Calculus

Calculate the average rate of change of a function over an interval using the difference quotient. Enter any f(x) to get the secant slope, a step-by-step solution with MathJax formulas and a graph of the secant line and curve.

Solve Bernoulli differential equations step by step. Applies the v = y^(1-n) substitution to linearize the equation, builds the integrating factor, finds the closed-form solution and plots the particular curve against the slope field.

Calculate linear, circular, and continuous convolution of signals and functions with interactive visualizations, detailed step-by-step solutions, and comprehensive mathematical analysis.

Calculate the curl of any 2D or 3D vector field with a step-by-step determinant expansion. Enter the component functions, get the symbolic curl, evaluate it at a point, spot irrotational fields and view a vorticity overlay.

Calculate the curvature (κ) of a function y=f(x) or a parametric curve at a specific point, with step-by-step derivatives, osculating circle visualization, and radius of curvature.

Calculate derivatives instantly with step-by-step solutions. Supports single-variable, partial, implicit, and directional derivatives with interactive visualizations.

Plot the slope field of any first-order ODE over a custom x-y region. Click the canvas to spawn solution curves, watch particles flow along the field, and see equilibrium nullclines, rendered as a pure SVG you can save or share.

Calculate directional derivatives of multivariable functions with step-by-step solutions, gradient computation, unit vector normalization, and interactive 3D surface visualization.

Calculate the divergence of any 2D or 3D vector field with step-by-step partial derivatives. Enter the component functions, get the symbolic divergence, evaluate it at a point, identify sources and sinks, and view a heat map.

Compute double integrals with detailed step-by-step solutions and 3D surface visualization. Supports definite and indefinite integrals with polynomial, trigonometric, exponential, and hyperbolic functions.

Solve any first-order ODE numerically with Euler's method. See the iteration table, the Euler polygon drawn on the slope field, and a convergence comparison at h, h/2 and h/4, with optional error analysis against a closed form.

Compute the discrete FFT of a real or complex signal sequence. Apply common window functions, choose FFT length and zero padding, inspect magnitude, phase, frequency bins, dominant peaks, and copy the full complex spectrum.

Solve first-order ordinary differential equations symbolically and numerically. Auto-detects separable, linear, exact and autonomous forms, applies the right technique, and draws an interactive slope field with the solution curve.

Calculate the Fourier series coefficients a0, an and bn for any periodic function. See the full integral computations, a coefficient table, the partial sum formula and a graph comparing the function with its Fourier approximation.

Calculate the gradient vector of a multivariable function. Enter f(x, y) or f(x, y, z) to get every partial derivative, evaluate the gradient at a point, read its magnitude and direction, and view a 2D gradient field.

Compute derivatives of implicit functions F(x,y)=0 with detailed step-by-step solutions. Supports polynomials, trigonometric, exponential, and logarithmic functions up to 5th order derivatives.

Evaluate improper integrals with infinite limits or discontinuities. Supports Type I infinite bounds and Type II unbounded integrands, with step-by-step solutions, convergence analysis, visualizations and truncation-limit comparison.

Calculate the exact sum of convergent infinite series including geometric, telescoping, p-series, and well-known special series. Get step-by-step convergence proofs with animated partial sum visualizations.

Calculate the instantaneous rate of change of any function at a point using the limit definition. Get step-by-step solutions with MathJax formulas, an interactive tangent line graph and a convergence table as h approaches zero.

Calculate definite and indefinite integrals with detailed step-by-step solutions, interactive function visualization, and comprehensive explanations for calculus students and professionals.

Compute the inverse Laplace transform of F(s) to find f(t). Get step-by-step solutions, visualizations, and understand the transformation from frequency domain to time domain.

Evaluate limits of indeterminate forms (0/0, ∞/∞) using L'Hôpital's rule with step-by-step differentiation, interactive graph visualization, and detailed explanations.

Compute Laplace transforms instantly with detailed step-by-step solutions, interactive function presets, and dual visualization of time-domain and frequency-domain functions.

Calculate limits of mathematical functions with detailed step-by-step solutions. Supports one-sided limits, indeterminate forms, and L'Hospital's Rule.

