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Math Notes
Algebra
Numbers
Natural Numbers
Arithmetic Operations over Natural Numbers
Ordering of Natural Numbers
Division with Remainder
Prime Factorization of Natural Number
Greatest Common Divisor
Least Common Multiple
Criteria of Divisibility
Using Letters in Algebra. Variables
Rational Numbers
Fractions. Proper and Improper Fractions. Mixed Numbers
Equality of Fractions. Properties of Fractions. Reduction of Fractions.
Finding Common Denominator of Fractions
Arithmetic Operations over Fractions
Decimals
Arithmetic Operations over Decimals
Percents
Converting Proper Fraction into Infinite Periodic Decimal
Converting Infinite Periodic Decimal into Proper Fraction
Set of Rational Numbers
Coordinate Line
Real Numbers
Number Plane.Cartesian Coordinate System in the Plane and Space
Polar Coordinate System
The Designations of Certain Number Sets. The Basic Concepts Associated with Sets
Comparison of Real Numbers
Properties of the Numerical Inequalities
Numerical Intervals
Absolute Value of Real Numbers
Formula for Distance Between Two Points
Arithmetic Operations on Real Numbers
Properties of Arithmetic Operations on Real Numbers
Proportions
Integer Part of Numbers. Fractional Part of Number
The Power with Natural Exponent
The Power with Zero Exponent. The Power with Negative Exponent
The Standard Form of Positive Real Numbers
Definition of Arithmetic Root. Arithmetic Properties of the Roots
The Root of Odd Degree n From Negative number a
Degree with Fractional Exponents
The Properties of Powers with the Rational Exponents
Approximate Values of the Number. Absolute and Relative Errors
Decimal Approximations of the Real Number by Excess and Defect
The Degree with the Irrational Exponent
Properties of Powers with the Real Exponents
Complex Numbers
Notion of Complex Numbers
Algebraic Form of Complex Numbers
Arithmetic operations over Complex Numbers
Geometric Representation of Complex Numbers. Trigonometric Form of Complex Numbers
Operations over Complex Numbers in Trigonometric Form. De Moivre's Formula
Whole Numbers
Natural Numbers
What is Whole Number
Adding Whole Numbers
Subtracting Whole Numbers
Multiplying Whole Numbers
Multiplication Table
Dividing Whole Numbers
Expressions
Basic Concepts
The Types of Algebraic Expression
Allowable Value of Variables. Domain of Algebraic Expression
The Concept of Identity Transformation Expression. Identity
Rational Expressions
Monomials and Operations on them
Polynomials. Transformation of Polynomials to the Standart Form
Short Multiplication Formulas
Factoring Polynomials
Polynomial of One Variable
Division of Polynomials. Horner′s Scheme. Bezout′s Theorem
Factoring Quadratic Polynomials into Linear Factors
Factoring Binomials `x^n-a^n`
Raising Binomial to the Natural Power (Newton's Binom Formula)
Rational Fractions
Rational Fraction and its Basic Property
Reducing of Rational Fractions
Reducing Rational Fractions to the Common Denominator
Addition and Subtraction of Rational Fraction
Multiplication and Division of Rational Fractions
Raising Rational Fraction to the Integer Power
Transformation of Rational Expressions
Irrational Expressions
The Simplest Transformations of Arithmetic Root (Radical)
The Identity `sqrt(a^2)=|a|`
Transformation of Irrational Expressions
Logarithmic Expressions
Notion of Transcedental Expression
Definition of Logarithm of Positive Number with Given Base
Properties of Logaritms
Taking Logarithm and Exponentiating
Decimal Logarithm. Characteristic and Mantissa of Decimal Logarithm
Trigonometric Expressions
Definition of Trigonometric Expression
Formulas for Adding and Subtracting Arguments
Reduction Formulas
Relation between Trigonometric Functions of Same Argument
Double Angle Formulas
Power-Reduction Formulas
Expressions for `sin(t),cos(t),tan(t)` through `tan(t/2)`
Converting Sum of Trigonometric Functions into Product
Converting Product of Trigonometric Functions into Sum
Converting Expression `acos(t)+bsin(t)` into the Form `Asin(t+alpha)`
Transforming Expressions that contain Inverse trigonometric Functions
Integers
Negative Numbers
What is Integer Number
Adding Integers
Subtracting Integers
Multiplying Integers
Functions and Graphs
Properties of Functions
Definition of the Function
Analytical Representation of the Function
Tabular Representation of the Function
Graphical Representation of the Function
Graph of the Function Represented Analytically
Odd and Even Functions
Graphs of Even and Odd Functions
Periodic Functions
Increasing and Decreasing Functions
Types of Functions
Constant Function
Direct Proportionality
Linear Function
Relative Position of Graphs of the Linear Functions
Inverse Proportionality
Function `y=x^2`
Function `y=x^3`
Power Function with Natural Exponent
Power Function with Integer Negative Exponent
Function `y=sqrt(x)`
Function `y=root(3)(x)`
Function `y=root(n)(x)`
Power Function with Positive Fractional Exponent
Power Function with Negative Fractional Exponent
Function y=[x]
Function y={x}
Exponential Function
Inverse Function. Graph of the Inverse Function
Logarithmic Function
Number `e`. Function `y=e^x`. Function `y=ln(x)`
Definition of Trigonometric Functions
Signs of Trigonometric Functions
Even and Odd Trigonometric Functions
Periodicity of Trigonometric Functions
Properties and Graph of the Function y=sin(x)
Properties and Graph of the Function y=cos(x)
Properties and Graph of the Function y=tan(x)
Properties and Graph of the Function y=cot(x)
Function y=arcsin(x)
Function y=arccos(x)
Function y=arctan(x)
Function y=arccot(x)
Inverse Trigonometric Functions
Transformations of Graphs
Drawing Graph of the Function y=mf(x)
Graphs of Functions `y=ax^2,y=ax^3`
Drawing Graph of the Function y=f(x-a)+b
Graph of the Quadratic Function
Ways to Draw Graph of the Quadratic Function
Drawing Graph of the Function y=f(kx)
Stretching and Compressing Graphs of Trigonometric Functions
Graph of the Harmonic Oscillation `y=Asin(omega x+alpha)`
Equations and Systems of Equations
Equations with One Variable
Definition of the Equation. Roots of the Equation.
