Contents

1 Modeling Discrete Data

• 1.1 Introduction to the Issues

  1.1.1 Modeling with an Elementary Function
  1.1.2 Review of Modeling Issues So Far
  1.1.3 Fitting a Curve to Data: The Method of Least Squares
  1.1.4 An Agenda for Future Work
  

• 1.2 Lines in the Plane
• 1.3 Functions and Their Graphs
• 1.4 Defining New Functions from Old
• 1.5 Trigonometric Functions
• 1.6 Exponential and Logarithm Functions
• 1.7 Case Study: Modeling with Elementary Functions

2 Modeling Rates of Change

• 2.1 Introduction to the Issues

  2.1.1 Average Speed
  2.1.2 The Meaning of Constant Acceleration
  2.1.3 Hypotheses and Open Questions
  2.1.4 The Distance Function is Quadratic
  2.1.5 Average Rate of Change
  2.1.6 An Agenda for Future Work
  

• 2.2 The Legacy of Galileo, Newton, and Leibniz
• 2.3 Limits of Functions
• 2.4 Limits at Infinity
• 2.5 Continuity
• 2.6 Tangent Lines and Their Slopes
• 2.7 The Derivative
• 2.8 Differentiation Rules
• 2.9 Derivatives of the Trigonometric Functions
• 2.10 The Mean Value Theorem
• 2.11 Implicit Differentiation
• 2.12 Derivatives of Exponential and Logarithm Functions
• 2.13 Newton’s Method
• 2.14 Linear Approximations
• 2.15 Antiderivatives and Initial Value Problems
• 2.16 Velocity and Acceleration
• 2.17 Related Rates
• 2.18 Case Study: Torricelli’s Law

3 Modeling with Differential Equations

• 3.1 Introduction to the Issues

  3.1.1 Solution by Inspection
  3.1.2 Slope Fields
  3.1.3 An Analytical Tool: Separation of Variables
  3.1.4 An Agenda for Future Work
  

• 3.2 Exponential Growth and Decay
• 3.3 Separable Differential Equations
• 3.4 Slope Fields and Euler’s Method
• 3.5 Issues in Curve Sketching
• 3.6 Optimization
• 3.7 Case Study: Population Modeling

4 Modeling Accumulations

• 4.1 Introduction to the Issues

  4.1.1 The Area of a Circle
  4.1.2 What is the Area of a Circle?
  4.1.3 Another Calculation of the Area of a Circle
  4.1.4 The Method of Accumulations
  4.1.5 The Circumference of a Circle
  4.1.6 The Volume of Water in a River
  4.1.7 An Agenda for Future Work
  

• 4.2 The Definite Integral
• 4.3 Properties of the Definite Integral
• 4.4 The Fundamental Theorem of Calculus
• 4.5 Techniques of Integration
• 4.6 Trapezoid Rule
• 4.7 Areas Between Curves
• 4.8 Volumes of Solids of Revolution
• 4.9 Arc Length
• 4.10 Inverse Trigonometric Functions
• 4.11 Case Study: Flood Watch

5 Culminating Experience

• 5.1 Case Study: Sleuthing Galileo

Index Page