checking 9 exercises of hw of math 508 and solving 2 missing.

Unit 7 Homework Set 7
1. For this problem, use the coffee cooling rate of change equation
dC
dt = −0.4C + 28.
(a) Is there ever a time when two cups of coffee, one at initially 160◦F and one at 180◦F, are the
exact same temperature? Answer this question according to the uniqueness theorem. Comment
on whether your answer matches what you expect to happen in real life?
(b) How long will it take a cup of hot coffee that is initially 180◦F to cool down to 100◦F? Use the
reverse product rule to figure this out and then check the reasonableness of your answer with
Euler’s method.
2. For each part below you are provided with an autonomous derivative graph. Figure out the longterm behavior of every possible solution function. Illustrate your conclusions with representative y(t)
solution graphs and summarize your findings about the long-term behavior of different solutions in
paragraph form.
(a) (b)
3. For each part in problem 2, create a phase line and classify each equilibrium solution as either an
attractor, repeller, or node.
4. For problem 2b, use the uniqueness theorem to determine if any of the non-constant solution functions
ever reach the equilibrium solution of y(t) = 0 in a finite amount of time.
5. Given an autonomous differential equation dy
dt = f(y) , give a general strategy for how to use an
autonomous derivative graph to determine the long term behavior of solution functions.
Page 7.7Unit 7: Modeling with Autonomous Differential Equations
6. Suppose you wish to predict future values of some quantity, y, using an autonomous differential
equation (that is, dy/dt depends explicitly only on y). Experiments have been performed that give
the following information:
• The only equilibrium solutions are y(t) = 0, y(t) = 15, and y(t) = 60
• If the value of y is 100, the quantity decreases
• If the value of y is 30, the quantity increases
• If the value of y is negative, the quantity increases
(a) How many different phase lines match the above? Sketch all possible phase lines.
(b) Provide a rough sketch of an autonomous derivative graph for each of your phase lines in part
6a.
(c) For each of your different sketches in part 6b, develop a differential equation that fits the basic
features.
7. In what ways is the letter y in the differential equation dy
dt = .3y both a variable and a function? In
what ways is dy
dt a function?
8. Newton’s law of cooling is an empirical law that states that an object immersed in a constant, ambient
temperature will have its temperature change at a rate proportional to the difference between the its
temperature and the ambient temperature. Explain how the cooling coffee problem reflects Newton’s
law of cooling.
9. A body was found in a temperature controlled environment (i.e., you know the room temperature)
and is subject to Newton’s law of cooling. Explain why you only need the room temperature and
the measurement of the body’s temperature at two different times to give an estimate of the time of
death.
10. Are the following true or false statements? Explain your reasoning.
(a) If the autonomous derivative graph has a vertical tangent line at some point, then according to
the uniqueness theorem, solutions to the differential equation are guaranteed to touch or cross
at that point.
(b) If the autonomous derivative graph is a polynomial, then according to the uniqueness theorem,
solutions to the differential equation are everywhere unique.
11. (a) Go to the glossary and identify all terms that are relevant to this unit and list those terms here.
(b) Are there other vocabulary terms that you think are relevant for this unit that were not included?
If yes, list them.
Page 7.