Mathematical Modeling
1.2What is mathematical reasoning?
This section aims to introduce the idea of mathematical reasoning and give an
example of how it is used.
1.3Logical Deduction
This section analyzes the previous example in detail to develop a three phase
deductive process to develop a mathematical model.
1.5Is this actually math?
This section aims to show how mathematical reasoning is different than ‘typical
reasoning’, as well as showing how what we are doing is mathematical.
1.6Math as a Language
This section contains important points about the analogy of mathematics as a
language.
1.9Embrace Laziness!
This section aims to show the virtues, and techniques, in generalizing numeric models
into ‘generalized’ models.
Variables, Functions, Graphing, and Universal Properties
2Goals of this Section
This section is on functions, their roles, their graphs, and we introduce the Library of
Functions
2.2Relationship vs. Equations
In this section we discuss a very subtle but profoundly important difference between
a relationship between information, and an equation with information.
2.5Functions Require Context
In this section we demonstrate that a relation requires context to be considered a
function.
2.10Set Notation
In this section we cover how to actual write sets and specifically domains, codomains,
and ranges.
2.16Function Composition
We cover the idea of function composition and it’s effects on domains and
ranges.
3Graphing Introduction
This section introduces graphing and gives an example of how we intuitively use
it.
3.3Graphing To Relate Variables
This section describes how we will use graphing in this course; as a tool to visually
depict a relation between variables.
3.4Graphs Aren’t Precise
This section describes how accuracy and precision are different things, and how that
relates to graphs.
5Universal Properties
This section introduces the idea of studying universal properties to avoid memorizing
vast amounts of information.
5.2Geometric Vs Analytic Viewpoints
We discuss what Geometric and Analytic views of mathematics are and the different roles they play in learning and practicing
mathematics.
5.3Geometric Perspective
We discuss the geometric perspective and what its role is in learning and practicing mathematics.
5.4Analytic Viewpoint
We discuss the analytic view of mathematics such as when and where it is most useful or appropriate.
5.5Even Functions
This section describes the geometry and useful symmetry of even functions, as well as
how to test for them analytically.
5.6Odd Functions
This section describes the geometry and useful symmetry of odd functions, as well as how to test for them analytically.
5.8Rigid Translations: Geometrics
This section describes the geometric perspective of Rigid Translations.
5.10Rigid Translations: Analytics
This section describes the analytic perspective of what makes a Rigid Translation.
5.12Transforms: Geometric
This section describes the geometric interpretation of what makes a transformation
5.13Transforms: Analytic
This section describes the analytic interpretation of what makes a transformation and how to use the function notation to perform
(or read) a transformation quickly and easily.
5.15Transform And Translates
This covers doing transformations and translations at the same time. In particular we discuss how to determine what order to do
the translations/transformations in.
5.17Points of Interest on Graphs - Zeros
This section describes types of points of interest (PoI) in general and covers zeros of
functions as one such type.
5.18Sign Charts
This section describes one of the most useful tools in mathematics - especially in
calculus - the sign chart!
5.19Points of Interest on Graphs - Extrema
This section describes extrema of a function as points of interest (PoI) on a
graph.
5.22Points of Interest on Graphs - Discontinuities
This section describes discontinuities of a function as points of interest (PoI) on a
graph.
5.24Algebra with Functions
This section describes how to perform the familiar operations from algebra
(eg add, subtract, multiply, and divide) on functions instead of numbers or
variables.
5.26Equals Signs are Magic!
This section describes the very special and often overlooked virtues of the ‘equals
sign’. It also includes when and why you should “set something equal to zero” which
is often overused or used incorrectly.
5.27One and Zero; the Most Useful of Numbers
This section describes the very special and often overlooked virtue of the numbers
Zero and One.
5.31Inverse Function - Analytic View
This section introduces the analytic viewpoint of invertability, as well as one-to-one functions.
Exploration of Functions
6Polynomial Functions
This section is an exploration of polynomial functions, their uses and their
mechanics.
6.2What is (and isn’t) a Polynomial?
We know an awful lot about polynomials, but it relies on the very specific structure
of a polynomial, and thus it is paramount that one can correctly recognize what is,
and isn’t, a polynomial to use these tools.
6.4Fundamental Theorem of Algebra
This section covers one of the most important results in the last couple centuries in
algebra; the so-called “Fundamental Theorem of Algebra.”
6.6An Interjection into Polynomial History!
This section is a quick foray into math history, and the history of polynomials!
