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Differential Equations and Mathematical Modeling
Chapter Overview
Chapter Overview
Connections
Connections
The lovely image above is actually a two-dimensional projection of a three dimensional graph that is determined by three differential equations:
These equations were first used by Edward Lorenz (1917–2008), an MIT professor who had served as a meteorologist during World War II. He had hoped to use three-dimensional computer simulations of atmospheric convection to predict future weather conditions. What he discovered was somewhat discouraging for weather predictors but very exciting for mathematicians: the first insights into what is widely known today as Chaos Theory. You may have read of the “butterfly effect,” the whimsical name for the chaotic impact that small perturbations can have on the solutions to such systems. (The fact that this classic solution, known as the Lorenz attractor, resembles a butterfly is just a happy coincidence.) You will learn more about differential equations and their graphs in this chapter.
Chapter 7 Overview
Chapter 7 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 many situations in 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 2020. In this chapter, we examine the analytic, graphical, and numerical techniques on which such predictions are based.
Section 7.1: Slope Fields and Euler's Method
Section 7.1: Slope Fields and Euler's Method
Section 7.2: Antidifferentiation by Substitution
Section 7.2: Antidifferentiation by Substitution
Section 7.3: Antidifferentiation by Parts (BC Topic)
Section 7.3: Antidifferentiation by Parts (BC Topic)
Section 7.4: Exponential Growth and Decay
Section 7.4: Exponential Growth and Decay
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