PSK Lesson Notes

LESSON NOTES MENU

Algebra I / Elem. Algebra

Introduction to Algebra

1.1^s Basic Algebraic Processes
1.2^s 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^s Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1^s Intercept Points
4.2^s Slopes
4.3 Equations of Lines I - Slope-Intercept Formula
4.4 Equations of Lines II - Point-Slope Formula
4.5 Graphing Linear Inequalities

Linear Systems

5.1^s 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 Applications
5.5 3x3 Systems - Algebraically
5.6^s 3x3 Systems - Electronically

Algebra II / E&I Algebra

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

8.1 Operations I - Evaluate & Simplify
8.2 Operations II - Multiply & Divide
8.3 Operations III - Add & Subtract (same denom)
8.4^s Operations IV - Add & Subtract (different denom)
8.5^s Compound Fractions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

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

Nonlinear Equations and Applications

11.1^s 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^s Parabolas 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 Properties of Logarithms
13.4 Exponential & Logarithmic Equations
13.5^s Applications - Exponentials & Logarithms

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^s Circles

Conic Sections

Sequences and Series

8.1^s Introduction to Sequences and Series
8.2 Arithmetic Sequences and Series
8.3 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 Combination Graphs
4.6^s Harmonic Motion
4.7^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^s 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 Electronic Differentiation
2.10^s Derivatives of Inverse Functions
2.11^s Derivatives of Inverse Trig Functions
2.12 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^s Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

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

Calculus III

Sequences and Infinite Series

11.1^s Sequences
11.2^s Infinite Series
11.3 Special Series
11.4 n-th Term Test
11.5 Integral and p-Seies Tests
11.6^s Ratio and Root Tests
11.7 Comparison Tests
11.8 Alternating Series
11.9 Absolute and Conditional Convergence

Power Series

12.1^s Taylor Polynomials and Approximations
12.2^s Representing Functions as Power Series
12.3 Taylor, Maclaurin, and Binomial Series
12.4^s Calculus of Power Series

Vectors and Geometry of Space

13.1^s Vectors in 2D
13.2^s Rectangular Coordinates and Vectors in 3D
13.3^s Dot Product
13.4^s Cross Product
13.5 Lines and Curves
13.6 Planes
13.7^s Quadric Surfaces

Vector-Valued Functions

14.1 Intro to VV Functions
14.2 Calculus of VV Functions
14.3 Velocity and Acceleration
14.4 Arc Length
14.5 Curvature
14.6 Normal and Binormal Vectors
14.7 Cylindrical and Spherical Coordinates

Calculus IV

15.Functions of Several Variables
16.Multiple Integration
17.Vector Analysis

S = contains supplemental resources

Course: Calculus I
Topic: Limits and Continuity
Subtopic: Limits Graphically

Overview

The first big topic in calculus is limits. As x gets closer and closer to a number, f(x) gets closer and closer (approaches) a value. This is the concept of a limit. The answer from the limit is the value that f(x) is approaching.

In this lesson we analyze limits graphically. We'll observe the graph of f(x) and see what value the outputs are approaching as the input x gets closer and closer to a given number in an intuitive manner.

Objectives

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

1.1.1 Write limits using correct mathematical notation
1.1.2 Graphically determine the limit of a function when the limit is a real number
1.1.3 Graphically determine the limit of a function when the limit is a infinite
1.1.4 Graphically determine that a limit of a function does not exist
1.1.5 Understand the difference between a function being undefined at an x-value vs. the limit not existing at an x-value (e.g. holes vs. jumps)
1.1.6 Graphically determine a one-sided limit (limit from the left or a limit from the right)

Terminology

Define: limit, approach (in terms of a limit), D.N.E. = does not exist, one-sided limit

Text Notes

Chapter 1 is all review from a precalculus course. Skim and review if needed. If this material is not familiar you should retake College Algebra (Clark’s math 111) and/or College Trigonometry (Clark’s math 103) before taking this Calculus I course. Being an early transcendentals course, you must be confident with functions and graphs including exponential, logarithmic, and trigonometric starting on day one.
Chapter 2 begins with an overview of calculus and its applications. It serves as an important foundation for the entire course, but is a bit overwhelming for our first section of material. Many of the concepts are covered in more depth as we progress through chapter 2.

Supplemental Resources (recommended)

Watch MIT Prof. Gilbert Strang's The Big Picture of Calculus.

Why Do We Study Calculus? by Prof. Eric Schechter of Vanderbilt University is a worthy read.

Supplemental Resources (optional)

Video: Calculating Limits Intuitively, Selwyn Hollis's Video Calculus

Lesson: Slopes and Velocities, Dale Hoffman's Contemporary Calculus overviews several applications of calculus for motivational purposes. You are NOT expected to know solve these problems ... yet!

Prof. Keely's Lesson Notes for Mathematics Online