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Mathematics (All Of It)

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Syllabus

Introduction to Mathematics.
Addition and Subtraction of Small Numbers.
Multiplication and Division of Small Numbers.
Understanding Fractions, Improper Fractions, and Mixed Numbers.
Large Whole Numbers: Place Values and Estimating.
Decimals: Notation and Operations.
Working With Percentages.
Converting Between Fractions, Decimals, and Percentages.
Addition and Subtraction of Large Numbers.
The Distributive Property for Arithmetic.
Multiplication of Large Numbers.
Division of Large Numbers: Long Division.
Negative Numbers.
Understanding Exponents and Their Operations.
Order of Arithmetic Operations: PEMDAS.
Divisibility, Prime Numbers, and Prime Factorization.
Least Common Multiple (LCM).
Greatest Common Factor (GCF).
Addition and Subtraction of Fractions.
Multiplication and Division of Fractions.
Analyzing Sets of Data: Range, Mean, Median, and Mode.
Introduction to Algebra: Using Variables.
Basic Number Properties for Algebra.
Algebraic Equations and Their Solutions.
Algebraic Equations With Variables on Both Sides.
Algebraic Word Problems.
Solving Algebraic Inequalities.
Square Roots, Cube Roots, and Other Roots.
Simplifying Expressions With Roots and Exponents.
Solving Algebraic Equations With Roots and Exponents.
Introduction to Polynomials.
Adding and Subtracting Polynomials.
Multiplying Binomials by the FOIL Method.
Solving Quadratics by Factoring.
Solving Quadratics by Completing the Square.
Solving Quadratics by Using the Quadratic Formula.
Solving Higher Degree Polynomials by Synthetic Division and the Rational Roots Test.
Manipulating Rational Expressions: Simplification and Operations.
Graphing in Algebra: Ordered Pairs and the Coordinate Plane.
Graphing Lines in Algebra: Understanding Slopes and Y-Intercepts.
Graphing Lines in Slope-Intercept Form (y = mx + b).
Graphing Lines in Standard Form (ax + by = c).
Graphing Parallel and Perpendicular Lines.
Solving Systems of Two Equations and Two Unknowns: Graphing, Substitution, and Elimination.
Absolute Values: Defining, Calculating, and Graphing.
What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational.
Introduction to Geometry: Ancient Greece and the Pythagoreans.
Basic Euclidean Geometry: Points, Lines, and Planes.
Types of Angles and Angle Relationships.
Types of Triangles in Euclidean Geometry.
Proving Triangle Congruence and Similarity.
Special Lines in Triangles: Bisectors, Medians, and Altitudes.
The Triangle Midsegment Theorem.
The Pythagorean Theorem.
Types of Quadrilaterals and Other Polygons.
Calculating the Perimeter of Polygons.
Circles: Radius, Diameter, Chords, Circumference, and Sectors.
Calculating the Area of Shapes.
Proving the Pythagorean Theorem.
Three-Dimensional Shapes Part 1: Types, Calculating Surface Area.
Three-Dimensional Shapes Part 2: Calculating Volume.
Back to Algebra: What are Functions?.
Manipulating Functions Algebraically and Evaluating Composite Functions.
Graphing Algebraic Functions: Domain and Range, Maxima and Minima.
Transforming Algebraic Functions: Shifting, Stretching, and Reflecting.
Continuous, Discontinuous, and Piecewise Functions.
Inverse Functions.
The Distance Formula: Finding the Distance Between Two Points.
Graphing Conic Sections Part 1: Circles.
Graphing Conic Sections Part 2: Ellipses.
Graphing Conic Sections Part 3: Parabolas in Standard Form.
Graphing Conic Sections Part 4: Hyperbolas.
Graphing Higher-Degree Polynomials: The Leading Coefficient Test and Finding Zeros.
Graphing Rational Functions and Their Asymptotes.
Solving and Graphing Polynomial and Rational Inequalities.
Evaluating and Graphing Exponential Functions.
Logarithms Part 1: Evaluation of Logs and Graphing Logarithmic Functions.
Logarithms Part 2: Base Ten Logs, Natural Logs, and the Change-Of-Base Property.
