Differential Equations

1. The Gompertz differential equation dy dt y (a-blny), where y> 0 and a and b are parameters, is used in actuarial studies, and to model growth of objects as diverse as tumors and organizations. (a) Find all equilibrium solutions. (b) Use the substitution z = In y to find the general solution of the Gompertz differential equation. (c) Solve the IVP consisting of the Gompertz differential equation and the initial condition y(0) = yo > 0. (d) Describe the limiting value for y(t) as too. Assume that a > 0, and consider the cases b>0 and b < 0.

3. Let A and B denote n x n matrices. For each of the following, prove the statement in general or give a counterexample that shows it is not true. (a) (A + B) (AB) = A²-B² (b) (I + A)² = 1 + 2A + A²

1. Use limits involving to describing the asymptotic behavior of each function from its graph. 3x x-2 f(x) =

2. Find the derivative of the following function: f(x) = 4√x³-—+1

3. The manager of an electronics store estimates that the number of flash drives that they will sell at a price of dollars is: S(x): = 2250 x +9

4. Find the derivative of the following function: f(x) = 1 V(5x+1)2

dy 7. For the following equation, find dx evaluated at the given values: x²y² - xy = 2atx = -1, y = 1

6. A computer dealer can sell 12 personal computers per week at a price of $2,000 each. He estimates that each $400 price decrease will result in three more sales per week. If the computers cost him $1,200 each, what price should he charge to maximize his profit? How many will he sell at that price? dy

9. One bank offers 6% compounded quarterly and a second offers 5.98% compounded continuously. Where should you take your money?

8. For the cost function: C(x) = 135√2x +3 Where C is in dollars and x is the number produced in thousands, us C(12) and MC(12) to approximate the cost of producing 11,600 items. Interpret the marginal cost value.