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Chapter 6 Differential Equations and Mathematical Modeling O ne way to measure how light in the ocean di- minishes as water depth increases involves using a Secchi disk. This white disk is 30 centimeters in diameter, and is lowered into the ocean until it disappears from view. The depth of this point (in meters), divided into 1.7, yields the coeffi- cient k used in the equation l x l 0 e kx . This equation estimates the intensity l x of light at depth x using l 0 , the intensity of light at the surface. In an ocean experiment, if the Secchi disk disap- pears at 55 meters, at what depth will only 1% of surface radiation remain? Section 6.4 will help you answer this question. 320
Section 6.1 Slope Fields and Euler’s Method 321 Chapter 6 Overview One of the early accomplishments of calculus was predicting the future position of a planet from its present position and velocity. Today this is just one of a number of occa- sions on which we deduce everything we need to know about a function from one of its known values and its rate of change. From this kind of information, we can tell how long a sample of radioactive polonium will last; whether, given current trends, a population will grow or become extinct; and how large major league baseball salaries are likely to be in the year 2010. In this chapter, we examine the analytic, graphical, and numerical tech- niques on which such predictions are based. Slope Fields and Euler’s Method Differential Equations We have already seen how the discovery of calculus enabled mathematicians to solve problems that had befuddled them for centuries because the problems involved moving objects. Leibniz and Newton were able to model these problems of motion by using equa- tions involving derivatives—what we call differential equations today, after the notation of Leibniz. Much energy and creativity has been spent over the years on techniques for solv- ing such equations, which continue to arise in all areas of applied mathematics. 6.1 What you’ll learn about • Differential Equations • Slope Fields • Euler’s Method . . . and why Differential equations have always been a prime motivation for the study of calculus and remain so to this day. EXAMPLE 1 Solving a Differential Equation Find all functions y that satisfy dy dx sec 2 x 2 x 5. SOLUTION We first encountered this sort of differential equation (called exact because it gives the derivative exactly) in Chapter 4. The solution can be any antiderivative of sec 2 x 2 x 5, which can be any function of the form y tan x x 2 5 x C . That family of func- tions is the general solution to the differential equation. Now try Exercise 1. Notice that we cannot find a unique solution to a differential equation unless we are given further information. If the general solution to a first-order differential equation is continuous, the only additional information needed is the value of the function at a single point, called an initial condition . A differential equation with an initial condition is called an initial value problem . It has a unique solution, called the
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