Mathematical modeling and applied calculus / Joel Kilty, Alex McAllister.

Format:

Book

Author/Creator:

Kilty, Joel, author.

McAllister, Alex M., author.

Language:

English

Subjects (All):

Mathematical models.

Physical Description:

1 online resource (xiv, 717 pages) : illustrations

Edition:

1st ed.

Place of Publication:

Oxford, England : Oxford University Press, [2018]

Summary:

Mathematical Modeling and Applied Calculus is a modern take on modeling and calculus aimed at students who need some experience with these ideas.

Contents:

Cover

Mathematical Modeling and Applied Calculus

Copyright

Dedication

Contents

Preface

Overview of the Book

Pedagogical Features

Course Designs

Acknowledgments

Chapter 1. Functions for Modeling Data

1.1 Functions

Tabular Functions and Nonfunctions

Graphical Functions and Nonfunctions

Analytic Presentations of Functions

Piecewise Functions

Exercises

1.2 Multivariable Functions

1.3 Linear Functions

Parameters of Linear Functions

Monotonicity

1.4 Exponential Functions

Recognizing Exponential Functions Graphically

Parameters of Exponential Functions

Concavity

Algebra of Exponents

1.5 Inverse Functions

Tabular Inverses

Graphical Inverses

Existence of Inverses

Monotonicity and Inverses

Finding Inverses Algebraically

1.6 Logarithmic Functions

Logarithms as Parametrized Families of Functions

Algebra of Logarithms

Logarithms and Exponential Models

Semi-log Plots and Log-Log Plots

1.7 Trigonometric Functions

Measuring Angles

Right Triangle Definitions of Trigonometric Functions

The Unit Circle Definitions of Trigonometric Functions

Graphs of Trigonometric Functions

Trigonometric Identities

Chapter 2. Mathematical Modeling

2.1 Modeling with Linear Functions

Numerically Identifying Linear Data Sets

Conjecturing Linear Models

Best Possible Linear Models

2.2 Modeling with Exponential Functions

Numerically Identifying Exponential Data Sets

Conjecturing Exponential Models

Best Possible Exponential Models

2.3 Modeling with Power Functions

Graphically Identifying Power Functional Data Sets

Numerically Identifying Power Functional Data Sets

Conjecturing Power Function Models.

Best Possible Power Function Models

Logarithmic Scale Plots

2.4 Modeling with Sine Functions

The Sine Function

Parameters of Sine Functions

Conjecturing Sine Models

Best Possible Sine Models

2.5 Modeling with Sigmoidal Functions

Parameters of Sigmoidal Functions

Conjecturing Sigmoidal Models

Best Possible Sigmoidal Models

2.6 Single-Variable Modeling

Graphically Identifying Reasonable Models

Context and Choosing Models

Refining Models Using More Data

A Limitation of Mathematical Models

2.7 Dimensional Analysis

Fundamental and Derived Dimensions

Arithmetic with Dimensions and Units

Solving for Unknown Dimensions

Generalized Products

Dimensional Analysis

Chapter 3. The Method of Least Squares

3.1 Vectors and Vector Operations

Three Vector Operations

A Geometric Interpretation of Scalar Multiplication

A Geometric Interpretation of Vector Addition

An Introduction to Vector Fields

3.2 Linear Combinations of Vectors

Finding Desired Linear Combinations

Vector Equations as Matrix Equations

Matrix-Vector Multiplication

Solving Matrix Equations

3.3 Existence of Linear Combinations

Linear Combinations of Two Vectors

Linear Combinations of Three or More Vectors

Linear Combinations and Data Sets

3.4 Vector Projection

The Dot Product

Geometry of the Dot Product

Residual Vectors

Vector Projection

Understanding Vector Projection

3.5 The Method of Least Squares

Applying the Method of Least Squares

The Residual Vector of Minimal Length

Understanding the Method of Least Squares

Chapter 4. Derivatives

4.1 Rates of Change

Average Rate of Change

Instantaneous Rate of Change.

