The Sensible Calculus Book

© 2002  M. Flashman
Chapter IV  Differential Equations from an Elementary Viewpoint

Preface
IV.A. Differential Equations. Basic Concepts and Definitions.
IV. B. Solving Differential Equations\
    IV. B.1.  Beginning Theory.
    IV. B. 2. The Calculus of Indefinite Integrals: Basic Facts.
IV. C.  Motion and Differential Equations.
IV. D.  Geometry and Differential Equations: Tangent Fields and Integral Curves.
IV. E.  Approximating The Solution to A Differential Equation with Initial Condition.
            Euler's Method.
IV. F. Euler Meets The Position, Cost, and Area Problems:
An Introduction to the Fundamental Theorem of Calculus
IV. G. Differential Equations and Area: Variation on A Theme.
IV.H. The Fundamental Theorem of Calculus: A Differential Equations Approach
    IV.H. Appendix: Proof of The Fundamental Theorem of Calculus for Positive Continuous Functions.


Preface
  • Some Thought Experiments
  • Explaining velocity
  • Variables and Rates of Change
  • Purpose and organization of the chapter 

    • IV.A. Differential Equations. Basic Concepts and Definitions.
  • Definition of a Differential Equation.
  • Interpretation of a Differential Equation.
  • What is a Solution ?

    • IV. B. Solving Differential Equations
    Preface
    IV. B.1.  Beginning Theory. 
  • What if K'(t)=0? Theorem 4.1: Proof
  • Theorem 4.1: Interpretations
  • What if f'(x)=g'(x)? Theorem 4.2 : Proof
  • Theorem 4.2: Interpretations
    • IV. B. 2. The Calculus of Indefinite Integrals: Basic Facts.
    Preface
  • The Constant of Integration
  • Indefinite Integral Calculus
  • Core Functions
  • Linearity
  • Problems

    • IV. C. Motion and Differential Equations.
    Preface
  • Constant Velocity
  • Constant Acceleration
  • Problems

    • IV. D. Geometry and Differential Equations: Tangent Fields and Integral Curves.
    Preface
  • What is a Tangent (Direction) Field?
  • What is an Integral Curve?
  • Examples where y' = dy/dx = P(x)
  • Example where y' = dy/dx = P(x, y)
  • Problems

    • IV. E.  Approximating The Solution to A Differential Equation with Initial Condition.
                                                     Euler's Method.
    Preface
  • Review of the Differential
  • A Motivating Example:
  • estimating the solution to an initial value problem
  • Interpretations
  • Euler's Method
  • An Example where y' = P(x,y)
  • Problems

    • IV. F.  Euler's Method Meets The Position, Cost, and Area Problems
    Preface
  • Motion and Euler Revisited
  • Costs and Euler Revisited
  • Area
  • Area Meets Velocity
  • Area Meets Marginal Cost
  • Theorem 4.3

    • (The Fundamental Theorem, Evaluation Form to Find Area)
  • Problems

    • IV. G.  Differential Equations and Area: Variation on A Theme.
    Preface
  • Motion and Area
  • An Area Function Solves A Differential Equation
  • Justification
  • Problems

    • IV.H. The Fundamental Theorem of Calculus: A Differential Equations Approach 

    IV.H. Appendix: Proof of The Fundamental Theorem of Calculus for Positive Continuous Functions.