Calculate line integrals of scalar and vector fields along parametric curves in 2D and 3D. Enter the field, parametric equations and bounds for symbolic results with step-by-step solutions, arc length and a curve visualization.

Compute the Maclaurin series expansion of common functions at x = 0. Get the nth-order polynomial terms, a Lagrange remainder estimate, the radius of convergence and an animated graph of partial sums converging to the function.

Find roots of equations with the Newton-Raphson method. Enter any function and an initial guess to see step-by-step iterations with tangent line approximations, convergence analysis and a graph tracing the path to the root.

Approximate definite integrals with Gauss-Legendre quadrature, Romberg extrapolation and adaptive Simpson quadrature. Compare estimates, error signals, function evaluations, convergence and adaptive interval placement in one workspace.

Find the maximum and minimum values of any function using the first and second derivative tests. Get critical points, intervals of increase and decrease, concavity analysis, inflection points and an interactive graph with worked steps.

Compute partial derivatives of multivariable functions with detailed step-by-step solutions, interactive examples, and geometric visualization of tangent planes.

Find the power series representation of a function centred at any point. Compute Taylor and Maclaurin coefficients, determine the radius and interval of convergence with endpoint analysis, and watch partial sums converge.

Determine the radius and interval of convergence for power series using the Ratio Test or Root Test, with step-by-step solutions, convergence visualization, and endpoint analysis.

Set up and solve related rates problems step by step with implicit differentiation and the chain rule. Covers the expanding sphere, sliding ladder, filling cone, ripple, shadow length, approaching cars and inflating balloon.

Approximate definite integrals using Riemann sums with left endpoint, right endpoint, midpoint, trapezoidal, and Simpson's rule. View animated rectangle visualizations, step-by-step solutions, and convergence analysis.

Solve ordinary differential equations numerically with the classic 4th-order Runge-Kutta method. Enter the equation, initial conditions and step size to see the k1 to k4 computations, a solution table and the solution curve.

Solve second-order linear ODEs with constant coefficients, homogeneous or not. Derives the characteristic equation, classifies the damping as over, critical or under-damped, gives the closed form and plots the phase-plane trajectory.

Test whether an infinite series converges using the Ratio, Root, Integral, Comparison, Limit Comparison, Alternating Series and p-Series tests, with step-by-step solutions and animated partial sum graphs.

Approximate definite integrals with Simpson's 1/3 rule, 3/8 rule and the composite rule. Includes an interactive parabolic visualization, error estimation, convergence analysis, method comparison and step-by-step solutions.

Compute derivatives of any single-variable function with step-by-step solutions, differentiation rule identification, interactive graph, and critical point analysis.

Evaluate surface integrals of scalar fields and flux integrals of vector fields over parametric surfaces. Choose a preset - sphere, cylinder, cone, paraboloid or torus - or enter your own, with normal vectors and a 3D visualization.

Calculate the surface area of a solid of revolution. Enter any function f(x), set integration bounds and axis of rotation, and get step-by-step solutions with interactive 3D visualizations using the disk and shell surface area formulas.

Solve systems of ordinary differential equations symbolically and numerically. Classifies the equilibrium as saddle, node, spiral or centre, derives eigenvalues step by step, and draws an interactive phase portrait with a trajectory.

Calculate the Taylor series expansion of any function around a point with step-by-step derivative calculations, interactive comparison graph, and educational explanations.

Approximate definite integrals with the trapezoidal rule, with an interactive trapezoid visualization, error estimation, Richardson extrapolation, convergence analysis and a per-trapezoid area breakdown. Accepts functions or data points.

Compute triple integrals with detailed step-by-step solutions and 3D visualization. Supports definite and indefinite integrals with symbolic computation.

Calculate the volume of a solid of revolution using the disk, washer, and shell methods. Enter your function, bounds, and axis of rotation to get step-by-step solutions with interactive 3D visualizations.

Calculate the Wronskian determinant of a set of functions to test linear independence. See the full matrix with derivatives, a step-by-step determinant expansion, and a verdict on whether the functions form a fundamental set.

Compute one-sided Z-transforms and causal inverse Z-transforms. Enter a sequence formula, finite sample list, or rational X(z) to get the transform, ROC, poles and zeros, sample table, and visual z-plane analysis.