MATHEMATICS REVIEWER

Limit

The value that a function approaches as the input gets arbitrarily close to a certain point, providing the foundation for derivatives and integrals.

Derivative

A measure of how a function changes at a given point, representing the instantaneous rate of change or slope of the function.

Integral

A mathematical operation that calculates the area under a curve, representing accumulation or total change.

Continuity

A property of a function where small changes in input result in small changes in output, meaning there are no sudden jumps or gaps.

Differential Equation

An equation involving derivatives that describe how a function changes over time, commonly used to model dynamic systems.

Chain Rule

A rule in differentiation used to find the derivative of a composite function, stating that the derivative of f of g of x is f prime g of x times g prime x.

Taylor Series

A representation of a function as an infinite sum of terms calculated from its derivatives at a single point.

Partial Derivative

A derivative taken with respect to one variable while treating other variables as constants, commonly used in multivariable functions.

Vector Calculus

A branch of calculus that extends differentiation and integration to vector fields, commonly used in physics and engineering.

A technique for evaluating limits of indeterminate forms by taking derivatives of the numerator and denominator.

L’Hopital’s Rule

Probability

A numerical measure of the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain)

Random Variable

A variable that takes on different numerical values based on the outcome of a random process, classified as discrete or continuous.

Normal Distribution

A probability distribution that follows a bell-shaped curve, characterized by its mean and standard distribution.

Standard Deviation

A measure of how spread out values are around the mean in a data set. A small SD means values are close to the mean, while a large SD indicates more variability.

Expected Value

The weighted average of all possible outcomes of a random variable, providing a long-term prediction of its behavior.

Baye’s Theorem

A formula that calculates the probability of an event based on prior knowledge of related conditions.

Confidence Interval

A range of values used to estimate an unknown population parameter, calculated based on sample data and a confidence level.

Hypothesis Testing

A statistical method for making decisions about a population based on sample data, involving null and alternative hypotheses.

Regression Analysis

A statistical technique used to model relationships between variables and make predictions based on data trends.

Central Limit Theorem

A fundamental statistical principle stating that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population’s orignial distribution.

Polynomial

A mathematical expression consisting of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. The highest exponent in a polynomial determines its degree.

Matrix

A rectangular array of numbers arranged in rows and columns, commonly used to solve systems of equations, perform transformations, and represent data.

Determinant

A special number associated with a square matrix, calculated using a specific formula, which helps determine if the matrix has an inverse.

Eigenvalue

A scalar that describes how a linear transformation changes a vector’s magnitude along certain directions. It is found by solving the characteristic equation of a matrix.

Group

A set with an operation that satisfies four properties: closure, associativity, identity element, and invertibility.

Ring

An algebraic structure with two operations, usually addition and multiplication, where addition forms a group, and multiplication is associative

Field

A mathematical system where addition, subtraction, multiplication, and division (except by zero) can be performed, with all operations following specific properties.

Factorization

The process of breaking down an algebraic expression or number into simpler factors, often to simplify equations or find roots.

Homomorphism

A function between two algebraic structures that preserves their operations, meaning the function maintains structure consistency.

Commutativity

A property where the order of applying an operation does not affect the result. This property holds for addition and multiplication but not for matrix multiplication.

Axiom

A fundamental assumption accepted as true without proof, serving as a foundation for mathematical reasoning.

TheoremA

A statement that has been logically proven using axioms and previously established theorems.

Parallel Postulate

A fundamental assumption in Euclidean geometry stating that exactly one parallel line can be drawn through a point not on a given line

Congruence

A property indicating two shapes are identical in size and shape, achieved through transformations like rotation, reflection, or translation.

Similarity

A property where two geometric figures have the same shape but different sizes, with corresponding angles equal and sides proportional.

Manifold

A mathematical space that locally resembles Euclidean space but can have a different overall shape, such as the surface of a sphere.

Symmetry

A property where a figure remains unchanged under specific transformations, such as reflections, rotations, and translations.

Geodesic

The shortest path between two points on a curved surface, such as a great circle on a sphere.

Transformation

A function that moves or alters a geometric figure while maintaining certain properties, such as rigid motions or scaling.

Euclidean Space

A mathematical space that follows the traditional rules of Euclidean geometry, where angles and distances behave as expected.

Graph Theory

The study of graphs, which are mathematical structures consisting of nodes (vertices) and edges (connections), used to model relationships and networks.

Combinatorics

The branch of mathematics that deals with counting, arranging, and analyzing different possibilities in a set.

Boolean Algebra

A mathematical system for working with binary variables (true or false, 1 or 0) and logical operations like AND, OR, and NOT

Set Theory

The study of collections of objects, called sets, and their properties, such as union, intersection, and complement.

Permutation

An arrangement of objects in specific order, calculated using factorial notation.

Combination

A selection of objects without regard to order, calculated using binomial coefficients.

Recursion

A process where a function is defined in terms of itself, often used in algorithms and mathematical sequences.

Big-O Notation

A mathematical notation used in computer science to describe the efficiency of an algorithm in terms of time or space complexity.

Game Theory

A mathematical framework for analyzing decision-making in strategic situations, often used in economics and political science

Cryptography

The study of secure communication methods, involving mathematical techniques such as encryption and decryption.