = 4 x 2 + y 2 + 2 whose graph is given in the picture on the right.

11 8. Consider the function f(x, y) = 4 x 2 + y whose graph is given in the picture on the right. (a) Find the z-coordinate z 0 of the point P on the graph of the function f(x, y) with x-coordinate x 0 = 1 and y-coordinate y 0 = 1. (b) Write the equation of the tangent plane to the graph of the function f(x, y) at the point P, as above, with coordinates x 0 = 1 and y 0 = 1. (c) Write the linear approximation, L(x, y), of the function f at the point with x 0 = 1 and y 0 = 1, as above, and use it to approximate f(1.1, 0.9). Compare this approximate value to the exact value f(1.1, 0.9).

12 9. If a large block of ice is placed in a room we can describe how the temperature of the block of ice and room are changing using the system of differential equations di dt = α(r I) dr dt = β(i R), where I is the temperature of the block of ice, R is the temperature of the room and α and β are positive constants that determine the relationship between the rates of change of the temperatures of the block of ice and room and the difference between these temperatures. (All temperatures are measured in degrees Fahrenheit ( F).) (a) Find the solution of the given system of linear differential equations in the case that α = 0.5, β = 0.25, I(0) = 32, and R(0) = 74. That is I I I(0) 32 d = = dt R R R(0) 74

13 (b) Describe the long term behavior of your solution. In particular what happens to the temperature of the room and to the block of ice? (c) Which of the pictures below describes the behavior of the two temperatures?

14 10. Suppose a habitat is divided up into patches and each patch can be occupied by at most one individual. If two species A and B live in this habitat, the growth of the population of A is controlled by the internal dynamics of the population growth of A and the interactions between A and B. The situation for B is similar. Suppose the members of species A are able to outcompete members of species B, that is, the members of A are able to invade patches that are occupied by species B and displace the resident. If p 1 is the fraction of the sites occupied by A and p 2 is the fraction of the sites occupied by B, this situation is described by the differential equations dp 1 dt = c 1p 1 (1 p 1 ) m 1 p 1 dp 2 dt = c 2p 2 (1 p 1 p 2 ) m 2 p 2 c 1 p 1 p 2 where c 1, c 2, m 1, and m 2 are the colonization and mortality rates of species A and B, respectively. Suppose c 1 = 2, c 2 = 10, m 1 = 1, and m 2 = 2. Hence, after some algebra, the above system of nonlinear differential equations can be written as dp 1 dt = p 1 (1 2p 1 ) dp 2 dt = p 2 (8 12p 1 10p 2 ) (a) There is only one equilibrium point ( p 1, p 2 ) where both species are present. Identify that point and linearize the given system of differential equations at that point. Classify the type of equilibrium. p 2 Plot here the nullclines and the nontrivial equilibrium point p 1

15 (b) Choose the direction field that describes the system of nonlinear differential equations considered in (a). Direction Field A Direction Field B

16 Bonus. There are several ways that two species can interact in accordance with the Lotka-Volterra model. The direction fields below correspond to four different behaviors. For each direction field choose the type of interaction 1,2 between the two species and describe the long term behavior of the two populations. Type of interaction: Coexistence Founder control Competitive exclusion Predator-prey Long term behavior: Type of interaction: Coexistence Founder control Competitive exclusion Predator-prey Long term behavior: 1 Founder control: No species can invade the other. The competition outcome depends on the initial densities. 2 Competitive exclusion: One species, the strong one, outcompetes the other, the weak one.

17 Type of interaction: Coexistence Founder control Competitive exclusion Predator-prey Long term behavior: Type of interaction: Coexistence Founder control Competitive exclusion Predator-prey Long term behavior:

Spring /11/2014

Spring /11/2014 MA 138 Calculus 2 for the Life Sciences SECOND MIDTERM Spring 2014 3/11/2014 Name: Sect. #: Answer all of the following questions. Use the backs of the question papers for scratch paper. No books or notes