MATH 228 Review Exercises for Test 1

Part 1 – No Calculator

1 Find the derivative of each of the following functions:

2. Find the following limits:

Part 2 – Calculator may be used for all problems

1. Suppose a discrete time dynamical system is defined by c₀ = 2, and c_t+1 = 0.5c_t + 2.

a) Compute c₁, c₂, and c₃.

b) Find the equilibrium c*.

c) Find a general expression for c_t. (Hint: Consider c* - c₀, c* - c₁, etc.)

2. Let x_t+1 = (2.2)x_t(1-x_t). Find all the equilibria of the system and test them for stability using the derivative. Then cobweb the system to show that your answer is correct.

3. A radioactive element has a half life of 4 years. How much of an original sample of 18 grams will be left after 10 years?

4. Recall the model of the lung: c_t+1 = (1-q)c_t + qg, where q = W/V, W being the volume of air exhaled at each breath and V is the volume of the full lung. g is the concentration of some chemical in the ambient air. Suppose that W = 0.8 L while V = 5.0 L and that g represents Oxygen, that is g = 0.21. Find the eventual concentration of Oxygen in the lungs based on this model.

5. Find the leading behavior of f(x) = 1 + 2x + 3e^x at 0 and at ¥.