Understanding Calculus

• Calculus is the mathematical study of continuous change, similar to how geometry studies shape and algebra studies generalizations of arithmetic operations.
• It has two major branches: differential calculus and integral calculus.
• Differential calculus deals with instantaneous rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and areas under or between curves.
• These two branches are linked by the fundamental theorem of calculus.
• Calculus is used to find the maximum or minimum value of a function.

Differential Calculus

• Differential calculus studies the definition, properties, and applications of the derivative of a function.
• Finding the derivative is called differentiation.
• Given a function and a point in its domain, the derivative at that point is the slope of the tangent line to the function's graph at that point.
• The derivative helps analyze a function's behavior, including finding its maxima, minima, concavity, and inflection points.
• Derivatives are used in many areas of mathematics, science, and engineering.
• The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Integral Calculus

• Integral calculus studies the definitions, properties, and applications of indefinite and definite integrals.
• Finding the value of an integral is called integration.
• The indefinite integral, or antiderivative, reverses the operation of differentiation.
• F(x) is an indefinite integral of f(x) if f(x) is a derivative of F(x).
• The definite integral takes a function as input and returns a number representing the area between the input's graph and the x-axis.
• Integration is used to find area, volume, center of mass, and probability.

Fundamental Theorem of Calculus

• This theorem connects differentiation and integration.
• Part 1 states that if f is a continuous real-valued function on a closed interval [a, b], then the function F defined for x in [a, b] by F(x) = ∫ₐˣ f(t) dt is continuous on [a, b], differentiable on the open interval (a, b), and F'(x) = f(x) for all x in (a, b).
• Part 2 states that if f is a real-valued function continuous on [a, b] and F is an antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) - F(a).
• A consequence is that definite integration can be reversed by differentiation.
• The theorem is applied to solve many differential equations.

Limits and Continuity

• The concept of a limit is fundamental to calculus.
• Differentiation and integration rely on limits to find instantaneous rates of change or to sum infinite infinitesimal quantities.
• A function's limit is the value it approaches as the input approaches some value.
• A function is continuous at a point if the limit of the function as x approaches that point equals the function's value at that point.
• If a function is continuous at all points in its domain, it is said to be continuous.
• Most functions in calculus are continuous at all points in their domain.

Applications of Calculus

• Calculus has wide-ranging applications across various fields.
• In physics, it calculates velocity, acceleration, models gravity, and fluid flow.
• In engineering, it designs structures like bridges and airplanes.
• In economics, it models supply and demand and maximizes profits.
• In statistics, it calculates probabilities and tests hypotheses.
• Calculus finds the area of irregular shapes, the length of curves, and the volume of solids.
• In optimization, calculus helps find the maximum or minimum value of a function, subject to constraints.