Exam 9: Mathematical Modeling Using Differential Equations

The solution to the differential equation subject to the initial condition is , where k = _____ and C = _____. Round answers to 2 decimal places.

A cup of green tea contains 32 mg of caffeine when you are using the tea leaves for the first time. A cup from the second brew contains 12 mg of caffeine, while a cup from the third brew contains only 4 mg of caffeine. Caffeine leaves the body at a continuous rate of about 17% per hour. a) Write a differential equation for the amount, C, of caffeine in the body at time t hours after drinking the green tea. b) Use the differential equation to find at the start of the first hour (right after drinking the tea) for a cup from the first brew, and use your answer to estimate the change in caffeine in the body during the first hour. c) Does the initial amount of caffeine in the body (whether from the first, second or third brew) change the differential equation?

The slope fields for and are shown in the following figure. Which slope field goes with the differential equation = x ?

If the slope field for has constant slopes where x is fixed, what do we know about ?

Is a solution to the differential equation ?

If is a solution to the differential equation , then k = _____.

Given that and , estimate by first estimating y(1) and y(2). Assume that the rate of growth given by is approximately constant over each unit time interval.

A quantity y satisfies the differential equation . Thus, y is decreasing when y is ________(less/greater) than _____.

Sketch a slope field for the differential equation using the points indicated on the axes.

Consider the slope field for . What is the slope at the point (0,0)?

The equilibrium solution for is P = _____. This solution is ________ (stable/unstable).

A company earns a continuous annual rate of 11% of its net worth. At the same time, it has expenses of 6.2 million dollars per year. If the company's net worth at time t = 0 is 50 million, how many years will it take to go bankrupt? Round to the nearest year.

An anti-inflammatory drug has a half-life in the human body of about 8 hours. A. Use the half-life to find the value of k in the differential equation , where Q is the quantity of the drug in the body t hours after the drug is administered. Round to 4 decimal places. B. After how many hours will 45% of the original dose remain in the body? Round to 2 decimal places.

Which one(s) of the following are solutions to the differential equation ?

What is the solution of the differential equation if when ?

What is the threshold value for the strain of the flu modeled by the differential equations ? Round to the nearest whole number.

On the slope field for the differential equation , sketch the solution curve in the fourth quadrant that goes through the point (0, -1).

Consider three strains of the flu modeled by the following sets of differential equations. Which has the infecteds being removed the slowest? I. II. III.

Owls need trees to survive, but trees don't care one way or the other about owls. Which (if any) of the following systems of differential equations could model the interaction between owls and trees, with trees as x and owls as y? Select all that apply.

Money in a bank account earns interest at a continuous rate of 6% per year, and payments are made continuously out of the account at the rate of $10,000 per year. The account initially contains $100,000. Write a differential equation for the balance, B, in the account in t years and use it to find how many years it will take for the account to run out of money. Round to 1 decimal place.