What is this glossary?
It explains 44 of the most important calculus words in plain English, from antiderivative to Wronskian. Each term has a short definition, a small example, and links to a free calculator or a guide where you can practice it.
- Antiderivative
- A function \(F(x)\) whose derivative is \(f(x)\). For example, \(x^2\) is an antiderivative of \(2x\). Every antiderivative can have any constant added, so we write \(x^2 + C\). Calculator: Antiderivative calculator · Guide: What is an antiderivative?
- Asymptote
- A line that a graph gets closer and closer to but does not reach as it heads off to infinity or toward a break. Asymptotes can be vertical, horizontal, or slanted. Calculator: Asymptote calculator
- Chain rule
- The rule for differentiating a function inside another function: \(\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)\). For example, the derivative of \(\sin(x^2)\) is \(2x\cos(x^2)\). Calculator: Derivative calculator
- Concavity
- The way a graph bends. It is concave up (like a cup) where \(f”(x) > 0\) and concave down (like a cap) where \(f”(x) < 0\). Calculator: Concavity calculator
- Constant of integration
- The \(+\,C\) added to every indefinite integral. It is there because constants disappear when you take a derivative, so many functions share the same derivative. Guide: Why the constant of integration matters
- Continuity
- A function is continuous at a point if you can draw its graph there without lifting your pencil. In symbols, \(\lim_{x \to a} f(x) = f(a)\). Calculator: Limit calculator
- Critical point
- A point where the derivative is 0 or does not exist. Peaks, valleys, and flat spots of a graph happen at critical points. Calculator: Derivative calculator
- Curl
- A vector that measures how much a 3D vector field spins around each point. If the curl is zero everywhere, the field is irrotational. Calculator: Curl calculator
- Definite integral
- An integral with limits, \(\int_a^b f(x)\,dx\). Its answer is a number: the signed area between the graph and the x-axis from \(a\) to \(b\). Calculator: Definite integral calculator · Guide: Definite vs indefinite integral
- Derivative
- The instantaneous rate of change of a function, which is also the slope of its tangent line. The derivative of \(x^3\) is \(3x^2\). Calculator: Derivative calculator · Guide: Antiderivative vs derivative
- Determinant
- A single number computed from a square matrix. For a 2×2 matrix it is \(ad – bc\). A determinant of 0 means the matrix has no inverse. Calculator: Determinant calculator
- Differential equation
- An equation that links a function with its derivatives, such as \(y’ = 3y\). Its solution is a function, not a number. Calculator: Differential equation solver
- Divergence
- A number that measures how much a vector field spreads out from a point, \(\nabla\cdot\mathbf{F} = P_x + Q_y + R_z\). Positive divergence means the point acts like a source. Calculator: Curl and divergence calculator
- Eigenvalue
- A number \(\lambda\) for which a matrix just stretches some nonzero vector: \(A\mathbf{v} = \lambda\mathbf{v}\). The vector \(\mathbf{v}\) is the eigenvector. Calculator: Eigenvalue and eigenvector calculator
- Euler’s method
- A way to approximate the solution of a differential equation by taking many small straight-line steps along the slope. Smaller steps usually give a more accurate answer. Calculator: Euler’s method calculator
- Exponential growth
- Growth where an amount is multiplied by the same factor in each time period, \(P(t) = P_0(1 + r)^t\). When the factor is less than 1, it is exponential decay. Calculator: Exponential growth calculator
- Function
- A rule that gives exactly one output for each input, like \(f(x) = x^2\). Calculus studies how functions change and add up.
