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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

10.1 Calculus of Parametric Curves
10.2 Calculus of Polar Curves
10.3^s Polar Equations of Conic Sections

Calculus III

Under Construction

S = contains supplemental resources

Course: Calculus II
Topic: Parametric, Polar, and Conic Curves
Subtopic: Polar Equations of Conic Sections

Overview

Start by reviewing conic sections from a pre-calculus course. See Lesson Notes - College Algebra: Introduction to Conics | Parabolas | Ellipses | Hyperbolas | General & Degenerate Conics but ignore the "text notes". Work conics that are centered at the origin as well as those translated away from the origin.

Our main goal in this lesson is to concentrate performing calculus on the conic sections. Many calculus operations are simplified by converting the equations of the conics from rectangular form to polar form. This transformation is accomplished through an ingenious formula. We can then apply calculus to the polar form to find the area inside a bounded conic, the arc length of the conic, etc. Study conics that have horizontal or vertical axes as well as those that have a rotated axis.

Objectives

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

10.3.1 Understand the rectangular equations and graphs of conic sections including those with centers translated away from the origin
10.3.2 Use the formula r = (e·d)/(1±cosθ) or r = (e·d)/(1±sinθ) to convert equations of conics from rectangular to polar form
10.3.3 Identify a conic from its polar form
10.3.4 Understand the eccentricity of the conic sections and identify a conic given its eccentricity
10.3.5 Graph a polar equation of a conic manually and electronically
10.3.6 Find the polar equation of a conic given its graph or information about its graph
10.3.7 Understand some applications of conics in polar form including Kepler's Laws

Terminology

Define: focal point, major and minor axes, directrix, pole, eccentricity, Kepler and his laws

Supplemental Resources (recommended)

Explore the connection between eccentricity and conics via the java applet at www.mathwords.com/multimedia/Conics%20defined%20by%20eccentricity.htm.

Supplemental Resources (optional)

These are both quite good reads. The first reviews concepts from Pre-Caluclus and the second covers Calculus concepts.

Lesson: Conic Sections, Dale Hoffman's Contemporary Calculus

Lesson: Properties of the Conic Sections, Dale Hoffman's Contemporary Calculus

Prof. Keely's Lesson Notes for Mathematics Online