Glossary of calculus

Glossary of Calculus

Calculus is a branch of mathematics that studies continuous change. It is divided into two main branches: differential calculus and integral calculus. This glossary provides definitions and explanations of key terms and concepts in calculus.

A

Antiderivative

A function \( F(x) \) is an antiderivative of \( f(x) \) if \( F'(x) = f(x) \). The process of finding an antiderivative is called integration.

Asymptote

A line that a graph approaches but never touches. Asymptotes can be vertical, horizontal, or oblique.

B

Boundary value problem

A differential equation together with a set of additional constraints, called boundary conditions.

C

Chain rule

A formula for computing the derivative of the composition of two or more functions. If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Continuity

A function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point.

D

Derivative

A measure of how a function changes as its input changes. The derivative of a function \( f(x) \) is denoted \( f'(x) \) or \( \frac{df}{dx} \).

Differential equation

An equation that involves the derivatives of a function.

E

Euler's method

A numerical method for solving ordinary differential equations with a given initial value.

F

Fundamental theorem of calculus

Links the concept of the derivative of a function with the concept of the integral. It has two parts: the first part provides an antiderivative for a function, and the second part states that the integral of a function over an interval can be computed using one of its antiderivatives.

I

Integral

A mathematical object that represents the area under a curve. The process of finding integrals is called integration.

Integration by parts

A technique for finding integrals, based on the product rule for differentiation.

L

Limit

The value that a function or sequence "approaches" as the input or index approaches some value.

M

Mean value theorem

A theorem that states that for a continuous function on a closed interval, there exists a point at which the derivative of the function is equal to the average rate of change over the interval.

N

Newton's method

An iterative numerical method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

P

Partial derivative

The derivative of a function of several variables with respect to one variable, with all other variables held constant.

R

Riemann sum

An approximation of the integral of a function, calculated by dividing the region under the curve into shapes (usually rectangles) and summing their areas.

S

Series

The sum of the terms of a sequence.

Slope

The measure of the steepness or incline of a line, often represented as the ratio of the rise over the run.

T

Taylor series

An infinite sum of terms that are expressed in terms of the function's derivatives at a single point.

V

Vector calculus

A branch of calculus that deals with vector fields and differentiable functions of several variables.

W

Weierstrass function

An example of a function that is continuous everywhere but differentiable nowhere.

See also

• List of calculus topics
• History of calculus

References

• Stewart, James. Calculus: Early Transcendentals. Cengage Learning.
• Thomas, George B. Thomas' Calculus. Pearson.