Chap9_Sec1 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS 9
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Perhaps the most important of all the applications of calculus is to differential equations. DIFFERENTIAL EQUATIONS
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When physical or social scientists use calculus, more often than not, it is to analyze a differential equation that has arisen in the process of modeling some phenomenon they are studying. DIFFERENTIAL EQUATIONS
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It is often impossible to find an explicit formula for the solution of a differential equation. Nevertheless, we will see that graphical and numerical approaches provide the needed information. DIFFERENTIAL EQUATIONS
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9.1 Modeling with Differential Equations In this section, we will learn: How to represent some mathematical models in the form of differential equations. DIFFERENTIAL EQUATIONS
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In describing the process of modeling in Section 1.2, we talked about formulating a mathematical model of a real-world problem through either: Intuitive reasoning about the phenomenon A physical law based on evidence from experiments MODELING WITH DIFFERENTIAL EQUATIONS
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The model often takes the form of a differential equation. This is an equation that contains an unknown function and some of its derivatives. DIFFERENTIAL EQUATION
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This is not surprising. In a real-world problem, we often notice that changes occur, and we want to predict future behavior on the basis of how current values change. MODELING WITH DIFFERENTIAL EQUATIONS
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Let’s begin by examining several examples of how differential equations arise when we model physical phenomena. MODELING WITH DIFFERENTIAL EQUATIONS
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One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population. MODELS OF POPULATION GROWTH
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That is a reasonable assumption for a population of bacteria or animals under ideal conditions, such as: Unlimited environment Adequate nutrition Absence of predators Immunity from disease MODELS OF POPULATION GROWTH
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Let’s identify and name the variables in this model: t = time (independent variable) P = the number of individuals in the population (dependent variable) MODELS OF POPULATION GROWTH
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The rate of growth of the population is the derivative dP / dt . MODELS OF POPULATION GROWTH
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Hence, our assumption that the rate of growth of the population is proportional to the population size is written as the equation where k is the proportionality constant. dP kP dt = POPULATION GROWTH MODELS Equation 1
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Equation 1 is our first model for population growth. It is a differential equation because it contains an unknown function P and its derivative dP / dt. POPULATION GROWTH MODELS
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a model, let’s look at its consequences. POPULATION GROWTH MODELS
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

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Chap9_Sec1 - 9 DIFFERENTIAL EQUATIONS DIFFERENTIAL...

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