Equivalence of the Equations
Linear Equations
Quadratic Equations
Incomplete Quadratic Equations
Viet Theorem
Systems and Sets of Equations
Equations Involving Absolute Values
Notion of the Consequence of Equation. Extraneous Roots.
Equations with Variable in Denominator
Domain of the Equation
Rational Equations
Solving of Equation p(x)=0 by Factoring Its Left Side
Solving of Equations with Method of Introducing New Variable
Biquadratic Equation
Equations of Higher Degrees
Solving of Word Problems with the Help of Equations
Irrational Equations
Exponential Equations
Logarithmic Equations
Exponential Logarithmic Equations
The Simplest Trigonometric Equations
Solving of Trigonometric Equations by Factoring
Solving of Trigonometric Equatipns with a Method of Introducing New Variable
Homogeneous Trigonometric Equations
Universal Substitution for Rational Trigonometric Equations
Method For Solving Trigonometric Equations by Introducing Auxiliary Argument
Graphical Solving of Equations
Parametric Equations
Equations with Two Variables
Solving of Equation with Two Variables
Graph of the Function with Two Variables
Linear Equation with Two Variables and Its Graph
Systems of Equations
Systems of Two Equations with Two Variables. Equivalent Systems
Solving of System of Two Equation with Two Variables. Substitution Method
Solving of System of Two Equation with Two Variables. Addition Method
Solving of System of Two Equation with Two Variables. Method of Introducing New Variables
System of Two Linear Equations with Two Variables. Second-Order Determinants
Symmetric Systems
Graphical Solving of the System of Two Equations with Two Variables
Systems of Three Equations with Three Variables
Systems of Three Linear Equations with Three Varaibles. Third-Order Determinants
Systems of Exponential and Logarithmic Equations
Systems of Trigonometric Equations
Inequalities
Solving Inequalities
Basic Concepts Connected with Solving Inequalities
Graphical Method for Solving Inequality with One Variable
Linear Inequalities with One Variable
Systems of Inequalities with one Variable
Set of Inequalities with One Variable
Fractional-Linear Inequalities
Quadratic Inequalities
Solving Quadratic Inequalities Graphically
Inequalities Involving Absolute Values
Solving Rational Inequalities Using Method of Intervals
Exponential Inequalities
Logarithmic Inequalities
Irrational Inequalities
Trigonometric Inequalities
Inequalities and System of Inequalities with Two Variables
Proving Inequalities
Method of Estimating of the Sign of Difference
Synthetic Method of Proving Inequalities
Proving of Inequalities by Contradiction
Using Inequalities to Solve Equations
Elements of Combinatorics
Permutations and Combinations
Permutations
Arrangements
Combinations and their Properties. Pascal's Triangle
Binom of Newton (Binomial Theorem)
Binom of Newton
Properties of Newton's Binom Formula
Calculus I
What are Limits in Calculus?
