# Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 1.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 6 Differential Equations and Mathematical Modeling

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6.1 Slope Fields and Euler’s Method

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 4 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 5 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 6 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 7 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 8 What you’ll learn about Differential Equations Slope Fields Euler’s Method … and why Differential equations have been a prime motivation for the study of calculus and remain so to this day.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 9 Differential Equation An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 10 Example Solving a Differential Equation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 11 First-order Differential Equation If the general solution to a first-order differential equation is continuous, the only additional information needed to find a unique solution 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 particular solution to the differential equation.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 12 Example Solving an Initial Value Problem

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 13 Example Using the Fundamental Theorem to Solve an Initial Value Problem

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 14 Slope Field The differential equation gives the slope at any point (x, y). This information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. Repeating that process at many points yields an approximation called a slope field.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 15 Example Constructing a Slope Field

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 16 Euler’s Method for Graphing a Solution to an Initial Value Problem 5. To construct the graph moving to the left from the initial point, repeat the process using negative values for. 3. Increase by. Increase by, wherex x y y  2. Use the differential equation to find the slope / atthe point. 1. Begin at the point (, ) specified by the initial condition.x y dy dx that lies along the linearization. 4. Using this new point, return to step2. Repeating the process constructs the graph to the right of the initial point. y dy dx x  ( / ). Thisdefines a new point (x  x, y  y) x 

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 17 Example Applying Euler’s Method (1, 2)20.10.2(1.1, 2.2) 2.10.10.21(1.2, 2.41) 2.20.10.22(1.3, 2.63) 2.30.10.23(1.4, 2.86) 2.40.10.24(1.5, 3.1)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6.2 Antidifferentiation by Substitution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 19 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 20 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 21 What you’ll learn about Indefinite Integrals Leibniz Notation and Antiderivatives Substitution in Indefinite Integrals Substitution in Definite Integrals … and why Antidifferentiation techniques were historically crucial for applying the results of calculus.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 22 Indefinite Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 23 Example Evaluating an Indefinite Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 24 Properties of Indefinite Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 25 Power Functions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 26 Trigonometric Formulas

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 27 Exponential and Logarithmic Formulas

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 28 Example Paying Attention to the Differential

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 29 Example Using Substitution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 30 Example Using Substitution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 31 Example Setting Up a Substitution with a Trigonometric Identity

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 32 Example Evaluating a Definite Integral by Substitution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6.3 Antidifferentiation by Parts

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 34 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 35 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 36 What you’ll learn about Product Rule in Integral Form Solving for the Unknown Integral Tabular Integration Inverse Trigonometric and Logarithmic Functions … and why The Product Rule relates to derivatives as the technique of parts relates to antiderivatives.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 37 Integration by Parts Formula

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 38 Example Using Integration by Parts

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 39 Example Repeated Use of Integration by Parts

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 40 Example Solving for the Unknown Integral

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 41 Example Antidifferentiating ln x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 42 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 43 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 44 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 45 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 46 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 47 Quick Quiz Sections 6.1-6.3

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6.4 Exponential Growth and Decay

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 49 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 50 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 51 What you’ll learn about Separable Differential Equations Law of Exponential Change Continuously Compounded Interest Modeling Growth with Other Bases Newton’s Law of Cooling … and why Understanding the differential equation gives us new insight into exponential growth and decay.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 52 Separable Differential Equation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 53 Example Solving by Separation of Variables

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 54 The Law of Exponential Change

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 55 Continuously Compounded Interest

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 56 Example Compounding Interest Continuously

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 57 Example Finding Half-Life

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 58 Half-life

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 59 Newton’s Law of Cooling

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 60 Example Using Newton’s Law of Cooling A temperature probe is removed from a cup of coffee and placed in water that has a temperature of T = 4.5 C. Temperature readings T, as recorded in the table below, are taken after 2 sec, 5 sec, and every 5 sec thereafter. Estimate (a)the coffee's temperature at the time the temperature probe was removed. (b)the time when the temperature probe reading will be 8 C. o S o

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 61 Example Using Newton’s Law of Cooling

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 62 is a model for the  t, T – T  = ( t,T  4.5) data. Thus,  4.5  61.66 0.9277 Example Using Newton’s Law of Cooling  Use exponential regression to find that According to Newton's Law of Cooling, T  T =  T – T  e  kt S O S where T = 4.5 and T is the temperature of the coffee at t  0. S O t S T  T  4.5 + 61.66 0.9277 is a model of the  t,T , data.  t (b) The figure below shows the graphs of y  8 and y  T  4.5 + 61.66  0.9277  t (a)At time t  0 the temperature was T  4.5 + 61.66  0.9277   66.16 C 

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 6.5 Logistic Growth

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 64 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 65 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 66 Quick Review Solution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 67 What you’ll learn about How Populations Grow Partial Fractions The Logistic Differential Equation Logistic Growth Models … and why Populations in the real world tend to grow logistically over extended periods of time.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 68 Partial Fraction Decomposition with Distinct Linear Denominators

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 69 Example Finding a Partial Fraction Decomposition

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 70 Example Antidifferentiating with Partial Fractions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 71 Logistic Differential Equation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 72 Example Logistic Differential Equation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 73 The General Logistic Formula

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 74 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 75 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 76 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 77 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 78 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 79 Quick Quiz Sections 6.4 and 6.5

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 80 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 81 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 82 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 6- 83 Chapter Test Solutions

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