Introduction to Calculus

• Math is the abstract science of number, quantity, and space, studied either as abstract concepts or as applied to other disciplines such as physics and engineering.
• Core areas include arithmetic, algebra, geometry, trigonometry, calculus, statistics and analysis.

Calculus

• Calculus is the study of continuous change, in the same way geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
• Has two major branches: differential calculus and integral calculus.
• Differential calculus concerns instantaneous rates of change and slopes of curves.
• Integral calculus concerns the accumulation of quantities and areas under or between curves.
• Calculus is used to find the equations for tangent lines and extreme values.
• It is used extensively in science, engineering, and economics.
• It can solve many problems that algebra alone cannot.

Differential Calculus

• Differential calculus involves the study of rates at which quantities change.
• It is one of the two main divisions of calculus.
• Focuses on finding the derivative of a function.
• The derivative measures the instantaneous rate of change of the function, which is also the slope of the tangent line to the function's graph at a given point.
• Key concepts include limits, derivatives, and differentiation rules.
• Applications include optimization problems (finding maximum and minimum values), related rates problems, and curve sketching.

Integral Calculus

• Integral calculus is the other main division which deals with the accumulation of quantities.
• Involves finding the integral of a function.
• The integral can be interpreted as the area under the curve of the function.
• Key concepts include antiderivatives, definite integrals, and the fundamental theorem of calculus.
• Applications include finding areas, volumes, and solving differential equations.
• Integration is the inverse process of differentiation.

Geometry

• Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Includes Euclidean geometry, which studies shapes and space based on a set of axioms.
• Also includes non-Euclidean geometries (e.g., hyperbolic and elliptic geometry), which differ from Euclidean geometry in their axioms regarding parallel lines.
• Key concepts include points, lines, angles, polygons, circles, and solids.
• Used in computer graphics, engineering, and physics.

Trigonometry

• Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles.
• Focuses on trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant.
• Trigonometric functions relate angles to ratios of sides in right triangles.
• Key concepts include trigonometric identities, laws of sines and cosines, and trigonometric equations.
• Used in navigation, surveying, engineering, and physics.
• Trigonometry is often used to calculate unknown distances or angles.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Involves methods for summarizing and describing data (descriptive statistics) and for making inferences and generalizations about a population based on a sample (inferential statistics).
• Key concepts include probability, distributions (e.g., normal distribution), hypothesis testing, confidence intervals, and regression analysis.
• Used in a wide range of fields, including science, business, and social sciences.

Descriptive Statistics

• Descriptive statistics involves methods for summarizing and describing data.
• Common measures include mean, median, mode, standard deviation, and variance.
• Used to present data in a meaningful way through tables, charts, and graphs.

Inferential Statistics

• Inferential statistics involves making inferences and generalizations about a population based on a sample.
• Relies on probability theory to assess the uncertainty associated with these inferences.
• Key techniques include hypothesis testing, confidence intervals, and regression analysis.

Algebra

• Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols.
• It is a generalization of arithmetic, where letters and symbols represent numbers and quantities.
• Includes basic algebra, which covers topics such as solving equations and inequalities, and abstract algebra, which studies algebraic structures such as groups, rings, and fields.
• Key concepts include variables, expressions, equations, functions, and polynomials.
• Used in all areas of mathematics and in many other fields, including computer science, engineering, and economics.

Basic Algebra

• Basic algebra covers topics such as solving linear equations, quadratic equations, and systems of equations.
• Also includes manipulation of algebraic expressions, such as polynomials and rational expressions.
• Key concepts include variables, constants, coefficients, and exponents.

Abstract Algebra

• Abstract algebra studies algebraic structures such as groups, rings, and fields.
• Provides a foundation for advanced mathematics and is used in areas such as cryptography and coding theory.
• Key concepts include group theory, ring theory, and field theory.