Business Calculus

Functions
Functions Fundamentals
3 Topics
Functions and Their Representations: Potato Diagrams, Tables, Graphs, and Rules
Drawing Graphs by Point-Plotting
Function Domains, Codomains, and Ranges
PRAXIS: Functions Fundamentals
Linear Functions
2 Topics
Graphing Lines
Finding Equations of Lines
Quadratic and Polynomial Functions
2 Topics
Quadratic Functions and Their Graphs
Polynomials and Degree
PRAXIS: Linear, Quadratic, and Polynomial Functions
Exponential Functions
2 Topics
Exponential Functions: What They Are, and How to Evaluate Them
Graphing (Simple) Exponential Functions
Logarithmic Functions
3 Topics
Logarithmic Functions: What They Are, and How to Evaluate Them
Change-of-Base Formula and Computing Logs with a Calculator
Graphing (Simple) Logarithmic Functions
Constructing New Functions From Old Ones
4 Topics
Basic Function Arithmetic Using Function Rules, and Numerical Substitutions
Function Arithmetic: Using Tables to Evaluate Various Combinations of Functions at a Given Value
Function Arithmetic: Using Graphs to Evaluate Various Combinations of Functions at a Given Value
Piecewise Functions
PRAXIS: Exponential Functions, Logarithms, and Building New Functions From Old
Solving Equations with Functions
3 Topics
The Guess and Check Method for Solving Equations
Solving Equations Using Graphs
Algebraically Solving Equations
PRAXIS: Solving Functional Equations
Limits and Derivatives
Limits – An Overview
Left, Right, and Two-Sided Limits
3 Topics
Finding Limits Using Graphs
Guessing Limits Using Tables
Finding Limits Of Polynomials
PRAXIS: Limits
The Derivative as a Limit of an Average Rate of Change
4 Topics
Average Rates of Change
Finding Derivatives as a Limit of an Average Rate of Change
The h-Form of the Derivative and Finding Tangent Lines
The Derivative as a Function
PRAXIS: Derivatives Basics
Basic Derivative Rules
6 Topics
Deriving Polynomials Using the Power Rule and Simple Derivative Properties
Derivatives of Exponential Functions of the Form \(c\cdot e^x\)
Product Rule
Quotient Rule
Chain Rule
Derivatives of Logarithmic Functions
Finding Derivatives Using Several Different Rules
PRAXIS: Derivative Rules
Applications of Differentiation
Extrema: Finding Minima and Maxima
3 Topics
Finding Extrema by Graphing
First Derivative Test for Finding Local Mins and Maxes
First Derivative Test and Global Mins and Maxes
PRAXIS: Extrema
Concavity, Inflection Points, and the Second Derivative Test
3 Topics
Concavity and Inflection Points With Graphs
Concavity and Inflection Points Without Graphs
Second Derivative Test for Classifying Critical Points
PRAXIS: Concavity, Inflection Points, and Second Derivative Test
Optimization Applications
PRAXIS: Optimization Applications
Integration
Antiderivatives and Basic Rules for Such
2 Topics
The Power Rule for Antiderivatives
Initial Value Problems: Solving for C
The U-Substitution Method
3 Topics
How the Method Works
Using U-Sub with Exponential Functions
Antiderivatives that Result in Logarithms
PRAXIS: Antiderivatives
Area, Continuous Accumulation, and Definite Integrals
3 Topics
What Integrals Are and the Fundamental Theorem of Calculus
Computing Integrals Involving U-Substitution
Areas Between Curves
Average Value of a Function, and Moving Averages
2 Topics
Finding a Function’s Average Value Using Integrals
Moving Averages
PRAXIS: Area, Continuous Accumulation, and Definite Integrals
Compound Interest and Continuous Compounding
2 Topics
Compound Interest – Multiplying Your Money
The Natural Exponential Function and Continuous Compounding
Future Value of Income Streams
PRAXIS: Applications of Integration – Compound Interest and Future Value of Income Streams
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Constructing New Functions From Old Ones

Business Calculus Constructing New Functions From Old Ones

In what follows…

We will be talking about how we can build new functions from ones we already know of or have in hand. This is one of the key reasons why we use functions at all. They offer us the ability to “package up” a formula, graph, table, etc. in a convenient notational package that can be used in other usually more complicated formulas. And that is exactly what we will be doing in the topics below!

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