Contents

• Models, Data, and Curve Fitting
• Estimation and Limits
• Sequences and Discrete Dynamical Systems
• Derivatives and Differention
• Graphing
• Optimization
• Integrals and Integration
• Differential Equations and Continuous Models
• Series
• Numerical Methods

Models, Data, and Curve Fitting -- A Guide to this Chapter.

• The Mean and the Median
• Linear Models
• Linear Regression
• Quadratic Models
• Exponential Models
• Logistic Models
• Periodic Models
• Contents

• Introduction -- When is it Safe to Drink the Water?
• A Descriptive Look
• Estimation and Limits -- Area
• Estimation and Limits -- Tangents
• A More Precise Look
• Estimation and Limits -- A Graphic Approach
• Limits as x --> Infinity
• One-sided Limits
• Limits and Errors -- Multiplication by a Constant
• Limits and Errors -- Addition
• Limits and Errors -- Multiplication
• Limits and Errors -- Division
• Continuity
• Limits and Errors -- Composition
• Some Limits Involving Sines and Cosines
• Contents

• Quickstart
• Population Models -- Exponential Models
  • Exponential Models with Immigration -- Linear Dynamical Systems
  • Discrete Logistic Models
  • Pack-Hunters
• Discrete Dynamical Systems and Difference Equations
  • A Case Study -- Newton's Model of Cooling
• Equilibrium Points
• Cobweb Diagrams
• Longterm Behavior
  • Limits
  • Cycles
  • Chaos
• Classifying Equilibrium Points
• Classifying Cycles
• Period-Doubling and the Road to Chaos
• Contents

• Introduction -- Curved Mirrors
• Comparing a Function with its Derivative
• Computing the Derivative as a Limit
• Visualizing the Derivative as a Limit
• Sometimes There Isn't a Tangent
• Interpretation
• Powers
• Linearity of the Derivative
• Polynomials
• Derivatives of the Trigonometric Functions
• Product Rule
• Quotient Rule
• Chain Rule
• Inverse Function Rule
• Derivatives of the Inverse Trigonometric Functions
• The Exponential Function
• Logarithms
• Implicit Differentiation
• Related Rates
• Mean Value Theorem
• Higher Order Derivatives
• Partial Derivatives
• Directional Derivatives
• The Gradient
• Contents

• Interpreting the First Derivative -- Increasing and Decreasing, Local and Global Extremums
• Interpreting the Second Derivative -- Concavity and Points of Inflection
• Symmetry
• Cartesian Coordinates
• Polar Coordinates
• Parametric Equations
• Implicit Functions
• Contents

• Optimization on a Closed and Bounded Interval -- The Extreme Value Theorem
• Optimization on a Closed and Bounded Interval -- A Recipe
• Optimization on a Close and Bounded Interval -- Examples
• Optimization on Other Domains -- Techniques
• Optimization on Other Domains -- Examples
• Optimization -- Using A Computer or Graphing Calculator
• Optimizing a Function of More than One Variable -- Using Partial Derivatives
• Optimizing a Function of More Than One Variable -- The Method of Steepest Descent
• Optimizing a Function of More Than One Variable -- Using a Computer or Graphing Calculator
• Optimizing a Function of More Than One Variable -- Examples
• Contents

• Area
• Summation or "Sigma" Notation
• From Velocity to Displacement
• Riemann Sums
• The Definite Integral -- Definition
• The Definite Integral -- Exotic Multiplication
• Accumulation Functions
• The Fundamental Theorem of Calculus -- the Ho-Hum Theorem of Calculus
• Antidifferentiation -- Basic Formulas
• Linearity of the Intgeral
• The Chain Rule Revisited -- Change of Variables
• Integration by Parts
• Trigonometric Substitution
• Contents

• The Language of Differential Equations
• Slope Fields
• Euler's Method in Pictures
• Euler's Method in Formulas
• Guess-and-Check
• General Solutions and Particular Solutions -- Initial Value Problems
• Constant Relative Growth or Decay
• Separable Differential Equations
• First Order Linear Differential Equations
• Spring-and-Mass Systems
• One-Dimensional Dynamical Systems -- A Graphical Approach
• Two Dimensional Dynamical Systems -- A Graphical Approach
• Classifying Equilibrium Points in One Dimension
• Classifying Equilibrium Points in Two Dimensions
• Contents

• Geometric Series
• Convergence
  • The One-Way Convergence Test
  • The Integral Test
  • Comparison Theorems
  • The Ratio Test
• Power Series
• Power Series Solutions of Initial Value Problems
• Taylor Series
• Taylor's Theorem
• Fourier Series
• Contents

• Solving Equations
  • The Bisection Method
  • Newton's Method
  • The Secant Method
• Estimating the First Derivative
• Estimating the Second Derivative
• Estimating the Integral
  • The Right Hand Rule
  • The Left Hand Rule
  • The Midpoint Rule
  • The Trapezoid Rule
  • Simpson's Rule
• Initial Value Problems
  • Euler's Method for One Variable
  • Euler's Method for Several Variables
  • The Modified Euler's Method
  • Fourth Order Runge-Kutta
• Round-Off Error
• Propogation of Errors
• Taylor Series
• Interpolation
  • Linear Interpolation
  • Splines
• Contents