Calculus and Geometry Concepts

Limits: Fundamental concept for defining continuity and derivatives.
- Notation: lim (x → a) f(x)
Derivatives: Measure of how a function changes as its input changes.
- Rules: Power rule, product rule, quotient rule, chain rule.
- Applications: Tangent lines, rates of change.
Integrals: Represents the area under a curve.
- Types: Definite (limits) and indefinite (antiderivatives).
- Fundamental Theorem of Calculus: Connects differentiation and integration.
Applications: Used in physics, engineering, economics for optimization, motion, and area calculations.

Basic Shapes:
- Points, lines, angles, triangles, quadrilaterals, circles.
Theorems:
- Pythagorean theorem: a² + b² = c² for right triangles.
- Area formulas:
  - Triangle: 1/2 * base * height
  - Circle: π * radius²
Congruence and Similarity:
- Congruent figures: identical shape and size.
- Similar figures: same shape but different sizes.
3D Geometry:
- Volume formulas:
  - Cube: side³
  - Sphere: 4/3 * π * radius³
  - Cylinder: π * radius² * height

Expressions and Equations:
- Variables, constants, coefficients, terms.
- Types of equations: linear, quadratic, polynomial.
Functions:
- Definition: a relation between a set of inputs and outputs.
- Notation: f(x), g(x), etc.
Solving Equations:
- Methods: factoring, using the quadratic formula, graphing.
Inequalities:
- Solving and graphing linear and quadratic inequalities.
- Notation: <, >, ≤, ≥.

Integers: Study of whole numbers and their properties.
Prime Numbers:
- Definition: Natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
- Fundamental Theorem of Arithmetic: Every integer > 1 can be uniquely factored into primes.
Divisibility:
- Concepts: Divisors, multiples, greatest common divisor (GCD), least common multiple (LCM).
Congruences:
- Modulo operation: a ≡ b (mod n) means a and b leave the same remainder when divided by n.
Applications: Cryptography, coding theory, and computer algorithms.

Limits: Essential for defining continuity and derivatives; notation expressed as lim (x → a) f(x).
Derivatives: Measure of a function's change relative to its input; utilizes rules like power, product, quotient, and chain.
Applications of Derivatives: Important for determining the slope of tangent lines, calculating rates of change in various contexts.
Integrals: Represents the accumulation of quantities, specifically the area under a curve; can be definite (with limits) or indefinite (antiderivatives).
Fundamental Theorem of Calculus: Establishes a vital connection between differentiation and integration processes.
Applications of Integrals: Widely used across fields such as physics, engineering, and economics for tasks involving optimization and area calculations.

Basic Shapes: Include fundamental figures such as points, lines, angles, triangles, quadrilaterals, and circles, each with unique properties.
Theorems: The Pythagorean theorem states that for right triangles, the relationship is a² + b² = c², crucial for geometry applications.
Area Formulas:
- Triangle area calculated as 1/2 * base * height.
- Circle area calculated using π * radius².
Congruence and Similarity:
- Congruent figures maintain identical shapes and sizes, while similar figures have the same shape but different sizes.
3D Geometry: Volume formulas include:
- Cube: side³ for finding space occupied.
- Sphere: 4/3 * π * radius³ to calculate spherical volume.
- Cylinder: volume given by π * radius² * height.

Expressions and Equations: Consist of variables, constants, coefficients, and terms; categorized into linear, quadratic, and polynomial forms.
Functions: Defined as a mapping from a set of inputs to outputs; commonly denoted as f(x), g(x), etc.
Solving Equations: Key methods include factoring, the quadratic formula, and graphing techniques to find solutions.
Inequalities: Involves solving and graphing both linear and quadratic inequalities; uses notations such as <, ≤, and ≥ to represent relationships.

Integers: Focus on whole numbers and their intrinsic properties.
Prime Numbers: Defined as natural numbers greater than 1 that are only divisible by 1 and themselves; significant in various mathematical frameworks.
Fundamental Theorem of Arithmetic: Asserts that every integer greater than 1 can be uniquely expressed as a product of prime numbers.
Divisibility: Concepts include divisors, multiples, greatest common divisor (GCD), and least common multiple (LCM), essential for number manipulation.
Congruences: Modulo operation represented as a ≡ b (mod n), indicating that a and b yield the same remainder upon division by n.
Applications: Number theory is foundational in cryptography, coding theory, and computer algorithms, enhancing data security and error detection.