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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
Derivatives and Integrals
Antiderivatives
Differentiable Functions
Definite Integrals and Antiderivatives
L'Hôpital's Rule
9
Calculus for Trigonometry
The Derivatives of Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
10
Other Stuff
Introduction to Optimization
Extreme Values and Optimization
Integrals and the Method of Substitution
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
Chapter
2
Sequences as Models
¶
2.1
Introduction to Sequences
PDF version of Section 2.1
2.2
Recursive Sequences and Projection Functions
PDF version of Section 2.2
2.3
Computing Sequence Values
PDF version of Section 2.3
2.4
Exponents, Inverses and Logarithms
PDF version of Section 2.4
2.5
Logarithms and Their Properties
PDF version of Section 2.5
2.6
Dynamic Models Using Sequences
PDF version of Section 2.6