Fundamentals
Real Numbers
Substraction isn't a fundamental operation, it's addition with a
negative a-b = a+(-b), use it your advantage.
Use prime factorization to sum fractions. Factoring each denominator
into prime factors for \frac{5}{36} + \frac{7}{120} gives 36 =
2^2.3^2
and 120 = 2^3.3.5. The product of the highest power ones, gives us
the LCD: 2^3.3^2.5 = 360.
References
Study Guide
Chapter 1 — Fundamentals
| Section | Best-matching video(s) |
|---|
| 1.1 Real Numbers | (not in this playlist — prerequisite algebra) |
| 1.2 Exponents and Radicals | (not in this playlist — prerequisite algebra) |
| 1.3 Algebraic Expressions | (not in this playlist — prerequisite algebra) |
| 1.4 Rational Expressions | (not in this playlist — prerequisite algebra) |
| 1.5 Equations | Introduction to Solving Quadratics (CA 16); The Square Root Method (CA 17); Using Factoring to Solve Quadratics (CA 18); Completing the Square Made Easy (CA 19); Proving the Quadratic Formula (CA 20); Using the Quadratic Formula (CA 21); Using Substitutions (CA 23) |
| 1.6 Complex Numbers | Introduction to Complex Solutions of Polynomials (CA 35) (complex-number basics) |
| 1.7 Modeling with Equations | Applications of Quadratic Functions (CA 26) (closest) |
| 1.8 Inequalities | Solving Quadratic Inequalities (CA 25) (closest; his inequality videos are function-based) |
| 1.9 The Coordinate Plane; Graphs of Equations; Circles | (not in this playlist — prerequisite algebra) |
| 1.10 Lines | (not in this playlist — prerequisite algebra) |
| 1.11 Solving Equations and Inequalities Graphically | Finding Intersections of Functions (CA 22) |
| 1.12 Modeling Variation | (not in this playlist) |
Chapter 2 — Functions
| Section | Best-matching video(s) |
|---|
| 2.1 Functions | Introduction to Functions (CA 2); How to Evaluate Functions (CA 3); Finding the Domain of Functions (CA 4) |
| 2.2 Graphs of Functions | Using the Vertical Line Test (CA 6); Features of Graphs, Domain, Range (CA 7); Graphs You Must Know (CA 13) |
| 2.3 Getting Information from the Graph | Even vs Odd (CA 8); Increasing vs Decreasing (CA 9); Extrema (CA 10); How to Graph Piecewise Functions (CA 12) |
| 2.4 Average Rate of Change of a Function | Average Rate of Change of a Function (CA 11) |
| 2.5 Linear Functions and Models | (not in this playlist — prerequisite algebra) |
| 2.6 Transformations of Functions | Introduction to Graph Transformations (CA 14); How to Graph with Transformations (CA 15) |
| 2.7 Combining Functions | Operations of Functions (CA 5); Composition of Functions (CA 48); Finding Domain of Composite Functions (CA 49) |
| 2.8 One-to-One Functions and Their Inverses | One to One Functions (CA 50); Finding Inverse Functions (CA 51) |
Chapter 3 — Polynomial and Rational Functions
| Section | Best-matching video(s) |
|---|
| 3.1 Quadratic Functions and Models | Graphing Quadratic Functions (CA 24); Applications of Quadratic Functions (CA 26) |
| 3.2 Polynomial Functions and Their Graphs | Introduction to Polynomial Functions (CA 27); Power Functions (CA 28); Multiplicity and End Behavior (CA 29); Creating Polynomials from Real Zeros (CA 30); How to Sketch Polynomial Functions (CA 31) |
| 3.3 Dividing Polynomials | Synthetic Division and Long Division of Polynomials (CA 32) |
| 3.4 Real Zeros of Polynomials | Descartes' Rule of Signs (CA 33); How to Use the Rational Zeros Theorem (CA 34); Finding ALL Solutions of Polynomials (CA 37) |
| 3.5 Complex Zeros and the Fundamental Theorem of Algebra | Introduction to Complex Solutions of Polynomials (CA 35); Creating Polynomials from Complex Solutions (CA 36) |
| 3.6 Rational Functions | Finding Vertical Asymptotes (CA 38); Graphing Rational Functions with Transformations (CA 39); Horizontal Asymptote (CA 40); Oblique Asymptote (CA 41); End Behavior (CA 42); Asymptotes and Holes (CA 43); Graphing Rational Functions (CA 44) |
| 3.7 Polynomial and Rational Inequalities | Inequalities with Functions – Graphically (CA 45); Inequalities with Polynomial Functions (CA 46); Inequalities with Rational Functions (CA 47) |
Chapter 4 — Exponential and Logarithmic Functions
| Section | Best-matching video(s) |
|---|
| 4.1 Exponential Functions | Graphs of Exponential Functions (CA 52); Graphing Exponential Functions with Transformations (CA 53) |
| 4.2 The Natural Exponential Function | Graphs of Exponential Functions (CA 52) (covered within; no dedicated "e" video) |
| 4.3 Logarithmic Functions | Introduction to Logarithms and Their Graphs (CA 55); Graphing Logarithms with Transformations (CA 56) |
| 4.4 Laws of Logarithms | Properties of Logarithms (CA 58); How to Expand Logarithms (CA 59); How to Combine Logarithms (CA 60); How to Change the Base of a Logarithm (CA 61) |