6.7Exponents and Extrema: An Example
This section contains a demonstration of how odd versus even powers can effect
extrema.
6.9Exponents and Extrema 2: Local Extrema
This section contains information on how exponents effect local extrema
6.13Factoring: Common Factors
We discuss step 0 of factoring, extracting any common factors to simplify the process going forward.
6.14Factoring: Round One!
First dive into factoring polynomials. This section covers factoring quadratics with
leading coefficient of by factoring the coefficients.
6.24Polynomial Long Division
In this section we explore how to factor a polynomial out of another polynomial using polynomial long division
6.25Polynomial Synthetic Division
Factor one polynomial by another polynomial using polynomial synthetic division
7Radical Functions
This section is an exploration of radical functions, their uses and their mechanics.
7.3Simplifying Radicals - Part 1
This section discusses how to simplify numeric and odd root-value radicals.
7.5Simplifying Radicals - Part 2
This section discusses some of the complications that arise when simplifying radicals.
7.7Square Root: the Inverse Function
This section views the square root function as an inverse function of a monomial. This is used to explain the dreaded symbol and
when to use (and not use) absolute values.
7.8Solving Unsimplified Radicals
This section shows techniques to solve an equality that has a radical that can’t be simplified into a non radical form. This has
potential drawbacks which is also covered in this section.
9Exponential Functions: Goals
This section is an exploration of exponential functions, their uses and their
mechanics.
9.2A Review of Exponential Functions
This section reviews the basics of exponential functions and how to compute numeric
exponentials.
9.4Properties of Exponentials
This section gives the properties of exponential expressions. Most of these should be
familiar, although we go into slightly more details as to how and why these properties
hold in some cases.
9.7Properties of the Exponential Function
This section gives the properties of exponential functions. There is a subtlety
between the function and the expression form which will be explored, as well as
common errors made with exponential functions.
9.9Exponential Growth and Decay
This section discusses the two main modeling uses of exponentials; exponential
growth, and exponential decay.
10Logarithmic Functions
This section is an exploration of logarithmic functions, their uses and their
mechanics.
10.2Introduction and Notation of Logarithms
This section is a quick introduction to logarithms and notation (and ways to avoid
the notation).
10.4Logs as Inverse Functions
We discuss logarithms as inverse functions, and how to understand logs from this
perspective.
10.5Properties Of Logs
This is one of the most vital sections for logarithms. We cover primary and secondary
properties of logs, which are pivotal in future math classes as these properties are
often exploited in otherwise difficult mechanical situations.
10.7Common Mistakes Of Logs
This is one of the most vital sections for logarithms. We cover primary and secondary
properties of logs, which are pivotal in future math classes as these properties are
often exploited in otherwise difficult mechanical situations.
10.8Change of Base formula
This is one of the most vital sections for logarithms. We cover primary and secondary
properties of logs, which are pivotal in future math classes as these properties are
often exploited in otherwise difficult mechanical situations.
10.10Examples of Logs
This is a demonstration of several examples of using log rules to handle logs
mechanically.
11Piecewise Functions
This section is an exploration of the piece-wise function; specifically how and why
they are used and their mechanics.
11.1Piecewise Functions: The Geometric View
This section discusses the geometric view of piecewise functions.
11.2Piecewise Functions: The Analytic View
This section discusses the analytic view of piecewise functions.
11.4Domains of Piecewise Functions
We discuss the domains of piecewise functions - which must be explicitly given.
11.6Piecewise Functions: Computation
This section discusses how to compute values using a piecewise function
12Absolute Value Functions
This section is an exploration of the absolute value function; specifically how and
why they are used and their mechanics.
12.1Absolute Value: Geometric View
This discusses Absolute Value as a geometric idea, in terms of lengths and distances.
12.3Absolute Value: Analytic View
This discusses the absolute value analytically, ie how to manipulate absolute values algebraically.
12.5Computing Absolute Values
We discuss how to compute absolute values, as well as some techniques to speed up the process.
12.7Graphing Absolute Values
We discuss how to graph absolute values, as well as some key features of the graph.
13Rational Functions
This section is an exploration of rational functions; specifically those functions that
are made by taking a ratio (ie fraction) of polynomials.
13.3Domain of rational functions
We discuss one of the most important aspects of rational functions; the domain restrictions.
13.5Vertical Asymptotes
We discuss the circumstances that generate vertical asymptotes in rational functions.
13.6Holes in Domains of Rational Functions
We discuss the circumstances that generate holes in the domain of rational functions rather than vertical asymptotes.