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A Modeling Approach to Calculus
D. Brian Walton
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Contents
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Front Matter
Preface
1
Overview to the Course
Learning Mathematics
Numbers and Measurements
Variables, Expressions, and Equations
Graphs and Relations between Variables
Models and Dynamics
2
Sequences as Models
Introduction to Sequences
Recursive Sequences and Projection Functions
Computing Sequence Values
Exponents, Inverses and Logarithms
Logarithms and Their Properties
Dynamic Models Using Sequences
3
Discrete Calculus
Introduction to Discrete Calculus
Increments of Sequences
Accumulation Sequences
Summation Formulas
Limits of Sequences
Calculating Sequence Limits
4
Functions as Continuous Models
An Introduction to Functions
Constructing Functions
Transformations of Functions
Functions Defined on Intervals
Limits of Functions
Continuity of Functions
Describing the Behavior of Functions
5
Continuous Accumulation and Integration
An Overview of Calculus
Accumulation of Change
Riemann Sums
Properties of Definite Integrals
Functions Defined by Accumulation
6
Modeling Rates of Change
Rate of Accumulation and the Derivative
Extreme Values
Instantaneous Rate of Change
The Derivative
The Fundamental Theorem of Calculus, Part One
7
Rules of Differentiation
Derivative Rules
Differentiation and Related Rates
The Chain Rule
The Derivative of Exponential Functions
Implicit Differentiation and Derivatives of Inverse Functions
Logarithmic Differentiation
8
Calculus for Trigonometry
The Derivatives of Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
9
Applications of Derivatives
Differentiable Functions
Consequences of the Mean Value Theorem
The Fundamental Theorem of Calculus
Introduction to Optimization
Extreme Values and Optimization
L'HÃ´pital's Rule
Antiderivatives
Integrals and the Method of Substitution
10
Other Stuff
Limits Involving Infinity
Continuous Functions
Applications Involving Densities
Functions Defined by Their Rates
Back Matter
A
Mathematics Foundations
Numbers, Sets and Arithmetic
Algebra Review
B
Trigonometry Basics
Right Triangles and Trigonometry
Measuring Arbitrary Angles
Unit Circle Trigonometry
Inverse Trigonometric Functions
Feedback
Authored in PreTeXt
A Modeling Approach to Calculus
D. Brian Walton
Department of Mathematics and Statistics
James Madison University
Harrisonburg, Virginia, USA
waltondb@jmu.edu
November 14, 2018
Preface