Chapter 4 Review
Here are some review problems.
Exercise 4.2 Let \(u(t)\) denote the temperature of an object at time \(t\). Let \(A\) denote the ambient temperature around the object, assumed constant. Consider the following physics law: The rate at which the object temperature changes is proportional to the difference between the object temperature and the ambient temperature. Translate the plain English below into math:
- What is a simple mathematical expression for the rate at which the object temperature changes?
- What is a simple mathematical expression for the difference between the object temperature and the ambient temperature?
- Write an equation to express that the two quantities in (1) and (2) are proportional to each other. Use \(k\) for your constant of proportionality.
- What should the sign of the constant \(k\) be to make sense, given the nature of cooling? Explain your reasoning.
Exercise 4.6 Let \(N\) be a Poisson random variable with rate \(\lambda\). Use the Poisson probability distribution, given in the previous lecture, to verify that all probabilities sum to 1. In other words, verify that it holds