Slope estimate
Then read what a derivative measures.
Tools after ideas
Calculators
Small tools that run in your browser. No accounts and no server math. Use them to test a number after you can explain the idea. Exact algebra now includes Gram Schmidt, LU factorization, and Jacobian matrices. Each tool page includes a short lesson, pitfalls, related reading, and a small FAQ. Educational estimates only.
Start here
A short pathway through the tools
Steepest ascent
Then read multivariable calculus.
Finite sums
Then read convergence tests.
2 by 2 toolkit
Then read Jacobians or eigenvalues.
Dot and cross
Useful next to Stokes theorem.
Bayes update
Then read Bayes theorem without the fog.
Calculus tools
Limits, slopes, and accumulation. Pair each tool with the matching explainer so the output has a sentence attached. The epsilon delta finder is the companion to the limit explainer. Series approximations are the Taylor polynomial calculator.
Epsilon delta finder
Search for a numerical delta given f, a point, a proposed limit, and epsilon. See epsilon delta intuition.
Numerical limit estimator
Sample f(x) as x approaches a or infinity. See epsilon delta intuition.
Numerical derivative
Forward, backward, or central differences. See what a derivative measures and implicit differentiation.
Gradient and directional derivative
Partials, steepest ascent, and the slope in a chosen direction. See multivariable and vector calculus.
Jacobian matrix
Symbolic partial derivatives, the Jacobian determinant, and the value at a point. See Jacobian matrices.
Divergence and curl
Source, sink, and rotation of a vector field at a point, with a picture of the arrows. See Stokes theorem intuition.
Riemann sum integrator
Left, midpoint, or right rectangles. See calculus foundations.
Partial fraction decomposition
Exact decomposition of a rational function, with division, factoring, and each algebra step. See calculus foundations.
Second order differential equations
Characteristic equation for a y'' + b y' + c y = 0, with exact steps and initial conditions. See differential equations.
Wronskian
Matrix and determinant for two or three functions of x, and a linear-independence reading. A zero at one point is not dependence. See differential equations.
Laplace transform
Forward and inverse transforms of common terms, with the table rule and each shift written out. See differential equations.
Integrating factor
Solve y' + P(x) y = Q(x), with μ(x), the integral, and an optional initial condition. A non-elementary integral is labelled as Runge-Kutta. See differential equations.
Quadratic formula solver
Roots and discriminant. Algebra that calculus still uses.
Matrices and vectors
Linear maps in small dimension. Good companions for the linear algebra pillar. The 3 by 3 inverse page keeps exact fractions and writes every minor. The 3x3 eigenvalue calculator expands the characteristic polynomial and row reduces for a real eigenvector. The row reduction page writes each elimination step the same way. The Gram Schmidt calculator builds an orthogonal basis, and the LU decomposition calculator saves the multipliers from elimination.
2 by 2 matrix toolkit
Determinant, inverse, and products AB or BA. See Jacobian matrices.
3 by 3 determinant and inverse
Cofactor expansion, the adjugate, and an exact fraction inverse when it exists. See linear algebra.
Reduced row echelon form
Gaussian elimination with each step in exact fractions, and the solution of a linear system. See linear algebra.
2 by 2 eigenvalues
Characteristic polynomial roots. See eigenvalues in plain language.
3x3 eigenvalues
Characteristic polynomial, eigenvalues, and a real eigenvector, with steps. See eigenvalues in plain language.
Vector operations
Dot product, cross product, magnitude, and unit vectors. See Stokes theorem intuition.
Gram Schmidt
Orthogonal vectors, unit vectors, and QR factors in exact fractions. See linear algebra.
LU decomposition
L and U factors from elimination, with optional row exchanges and a solution of A x = b. See linear algebra.
Probability and counting
Models you can type, not proofs. Update a belief with the Bayes theorem calculator, or read Bayes theorem without the fog first if the two directions of a conditional still feel the same.
Bayes theorem calculator
Posterior P(H|E) from a prior, a likelihood, and a false positive rate, or from natural frequencies. See Bayes theorem without the fog.
Binomial probability
n choose k and P(X equals k) for a binomial model.
Normal distribution calculator
Probability below a value, above a value, or between two values, from a mean and a standard deviation. Standard z scores and inverse percentiles too.
Combinatorics
Permutations, combinations, and factorials with overflow warnings.
Sequences
Finite arithmetic and geometric sums, and the Taylor polynomial that matches a function near a center. Infinite convergence needs a test, not a partial sum alone. See convergence tests for series.
Arithmetic and geometric sequences
Nth term and finite sum. Pair with sequences and series.
Taylor and Maclaurin polynomials
Degree-n polynomial with exact coefficients, a derivative table, and a Lagrange remainder bound. See sequences and series.
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