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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
Functions Defined on Sets
Describing the Behavior of Functions
Transformations of Functions
5
Continuous Accumulation and Integration
An Overview of Calculus
Accumulation of Change
Riemann Sums
Properties of Definite Integrals
6
Functions Defined by Accumulation
Functions Defined by Accumulation
Rate of Accumulation and the Derivative
7
Limits and Continuity of Functions
Continuity of Functions
Limits Involving Infinity
Limit Rules
Continuous Functions
8
Modeling Rates of Change by Differentiation
Introduction to Optimization
Functions Defined by Their Rates
Rates of Change
The Derivative
Differentiation
Derivatives Take Practice
The Chain Rule
The Derivative of Exponential Functions
Implicit Differentiation and Derivatives of Inverse Functions
Applications Involving Densities
9
Calculus for Trigonometry
The Derivatives of Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
10
Applications of Derivatives
Differentiable Functions
Consequences of the Mean Value Theorem
Extreme Values and Optimization
L'Hôpital's Rule
Antiderivatives
The Fundamental Theorem of Calculus
Integrals and the Method of Substitution
Applications Involving Densities
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
Reference material
Notation
Feedback
Authored in PreTeXt
Chapter
10
Applications of Derivatives
10.1
Differentiable Functions
PDF version of Section 10.1
10.2
Consequences of the Mean Value Theorem
PDF version of Section 10.2
10.3
Extreme Values and Optimization
PDF version of Section 10.3
10.4
L'Hôpital's Rule
PDF version of Section 10.4
10.5
Antiderivatives
PDF version of Section 10.5
10.6
The Fundamental Theorem of Calculus
PDF version of Section 10.6
10.7
Integrals and the Method of Substitution
PDF version of Section 10.7
10.8
Applications Involving Densities
PDF version of Section 10.8