PSK Lesson Notes

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Elementary Algebra

Prep for Algebra

Introduction to Algebra

1.1 Algebraic Expressions
1.2 Basic Algebraic Processes

Linear Equations

2.1 Introduction to Solving
2.2 Advanced Solving
2.3 Special Cases
2.4 Percent Problems
2.5 Introduction to Formulas
2.6 Applications (of Linear Equations)
2.7 Convert Repeating Decimal to Fraction

Inequalities in One Variable

Graphing

4.1 Introduction to Graphing
4.2 Graphing by Point-Plotting
4.3 Graphing Electronically
4.4 Solving Equations Graphically

Functions

5.1 Introduction, Relations, Notation, Evaluate
5.2 Graphs & VLT
5.3 Domain & Range
5.4 Applications (of Functions)

Lines and their Graphs

Linear Systems

7.1 Solve by Graphing
7.2 Solve by Substitution
7.3 Solve by Elimination
7.4 2x2 Systems Electronically via Grapher
7.5 2x2 Systems Electronically via RREF
7.6 Applications (of Linear Systems)
7.7 3x3 and Larger Systems Algebraically
7.8 3x3 and Larger Systems Electronically

Exponents

Polynomials

Intermediate Algebra

Factoring

Polynomial Equations

11.1 Zero-Product Rule
11.2 Solve by Factoring
11.3 Solve by Graphing Electronically

Rational Expressions

12.1 Evaluate & Restrictions
12.2 Reduce Like Factors
12.3 Reduce Opposite Factors
12.4 Multiply & Divide
12.5 Add & Subtract Same Denominators
12.6 Build Equivalent Expressions
12.7 Add & Subtract Different Denominators
12.8 Add & Subtract Opposite Denominators
12.9 Compound Fractions

Rational Equations

13.1 Solving Rational Equations
13.2 Formulas Involving Rationals
13.3 Applications of Work & Motion

Radical Expressions

Non-Real Numbers

15.1 Imaginary Numbers
15.2 Complex Numbers

Radical Equations

16.1 Solving Radical Equations
16.2 Solving Radical Eqs Requiring FOIL
16.3 Formulas Containing Radicals

Quadratic Equations

17.1 Root Method
17.2 Complete the Square
17.3 Quadratic Formula
17.4 Discriminant & Type of Solutions
17.5 Applications of Solving Quadratic Equations

Geometry

Parabolas

19.1 Algebraic Approach
19.2 Graphical Approach
19.3 Graphing Electronically
19.4 Applications of Optimization
19.5 Solving Quadratic Equations Graphically

Functions & Graphs

20.1 Combining
20.2 Transformations
20.3 Evaluate 4 Ways
20.4 Composition
20.5 One-to-One & HLT
20.6 Inverses

Exponentials and Logarithms

21.1 Exponential Functions & Graphs
21.2 Logarithmic Functions & Graphs
21.3 Natural Exponentials & Logarithms
21.4 Properties of Logarithms
21.5 Exponential & Logarithmic Equations
21.6 Applications in the Sciences

Precalculus I / College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

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

Exponential and Logarithmic Functions

Systems and Matrices

Geometry Basics

6.1 Distance and Midpoint Formulae
6.2 Circles

Conic Sections

Sequences and Series

Precalculus II / Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

2.1 soh-cah-toa Definition
2.2 Calculators & intro to Inverse Trig Functions
2.3 Solving Right Triangles
2.4 Applications

Circular Functions

Graphs of Trigonometric Functions

4.1 Basic Graphs
4.2 Amplitude & Period
4.3 Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5 Combination Graphs
4.6 Harmonic Motion
4.7 Inverse Trig Functions and their Graphs

Trigonometric Identities

5.1 Using Trig Identities
5.2 Proving Trig Identities
5.3 Sum/Difference Identities
5.4 Double-Angle Identities
5.5 Half-Angle Identities
5.6 Expressions involving Identities with Inverse Trig Functions

Trigonometric Equations

6.1 Solving Trig Equations
6.2 Equations involving Identities
6.3 Equations with Multiple Angles
6.4 Equations involving Inverse Trig Functions
6.4 Solving Graphically

Oblique Triangles and the Laws

Vectors

Complex, Polar, and Parametric 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 Polar Equations and Graphs
9.5 Parametric Equations and Graphs

Calculus I

Limits and Continuity

1.0 Introduction to Calculus
1.1 Limits Graphically
1.2 Limits Numerically
1.3 Limits Algebraically
1.4 Delta-Epsilon Proofs
1.5 Properties of Limits
1.6 One-Sided Limits
1.7 Infinite Limits & Vertical Asymptotes
1.8 Limits at Infinity & Horizontal Asymptotes
1.9 Limits Electronically
1.10 Continuity

Derivatives

2.1 Definitions of Derivative
2.2 Rates of Change
2.3 Differentiability
2.4 Derivatives of Basic Functions
2.5 Higher Order Derivatives
2.6 Motion and the Derivative
2.7 Product and Quotient Rules
2.8 Derivatives of Trigonometric Functions
2.9 Chain Rule
2.10 Implicit Differentiation
2.11 Derivatives of Exponential and Logarithmic Functions
2.12 Logarithmic Differentiation
2.13 Derivatives of Inverse Functions
2.14 Derivatives of Inverse Trigonometric Functions
2.15 Electronic Differentiation & Formula Sheets
2.16 Related Rate Applications
2.17 Linear Approx & Differentials
2.18 Newton's Method

Analysis of Curves

3.1 Extreme Values & CPs
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing/Decreasing Functions
3.4 Second Derivative Test, IPs, & Concavity
3.5 Curve Sketching
3.6 Electronically Analyzing Curves
3.7 Fun with Reform Calculus
3.8 Optimization Applications

Antiderivatives

4.1 Antidifferentiaton
4.2 Summation Notation
4.3 Area Under a Curve
4.4 Riemann Sum Definition of Definite Integral
4.5 Properties of Sums and Definite Integrals
4.6 Definite Integrals Geometrically
4.7 Fundamental Theorems of Calculus

Calculus II

Transcendental Functions

Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Sequences and Series

12.0 Intro to Seqs and Series
12.1 Infinite Sequences
12.2 Infinite Series
12.3 Special Series
12.4 Divergence (n-th Term) Test
12.5 Integral Test
12.6 p-Series Test
12.7 Comparison Tests
12.8 Alternating Series
12.9 Absolute and Conditional Convergence
12.10 Ratio Test
12.11 Root Test
12.12 General Strategy

Power Series

13.1 Introduction to PS
13.2 Representing Functions
13.3 Calculus of PS
13.4 Taylor and Maclaurin
13.5 Binomial Series
13.6 Remainder of Taylor Series
13.7 Approximation using Taylor Polynomial
13.8 Applications of Taylor Series

Vectors and Geometry of Space

14.1 Vectors in 2D
14.2 3D Rectangular Coordinates
14.3 Vectors in 3D
14.4 Dot Product
14.5 Projections
14.6 Cross Product
14.7 Lines
14.8 Planes
14.9 Distances
14.10 Quadric Surfaces

Vector Functions

15.1 Introduction and Space Curves
15.2 Derivatives and Integrals
15.3 Tangent Vectors
15.4 Arc Length
15.5 Curvature
15.6 Normal and Binormal Vectors
15.7 Velocity and Acceleration
15.8 Cylindrical and Spherical Coordinates

Prof. Keely's Mathematics Lesson Notes

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