Sequence and Limit
Number Sequence
Limit of a Sequence
Infinitely Small Sequence
Infinitely Large Sequence
Sequence Theorems
Squeeze (Sandwich) Theorem for Sequences
Algebra of Limit of Sequence
Indeterminate Form for Sequence
Stolz Theorem
Monotonic Sequence
Monotonic Sequence Definition
The Euler Number `e`
Notion of Function
Function Concept
Variable Interval
Dependence between Variables
Definition of Function
Representing a Function
Inverse of a Function
Function Types
Increasing and Decreasing Functions
Even Odd Function
Piecewise Function
Periodic Function
Common Functions
Linear Function
Polynomial Function
Power Function
Rational Function
Algebraic and Non-Algebraic Functions
Exponential Function
Logarithmic Functions
Trigonometric Functions
Inverse Trigonometric Functions
Hyperbolic Functions
Inverse Hyperbolic Functions
Creating New Functions from Old
Transformation of Functions
Combinations of Functions
Composition of Functions
Limit of the Function
Definition of Limit of the Function
One-Sided Limits
Sandwich Theorem
Properties of the Limits
Limits Involving Infinity
Asymptotes
Indeterminate Forms for Functions
Continuity of the Function
Definition of Continuous Function
One-Sided Continuity. Classification of Discontinuities
Theorems involving Continuous Functions
Derivatives and Differentials
Derivative
Definition of Derivative
Derivatives of Elementary Functions
Table of the Derivatives
Tangent Line, Velocity and Other Rates of Changes
Studying Derivative Graphically
Differentiation Rules
Constant Multiple Rule
Sum and Difference Rules
Product Rule
Quotient Rule
Chain Rule
Implicit Differentiation
Derivative of Inverse Function
Logarithmic Differentiation
Differentials
Linear Approximations
Differentials
Using Differentials to Estimate Errors
Theorems of Differential Calculus
Fermat's Theorem
Rolle's Theorem
Mean Value Theorem
Higher-Order Derivatives
Definition of Higher-Order Derivatives
Formulas for Higher-Order Derivatives
Higher-Order Differentials
Parametric Differentiating
Taylor Formula
Taylor Polynomial
Maclaurin Polynomials of Common Functions
Studying Function with Derivatives
Studying Changes of Function
Condition of Constancy of the Function
Condition of Monotony of the Function
Proving Inequalities Using Monotony of the Function
Maxima and Minima. Necessary Conditions
First Derivative Test
Second Derivative Test
Higher-Order Derivative Test
Closed Interval Method
Convex and Concave Functions
Definition of Convex and Concave Functions
Properties of Convex Functions
Conditions of Concavity (Convexity) of the Function
Inflection Points
Drawing Graphs of Functions
Introduction to Sketching Graph of Function
Steps for Sketching Graph of Function
L'Hopital's Rule
Indeterminate Form of Type `0/0`
Indeterminate Forms of Type `oo/oo`
Other Indeterminate Forms
Applications of Derivatives
Related Rates
Optimization Problems
Applications to Economics
Tangent Line to Parametric Curves
Tangent Line in Polar Coordinates
Calculus II
Antiderivative and Indefinite Integral
Concept of Antiderivative and Indefinite Integral
Properties of Indefinite Integrals
Table of Antiderivatives
Area Problem
Definite Integral
Area Problem Revisited
Concept of Definite Integral
Properties of Definite Integral
The Fundamental Theorem of Calculus
Integration Techniques
Substitution (Change of Variable) Rule
Integration By Parts
Integrals Involving Trig Functions
Trigonometric Substitutions
Integrals Involving Rational Functions
Table of the Indefinite Integrals
Integration Strategy
Numerical (Approximate) Integration
Left Endpoint, Right Endpoint and Midpoint Rules
Trapezoidal Rule
Simpson's Rule
Improper Integrals
Type I (Infinite Intervals)
Type II (Discontinuous Integrands)
Comparison Test for Improper Integrals
Applications of Integrals
Area between Curves
Total Change
Volumes, Solids of Revolution and Method of Rings/Disks
Volume of Solid of Revolution and Method of Cylinders
Volume of Solid of Revolution about Slant Line
Arc Length
Average Value of a Function
Consumer and Producer Surpluses
Work
Hydrostatic Pressure and Force
Moments and Centers of Mass
Probability
The Distance Problem
Area Enclosed by Parametric Curves
Area in Polar Coordinates
Arc Length in Polar Coordinates
Series and Sequences
Sequences
Differential Equations
Basic Concepts
Introduction
Existence and Uniqueness
First-Order ODE
Separable Differential Equations
Homogeneous Equations
Exact Equations
Linear Differential Equations
Bernoulli Equations
Applications of First-Order ODE
Growth and Decay Problems
Temperature Problems
Falling Body Problems
Dilution Problems
Electrical Circuits
Orthogonal Trajectories
Nth-Order Linear ODE
Basic Concepts
Linear Independence and Wronskian
Particular Solution
Nth-Order Linear ODE with Constant Coefficients
Method of Solution
The Method of Undetermined Coefficients
Variation of Parameters
Initial Value Problems
Applications of Second-Order ODE
Spring Problems
Laplace Transform
Definition of the Laplace Transform
Table of the Laplace Tranforms
Properties of the Laplace Transform
Inverse Laplace Transform
Convolution Integral
Unit (Heaviside) Step Function
Dirac Delta Function
Solving IVP's with Laplace Transform
Systems of ODE's
Questions
Algebra
Geometry
Calculus I
Calculus II
Calculus III
Differential Equations
Linear Programming
Linear Algebra
Probability/Statistics
Discrete Mathematics
Other
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