(b) Write the DTDS: s t+1 =

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1 Math 155. Homework 4. Sections 1.9, Do the following problems from the Adler text: 1.11: 2 (note: c = e ατ ), 6, 15, : 26, : 32, 34 Also do the following problems. 1. Three seals splash into their salt water pool at Sea World, spilling 20 gallons of water. Their pool usually holds 350 gallons of water. If one replaces the 20 gallons of water with pure water without adding salt, the concentration of salt will decrease. (a) Fill in the four blank boxes below to model the situation above. Let s t represent the concentration of salt in the pool after the seals have splashed t times, measured in mol/gal. Step Volume Total Salt (mol) Salt Concentration (mol/gal) H 2 0 in pool before seals jump in 350 gal 350s t s t Water lost 20 gal 20s t s t H 2 0 in pool after seals jump in 330 gal s t Pure water replaced 20 gal 0 H 2 0 in pool after adding pure water (b) Write the DTDS: s t+1 = For problem 2, you will need a computer with the Wolfram Mathematica Player installed so that you can make use of Mathematica Demonstrations. Instructions for downloading the free Mathematica Player are at the course website under the Study Resources tab; colostate.edu/~shipman/math155/study-resources.html. The Mathematica Demonstrations, which plot iterations and cobweb diagrams of DTDS s as you actively change parameters, are also available at this site. The Mathematica Player is also installed on the computers on the first floor of the Morgan Library. To use the player in the Morgan Library, you just download any of the Mathematica Demonstrations, and then double click on the demonstration you have downloaded (no Mathematica Player icon appears in the library computers and Mathematica Player is not in the list of programs on those computers, but the player is installed!).

2 2. For this exercise, we will use the Wolfram Mathematica Player Demonstration for the Heart Model from the Study Resources page of the course website. Set the player for the parameters τ = 0.4, α = 0.9, u = 16, V c = 47 and initial condition V 0 = 3. (a) What is the long-term behavior of a heart under such conditions? (b) Now decrease τ to 0.3. What long-term behavior do you observe? (c) Now decrease τ to 0.2. What long-term behavior do you observe? (d) What is the biological interpretation of a decrease in τ? (e) Using the parameters τ = 0.2, α = 0.9, V c = 47, can you find a value for u that makes the heart healthy? (f) A pacemaker monitors the heart to see if the heart beat is too slow. If the heart is beating too slow the pacemaker sends a signal to tell the heart to beat (i.e. sends a signal like the SA node sends to the AV node). What parameter or parameters in our model would be affected by a pacemaker?

3 3. Let V t represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c Let e ατ = 1 5, u = 10, and V c = 2. (a) If V 0 = 6, calculate V 1. Will the heart beat? Why or why not? (b) Does this system have an equilibrium? If so, find it algebraically; if not, explain why not. (c) Graph the updating function and cobweb from an initial value of V 0 = 6 to determine if this heart is i) healthy, ii) has a 2:1 block, or iii) has the Wenckebach phenomenon.

4 4. Let V t represent the voltage at the AV node in the heart model { e V t+1 = ατ V t + u, if V t e ατ V c e ατ V t, if V t > e ατ V c (a) For each of the following two graphs of the updating function, cobweb starting from an initial value of V 0 = 10, and determine if the heart i) is healthy, ii) has a 2:1 block, or iii) has the Wenckebach phenomenon. (b) Now let e ατ = 0.25, u = 10, and V c = 14. i) Does the system have an equilibrium? Justify your answer, and find the equilibrium if there is one. ii) Recall that e ατ = 0.25 determines the decay in voltage at the AV node between signals from the SA node. If α = 0.75, calculate the time τ between signals. Round to three decimal places.

5 5. Suppose that the population f(t) of a fungus as a function of time is given by f(t) = 3t (a) Find a formula for the slope of the secant line to f(t) that passes through (2, 13) and (2 + t, f(2 + t)). (b) Find a formula for the average rate of change of the fungus population between times 2 and 2 + t as a function of t. (c) What is the average rate of change in the fungus population between times t = 2 and t = 2.2? (d) Find the limit as t 0 of your formula in (b). What is the instantaneous rate of change of the fungus population at time t = 2? (e) Graph f(t) and indicate a graphical interpretation of the instantaneous rate of change at time t = 2.

6 6. Evaluate the following limits. 1. lim x 2 x 2. lim x 0 sin(x) 1 3. lim x 0 x 3 4. lim x 1 1 (x 1) 2 5. lim x x x 2 6. lim x x x 2 7. lim x 0 + ln(x) 2(x + x) 2 2 x 8. lim x 0 x 9. lim x 2 x 2 + x 6 x 2