| 4.5 Exponential and Logarithmic Equations | Solving Exponential Equations with Common Bases (CA 54); Introduction to Solving Logarithms and Exponentials (CA 57); Solving Logarithms with Common Bases (CA 62); Log Equations with Exponentials (CA 63); Exponential Equations with Logarithms (CA 64) |
| 4.6 Modeling with Exponential Functions | Review of Compound Interest (CA 65); Exponential Growth and Decay (CA 66) |
| 4.7 Logarithmic Scales | (not in this playlist) |
Chapter 5 — Trigonometric Functions: Unit Circle Approach
| Section | Best-matching video(s) |
|---|
| 5.1 The Unit Circle | Trigonometric Functions and the Unit Circle (Trig 6); How to Use the Unit Circle in Trigonometry (Trig 7) |
| 5.2 Trigonometric Functions of Real Numbers | Basic Properties of Trigonometric Functions (Trig 8); Reciprocal Identities (Trig 9); Pythagorean Identities (Trig 10) |
| 5.3 Trigonometric Graphs | The Graphs of Sine and Cosine (Trig 11); Graphing Transformations with Sine and Cosine (Trig 12) |
| 5.4 More Trigonometric Graphs | How to Graph Tangent and Cotangent (Trig 13); Transformations with Tangent and Cotangent (Trig 14); How to Graph Cosecant and Secant (Trig 15); Phase Shifts (Trig 16) |
| 5.5 Inverse Trigonometric Functions and Their Graphs | Introduction to Inverse Trig Functions (Trig 17); How to Use Inverse Trig Functions (Trig 18); How to Find Inverse Trig Functions (Trig 19) |
| 5.6 Modeling Harmonic Motion | Simple Harmonic Motion in Trig (Trig 35) |
Chapter 6 — Trigonometric Functions: Right Triangle Approach
| Section | Best-matching video(s) |
|---|
| 6.1 Angle Measure | Introduction to Angles (Trig 1); Converting Degrees, Minutes, Seconds (Trig 2); Introduction to Radians (Trig 3); Converting Radians and Degrees (Trig 4); Area of a Sector, Angular Velocity (Trig 5) |
| 6.2 Trigonometry of Right Triangles | Introduction to Right Triangle Trigonometry (Trig 30); Finding Sides and Angles with Right Triangle Trig (Trig 31) |
| 6.3 Trigonometric Functions of Angles | Basic Properties of Trigonometric Functions (Trig 8) (with Unit Circle, Trig 6–7) |
| 6.4 Inverse Trigonometric Functions and Triangles | How to Solve Basic Inverse Trig Functions (Trig 20); An In-depth Look at Using Inverse Trig Functions (Trig 21) |
| 6.5 The Law of Sines | How to Use the Law of Sines in Trigonometry (Trig 32) |
| 6.6 The Law of Cosines | How to Use the Law of Cosines in Trigonometry (Trig 33); How to Find the Area of a Triangle with Trigonometry (Trig 34) |
Chapter 7 — Analytic Trigonometry
| Section | Best-matching video(s) |
|---|
| 7.1 Trigonometric Identities | Introduction to Using Trigonometric Identities (Trig 23); How to Prove Trigonometric Identities (Trig 24) |
| 7.2 Addition and Subtraction Formulas | Introduction to Sum and Difference Formulas (Trig 25); Using Sum and Difference Formulas (Trig 26) |
| 7.3 Double-Angle, Half-Angle, and Sum-Product Formulas | Proving the Double and Half Angle Formulas (Trig 27); How to Use the Double and Half Angle Formulas (Trig 28); Product-to-Sum and Sum-to-Product (Trig 29) |
| 7.4 Basic Trigonometric Equations | How to Solve Trigonometric Equations (Trig 22) |
| 7.5 More Trigonometric Equations | How to Solve Trigonometric Equations (Trig 22) (same video covers both) |
Chapter 8 — Polar Coordinates and Parametric Equations
| Section | Best-matching video(s) |
|---|
| 8.1 Polar Coordinates | Introduction to Polar Coordinates (Trig 36); Convert Polar → Rectangular (Trig 37); Convert Rectangular → Polar (Trig 38) |
| 8.2 Graphs of Polar Equations | Convert Rectangular → Polar Equations (Trig 39); Convert Polar → Rectangular Equations (Trig 40); Graph Basic Polar Equations (Trig 41); Graph Advanced Polar Equations with Symmetry (Trig 42) |
| 8.3 Polar Form of Complex Numbers; DeMoivre's Theorem | (not in this playlist) |
| 8.4 Plane Curves and Parametric Equations | (not in this playlist) |
Chapter 9 — Vectors in Two and Three Dimensions
(Entire chapter not in this playlist — see Professor Leonard's Calculus 3 series for vectors.)
Chapter 10 — Systems of Equations and Inequalities
(Entire chapter not in this playlist — matrices, determinants, systems are in his Algebra course.)
Chapter 11 — Conic Sections
(Entire chapter not in this playlist.)
Chapter 12 — Sequences and Series
| Section | Best-matching video(s) |
|---|
| 12.1 Sequences and Summation Notation | Introduction to Sequences (CA 67); Introduction to Series and Summation Notation (CA 68) |
| 12.2 Arithmetic Sequences | Arithmetic Sequences (CA 69); Arithmetic Series (CA 70) |
| 12.3 Geometric Sequences | Geometric Sequences (CA 71); Geometric Series (CA 72) |
| 12.4 Mathematics of Finance | Review of Compound Interest (CA 65) (closest) |
| 12.5 Mathematical Induction | Proof by Mathematical Induction (CA 73) |
| 12.6 The Binomial Theorem | (not in this playlist) |
Chapter 13 — Limits: A Preview of Calculus
(Entire chapter not in this playlist — see Professor Leonard's Calculus 1 series for limits.)