PSK Lesson Notes

LESSON NOTES MENU

Algebra I / Elem. Algebra

The Basics

1.1 Intro to Algebra
1.2 Real Number System and Calculators

Linear Equations and Inequalities

Functions and Graphs I

3.1^s Introduction to Graphing
3.2 Solving Equations Graphically
3.3 Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1 Intercept Points
4.2^s Slopes
4.3 Equations of Lines
4.4 Graphing Linear Inequalities

Algebra II / E&I Algebra

Linear Systems

5.1 2x2 Systems of Equations I - Graphically
5.2 2x2 Systems of Equations II - Substitution Method
5.3 2x2 Systems of Equations III - Elimination Method
5.4^s 3x3 Systems of Equations
5.5 Applications
5.6 Systems of Linear Inequalities

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

10.1 Roots & Radicals
10.2 Fractional Exponents
10.3 Operations I - Simplify & Multiply
10.4 Operations II - Add, Subtract, & Divide
10.5 Operations III - Distribute, Foil, & Factor
10.6^s Complex Numbers

Nonlinear Equations and Applications

11.1 Radical Equations
11.2 Quadratic Equations I - Root Method
11.3^s Quadratic Equations II - Complete the Square
11.4^s Quadratic Equations III - Quadratic Formula
11.5^s Applications - Geometry & Pythagorean Theorem

Functions and Graphs II

12.1 Parabolas I - Algebraic Approach
12.2 Pareabolas II - Graphical Approach
12.3^s Functions III - Composite & Inverse

Exponential and Logarithmic Functions

13.1^s Exponential Functions & Graphs
13.2 Logarithmic Functions & Graphs
13.3 Exponential & Logarithmic Equations

Liberal Arts Maths

Under Construction

College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

3.1 Polynomial Functions I - Introduction, Zeros
3.2 Polynomial Functions II - Division, Remainder Thm, Factor Thm
3.3 Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule
3.4 Rational Functions & Graphs
3.5 Partial Fractions

Exponential and Logarithmic Functions

Systems and Matrices

Conic Sections

Sequences and Series

7.1^s Introduction to Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series

Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

Circular Functions

Graphs of Trigonometric Functions

4.1^s Basic Graphs
4.2^s Amplitude & Period
4.3^s Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5^s Combination Graphs & Harmonic Motion
4.6^s Inverse Trigonometric Functions

Trigonometric Identities

5.1 Proving Trig Identities
5.2 Sum/Difference Identities
5.3 Double-Angle Identities
5.4^s Half-Angle Identities
5.5 IDs Involving Inverse Trig Functions

Trigonometric Equations

Oblique Triangles and the Laws

Vectors

Complex, Parametric, and Polar Forms

9.1 Trigonometric Form of Complex Numbers
9.2 Products and Quotients in Trig Form
9.3 Powers and Roots in Trig Form
9.4 Parametric Equations and Graphs
9.5^s Polar Equations and Graphs

Calculus I

Limits and Continuity

Derivatives

2.1 Definition of Derivative
2.2 Derivatives of Basic Functions
2.3 Motion and the Derivative
2.4 Product Rule, Quotient Rules, & Deriv. of Trig Functions
2.5 Chain Rule
2.6 Implicit Differentiation
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Logarithmic Differentiation
2.9^s Derivatives of Inverse Functions
2.10^s Derivatives of Inverse Trig Functions
2.11 Related Rate Applications

Analysis of Curves

3.1 Extreme Values
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing-Decreasing Functions
3.4 Second Derivative Test & Concavity
3.5^s Curve Sketching
3.6 Fun with Reform Calculus
3.7 Optimization Applications

Antiderivatives

Calculus II

Transcendental Functions

5.0^s Review of Transcendental Functions and thier Derivatives
5.1 Integrals Resulting in Logarithmic Functions
5.2 Integrals Resulting in Inverse Trig Functions
5.3 Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Under Construction

S = contains supplemental resources

Course: Algebra II / Elementary Algebra
Topic: Exponents and Polynomials
Subtopic: Negative Exponents

Overview

Throughout the course we have been working with positive exponents, but in this lesson we formalize the processes and rules. We also add to our repertoire negative exponents (like 2^-4 and x^-3) and powers of zero (like 3⁰ and -y⁰). Negative powers mean to take a reciprocal (flip), so for example: x^-3 means 1/x³, (2x)^-4=1/(2x)⁴=1/(16x⁴), 2x^-4=2/x⁴, and (2/x)^-4=(x/2)⁴=x⁴/16.

Caution: Notice how important the parentheses are? They indicate exactly what is to the power and what isn't. Careful!

Scientific notation is a specific way of writing numbers and involves positive/negative powers. It allows us to easily deal with very large (and very small) numbers without having to write out and keep track of dozens of zeros. Scientific notation opens up a world of applications especially in the physical sciences!

Objectives

By the end of this topic you should know and be prepared to be tested on:

6.1.1 Understand the effect of a zero or negative power on a number or variable
6.1.2 Simplify expressions containing integer exponents (positive, negative, and zero)
6.1.3 Use the rules of exponents (product rule, power rule, and quotient rule) accurately
6.1.4 Algebraically convert numbers in decimal form into scientific notation and numbers in scientific notation into decimal form
6.1.5 Perform operations with numbers in scientific notation both algebraically and electronically

Terminology

Define: zero power, negative power, product rule of exponents, power rules of exponents, quotient rule of exponents (memorize all three rules!), decimal form (of a number)r, scientific notation

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 5.7

Review section 5.2 which covers the basic rules of positive exponents and section 5.5 which covers zero exponents before covering negative exponents in section 5.7.
The many rules of exponents displayed in purple boxes throughout 5.7 are worth memorizing!
This section includes scientific notation which is important to be able to compute manually and electronically.

Prof. Keely's Lesson Notes for Mathematics Online