Tangent Line Question and Fermat's Solution

4.2 The Derivative as a Function

Existence of Derivatives

Dimensions and Derivatives

Higher-Order Derivatives

Monotonicity of Functions

4.3 Derivatives of Modeling Functions

Derivatives of Linear Functions

The Power Rule

Di↵erentiation and Basic Arithmetic

Di↵erentiating Other Modeling Functions

Evidence for Di↵erentiation Rules

Tangent Lines and Linear Approximations

4.4 Product and Quotient Rules

The Product Rule

The Quotient Rule

More Trigonometric Derivatives

Differentiating with Multiple Rules

Tabular Functions

4.5 The Chain Rule

Composition of Functions

The Chain Rule

Di↵erentiating with Multiple Rules

Revisiting Extended Di↵erentiation Rules

Evidence for the Chain Rule

4.6 Partial Derivatives

Approximating Partial Derivatives

Differentiation Rules and Partial Derivatives

Higher-Order Partial Derivatives

Linear Approximation

4.7 Limits and the Derivative

Continuous Functions

Evaluating Limits at Discontinuities

Reprise: The Definition of the Derivative

Chapter 5. Optimization

5.1 Global Extreme Values

Extreme Value Theorem

Critical Numbers

Determining Critical Numbers from the Derivative

Locating Global Extreme Values

5.2 Local Extreme Values

Critical Numbers and Monotonicity

First Derivative Test

Revisiting Global Extreme Values

5.3 Concavity and Extreme Values

Concavity and Points of Inflection

Determining Concavity Graphically

Determining Concavity from the Second Derivative

Second Derivative Test

5.4 Newton's Method and Optimization

Applying Newton's Method.

Starting Values and Ending Criteria

Determining Points of Intersection

Optimization Using Newton's Method

Understanding Newton's Method

Newton's Method and Dimensions

5.5 Multivariable Optimization

Contour Plots and Extreme Values

Critical Points

Multivariable Second Derivative Test

Global Extreme Values

5.6 Constrained Optimization

The Gradient

Properties of the Gradient

Constrained Optimization

Understanding the Lagrange Multiplier

Method of Lagrange Multipliers

Understanding the Method of Lagrange Multipliers

Chapter 6. Accumulation and Integration

6.1 Accumulation

Left and Right Approximations

Midpoint Rule

Obtaining More Accurate Approximations

Riemann Sums

6.2 The Definite Integral

Net Accumulation and Geometry

Dimensions and Units of Definite Integrals

Properties of the Definite Integral

Understanding the Algebraic Properties of Integrals

6.3 First Fundamental Theorem

The Net Accumulation Function

The First Fundamental Theorem of Calculus

Antiderivatives of Modeling Functions

6.4 Second Fundamental Theorem

Using the Second Fundamental Theorem

Area Between Curves

Understanding the Fundamental Theorems

A Partial Proof of the Fundamental Theorems

6.5 The Method of Substitution

Antiderivatives via Substitution

Substitution with Intermediate Algebra

The Method of Substitution and Definite Integrals

Understanding Di↵erentials

6.6 Integration by Parts

Method of Integration by Parts

Choosing f(x) and g'(x)

Integration by Parts and Definite Integrals

Understanding Integration by Parts

Appendix A. Answer to Questions

Section 1.1 Questions

Section 1.2 Questions.

Section 1.3 Questions

Section 1.4 Questions

Section 1.5 Questions

Section 1.6 Questions

Section 1.7 Questions

Section 2.1 Questions

Section 2.2 Questions

Section 2.3 Questions

Section 2.4 Questions

Section 2.5 Questions

Section 2.6 Questions

Section 2.7 Questions

Section 3.1 Questions

Section 3.2 Questions

Section 3.3 Questions

Section 3.4 Questions

Section 3.5 Questions

Section 4.1 Questions

Section 4.2 Questions

Section 4.3 Questions

Section 4.4 Questions

Section 4.5 Questions

Section 4.6 Questions

Section 4.7 Questions

Section 5.1 Questions

Section 5.2 Questions

Section 5.3 Questions

Section 5.4 Questions

Section 5.5 Questions

Section 5.6 Questions

Section 6.1 Questions

Section 6.2 Questions

Section 6.3 Questions

Section 6.4 Questions

Section 6.5 Questions

Section 6.6 Questions

Appendix B. Answers to Odd-Numbered Exercises

Section 1.1 Exercises

Section 1.2 Exercises

Section 1.3 Exercises

Section 1.4 Exercises

Section 1.5 Exercises

Section 1.6 Exercises

Section 1.7 Exercises

Section 2.1 Exercises

Section 2.2 Exercises

Section 2.3 Exercises

Section 2.4 Exercises

Section 2.5 Exercises

Section 2.6 Exercises

Section 2.7 Exercises

Section 3.1 Exercises

Section 3.2 Exercises

Section 3.3 Exercises

Section 3.4 Exercises

Section 3.5 Exercises

Section 4.1 Exercises

Section 4.2 Exercises

Section 4.3 Exercises

Section 4.4 Exercises

Section 4.6 Exercises

Section 4.7 Exercises

Section 5.1 Exercises

Section 5.2 Exercises

Section 5.3 Exercises

Section 5.4 Exercises

Section 5.5 Exercises

Section 5.6 Exercises

Section 6.1 Exercises

Section 6.2 Exercises

Section 6.3 Exercises

Section 6.4 Exercises

Section 6.5 Exercises

Section 6.6 Exercises.

Appendix C. Getting Started with RStudio.

Notes:

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

9780192558138

0192558137

OCLC:

1119640075