Logarithms Part 3: Properties of Logs, Expanding Logarithmic Expressions.
Solving Exponential and Logarithmic Equations.
Complex Numbers: Operations, Complex Conjugates, and the Linear Factorization Theorem.
Set Theory: Types of Sets, Unions and Intersections.
Sequences, Factorials, and Summation Notation.
Theoretical Probability, Permutations and Combinations.
Introduction to Trigonometry: Angles and Radians.
Trigonometric Functions: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent.
The Easiest Way to Memorize the Trigonometric Unit Circle.
Basic Trigonometric Identities: Pythagorean Identities and Cofunction Identities.
Graphing Trigonometric Functions.
Inverse Trigonometric Functions.
Verifying Trigonometric Identities.
Formulas for Trigonometric Functions: Sum/Difference, Double/Half-Angle, Prod-to-Sum/Sum-to-Prod.
Solving Trigonometric Equations.
The Law of Sines.
The Law of Cosines.
Polar Coordinates and Graphing Polar Equations.
Parametric Equations.
Introduction to Calculus: The Greeks, Newton, and Leibniz.
Understanding Differentiation Part 1: The Slope of a Tangent Line.
Understanding Differentiation Part 2: Rates of Change.
Limits and Limit Laws in Calculus.
What is a Derivative? Deriving the Power Rule.
Derivatives of Polynomial Functions: Power Rule, Product Rule, and Quotient Rule.
Derivatives of Trigonometric Functions.
Derivatives of Composite Functions: The Chain Rule.
Derivatives of Logarithmic and Exponential Functions.
Implicit Differentiation.
Higher Derivatives and Their Applications.
Related Rates in Calculus.
Finding Local Maxima and Minima by Differentiation.
Graphing Functions and Their Derivatives.
Optimization Problems in Calculus.
Understanding Limits and L'Hospital's Rule.
What is Integration? Finding the Area Under a Curve.
The Fundamental Theorem of Calculus: Redefining Integration.
Properties of Integrals and Evaluating Definite Integrals.
Evaluating Indefinite Integrals.
Evaluating Integrals With Trigonometric Functions.
Integration Using The Substitution Rule.
Integration By Parts.
Integration by Trigonometric Substitution.
Advanced Strategy for Integration in Calculus.
Evaluating Improper Integrals.
Finding the Area Between Two Curves by Integration.
Calculating the Volume of a Solid of Revolution by Integration.
Calculating Volume by Cylindrical Shells.
The Mean Value Theorem For Integrals: Average Value of a Function.
Convergence and Divergence: The Return of Sequences and Series.
Estimating Sums Using the Integral Test and Comparison Test.
Alternating Series, Types of Convergence, and The Ratio Test.
Power Series.
Taylor and Maclaurin Series.
Hyperbolic Functions: Definitions, Identities, Derivatives, and Inverses.
Three-Dimensional Coordinates and the Right-Hand Rule.
Introduction to Vectors and Their Operations.
The Vector Dot Product.
Introduction to Linear Algebra: Systems of Linear Equations.
Understanding Matrices and Matrix Notation.
Manipulating Matrices: Elementary Row Operations and Gauss-Jordan Elimination.
Types of Matrices and Matrix Addition.
Matrix Multiplication and Associated Properties.
Evaluating the Determinant of a Matrix.
The Vector Cross Product.
Inverse Matrices and Their Properties.
Solving Systems Using Cramer's Rule.
Understanding Vector Spaces.
Subspaces and Span.
Linear Independence.
Basis and Dimension.
Change of Basis.
Linear Transformations on Vector Spaces.
Image and Kernel.
Orthogonality and Orthonormality.
The Gram-Schmidt Process.
Finding Eigenvalues and Eigenvectors.
Diagonalization.
Complex, Hermitian, and Unitary Matrices.
Double and Triple Integrals.
Partial Derivatives and the Gradient of a Function.
Vector Fields, Divergence, and Curl.
Evaluating Line Integrals.
Green's Theorem.
Evaluating Surface Integrals.
Stokes's Theorem.
The Divergence Theorem.

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