- Fundamental theorem of calculus
- The theorem that connects derivatives and integrals: if \(F’ = f\), then \(\int_a^b f(x)\,dx = F(b) – F(a)\). It is why antiderivatives let us compute areas. Calculator: Definite integral calculator
- Gradient
- The vector of all first partial derivatives of a function, such as \(\nabla f = \langle f_x, f_y\rangle\). It points in the direction where the function grows fastest. Calculator: Partial derivative calculator
- Improper integral
- An integral with an infinite limit, or with a function that blows up inside the interval. It is found with a limit and either converges to a number or diverges. Calculator: Improper integral calculator
- Indefinite integral
- The family of all antiderivatives of a function, written \(\int f(x)\,dx = F(x) + C\). It has no limits, and its answer is a function. Calculator: Integral calculator
- Inflection point
- A point on a graph where the concavity changes, from cup to cap or the other way. The second derivative changes sign there. Calculator: Concavity calculator
- Instantaneous rate of change
- How fast a function is changing at one exact point, found as the limit of average rates over smaller and smaller intervals. It equals the derivative there. Calculator: Instantaneous rate of change calculator
- Integral
- The main tool for adding up tiny pieces, such as area under a curve or total distance from speed. Integrals can be indefinite (a function) or definite (a number). Calculator: Integral calculator
- Integration by parts
- A method for integrating some products, based on the product rule: \(\int u\,dv = uv – \int v\,du\). The LIATE rule helps you choose \(u\). Calculator: Integration by parts calculator · Guide: Integration by parts
- Jacobian
- The matrix of all first partial derivatives of several functions. Its determinant tells how much a change of variables stretches area or volume, like \(r\) in polar coordinates. Calculator: Jacobian calculator
- L’Hôpital’s rule
- A rule for limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\): take the derivative of the top and of the bottom, then find the limit again. Calculator: Limit calculator
- Lagrange multiplier
- A method for finding the largest or smallest value of a function when the input must stay on a constraint. You solve \(\nabla f = \lambda\nabla g\) together with the constraint. Calculator: Lagrange multiplier calculator
- Laplace transform
- A tool that turns a function of time \(f(t)\) into a function \(F(s)\), so differential equations become algebra problems. For example, the transform of \(1\) is \(\frac{1}{s}\). Calculator: Laplace transform calculator
- Limit
- The value a function gets closer and closer to as the input gets closer to some number. Limits are the base of both derivatives and integrals. Calculator: Limit calculator
- Linear approximation
- Using the tangent line \(L(x) = f(a) + f'(a)(x – a)\) to estimate a function near \(x = a\). For example, it estimates \(\sqrt{4.1} \approx 2.025\). Calculator: Linear approximation calculator
- Matrix
- A rectangular grid of numbers arranged in rows and columns. Matrices can be added, multiplied, inverted, and row reduced. Calculator: Matrix calculator
- Partial derivative
- The derivative of a function of several variables with respect to one of them, treating the others as constants. It is written \(\frac{\partial f}{\partial x}\). Calculator: Partial derivative calculator
- Partial fractions
- Splitting a fraction of polynomials into simpler fractions that are easy to integrate, such as \(\frac{1}{x^2 – 1} = \frac{1/2}{x – 1} – \frac{1/2}{x + 1}\). Calculator: Partial fraction calculator
- Power rule
- For derivatives, \(\frac{d}{dx}x^n = nx^{n-1}\). For antiderivatives, \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\) when \(n \ne -1\). Guide: Power rule for antiderivatives
- Power series
- An infinite sum of the form \(\sum a_n (x – c)^n\). It works like a polynomial that never ends and only converges for some values of \(x\). Calculator: Interval of convergence calculator
- Product rule
- The rule for differentiating a product: \((fg)’ = f’g + fg’\). For example, \((x\sin x)’ = \sin x + x\cos x\). Calculator: Derivative calculator
- Quotient rule
- The rule for differentiating a fraction: \(\left(\frac{f}{g}\right)’ = \frac{f’g – fg’}{g^2}\). Calculator: Derivative calculator
- Radius of convergence
- How far from its center a power series still adds up to a finite number. It is often found with the ratio test. Calculator: Interval of convergence calculator
- Riemann sum
- An estimate of the area under a curve made by adding up the areas of thin rectangles. As the rectangles get thinner, the sum approaches the definite integral. Calculator: Definite integral calculator
- Second derivative
- The derivative of the derivative, written \(f”(x)\). It tells you about concavity, and for motion it is the acceleration. Calculator: Second derivative calculator · Acceleration calculator
- Tangent line
- The straight line that just touches a curve at one point and has the same slope there. Its slope is the derivative at that point. Calculator: Instantaneous rate of change calculator
- U-substitution
- An integration method that undoes the chain rule. You set \(u\) equal to an inside function, replace \(dx\) using \(du\), and integrate in terms of \(u\). Calculator: U-substitution calculator · Guide: U-substitution step by step
- Wronskian
- A determinant built from functions and their derivatives. If it is not zero, the functions are linearly independent, which matters when solving differential equations. Calculator: Wronskian calculator