9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

Explain how the growth rate function determines the solution of a population model.

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

33–42. Solving initial value problems Solve the following initial value problems.

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.

The population increases from 50 to 60 in the first month and eventually levels off at 150.

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.

An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C? 

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.