Algebra and Calculus Overview

Algebra

• Definition: A branch of mathematics involving symbols and the rules for manipulating those symbols to solve equations.
• Key Concepts:
  • Variables: Symbols that represent numbers (e.g., x, y).
  • Expressions: Combinations of variables and numbers (e.g., 2x + 3).
  • Equations: Mathematical statements that two expressions are equal (e.g., 3x + 2 = 11).
  • Functions: Relations where each input has exactly one output (e.g., f(x) = x²).
• Types:
  • Linear Algebra: Study of vectors, vector spaces, and linear transformations.
  • Quadratic Equations: Equations of the form ax² + bx + c = 0; solutions found using the quadratic formula.
  • Polynomials: Expressions that involve variables raised to whole-number exponents.

Calculus

• Definition: The mathematical study of continuous change, dealing with derivatives and integrals.
• Key Concepts:
  • Derivatives: Measure the rate of change of a function; represented as f'(x) or dy/dx.
  • Integrals: Represents the accumulation of quantities; the area under a curve.
  • Fundamental Theorem of Calculus: Connects differentiation and integration.
• Types:
  • Differential Calculus: Focuses on the concept of the derivative.
  • Integral Calculus: Focuses on the concept of integrals and their properties.
• Applications:
  • Used in physics, engineering, economics to model dynamic systems.

Statistics

• Definition: The study of data collection, analysis, interpretation, presentation, and organization.
• Key Concepts:
  • Descriptive Statistics: Summarizes and describes features of a dataset (mean, median, mode, variance).
  • Inferential Statistics: Makes predictions or inferences about a population based on a sample.
  • Probability: Measures the likelihood of events occurring, foundational for statistics.
• Types:
  • Population vs. Sample: Population encompasses the entire group; a sample is a subset used for analysis.
  • Hypothesis Testing: A method to test assumptions about a population parameter.
• Applications:
  • Used in research, business, social sciences to inform decision-making.

Algebra

• Branch of mathematics concerning symbols and rules for manipulating to solve equations
• Key Concepts:
  • Variables represent numbers (e.g., x, y)
  • Expressions combine variables and numbers (e.g., 2x + 3)
  • Equations state that two expressions are equal (e.g., 3x + 2 = 11)
  • Functions are relations where each input has a single output (e.g., f(x) = x²)
• Types:
  • Linear Algebra studies vectors, vector spaces, and linear transformations
  • Quadratic Equations are of the form ax² + bx + c = 0; solved using the quadratic formula
  • Polynomials are expressions with variables raised to whole-number exponents

Calculus

• Study of continuous change using derivatives and integrals
• Key Concepts:
  • Derivatives measure the rate of change of a function, represented as f'(x) or dy/dx
  • Integrals represent the accumulation of quantities, calculated as the area below the curve.
  • Fundamental Theorem of Calculus links differentiation and integration
• Types:
  • Differential Calculus focuses on derivatives
  • Integral Calculus focuses on integrals and their properties
• Applications:
  • Used to model dynamic systems in physics, engineering, and economics

Statistics

• Study of data collection, analysis, interpretation, presentation, and organization
• Key Concepts:
  • Descriptive Statistics summarizes and describes datasets (mean, median, mode, variance).
  • Inferential Statistics makes predictions about a population based on a sample.
  • Probability is a measure of event likelihood and a foundation for statistics.
• Types:
  • Population vs. Sample: Population encompasses the entire group; a sample is a subset for analysis.
  • Hypothesis Testing is a method to assess assumptions about population parameters.
• Applications:
  • Informs decision-making in research, business, and social sciences