Math
1,470 math calculators: percentages, fractions, equations, geometry, trigonometry, statistics and number theory. Each one shows the method in plain words, so you get the answer and understand how it was reached, free in your browser.
Implicit Derivative of an Ellipse Calculator
Get the tangent slope of an ellipse at a point on it: with semi-axes 5 and 3 and the point (4, 1.8), the implicit derivative gives a slope of −0.8 and the…
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Concave up intervals calculator
For x³ − 3x the second derivative is 6x, so the curve is concave up exactly on (0, +∞). Enter the coefficients, highest power first, to get both derivatives and…
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Critical Points of a Polynomial Calculator
The critical points of x³ − 3x + 2 are −1 and 1, where its derivative 3x² − 3 equals zero. Enter the coefficients, highest power first, and get the derivative…
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Derivative sign at a point calculator
Find out whether a polynomial is rising or falling at a chosen spot: for 3x³ − 2x² + 5x − 7 where x is 2 the derivative is 33, so it is increasing.
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Increasing or Decreasing at a Point Calculator
Check the direction of a polynomial at a single x-value: 2x² − 3x + 1 has derivative 1 where x is 1, so it is increasing there.
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Inflection points of a polynomial calculator
Find where a curve switches between bending up and bending down: x⁵ − 5x³ + 4x has three inflection points, one at 0 and two placed symmetrically on either…
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Minimum Sum with a Fixed Product Optimization Calculator
For a fixed product of 36, the smallest possible sum of two positive numbers is 12, reached with 6 and 6.
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Rolle's Theorem Point Finder Calculator
For x^2 - 4x + 3 on [1, 3], both ends are 0 and the derivative vanishes at 2, the Rolle point. Enter any polynomial up to degree 12 and its interval; every…
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Secant line slope calculator for two points
The points (1, 3) and (5, 11) give a secant slope of 2: y rises 8 while x moves 4. Enter any two points on a curve to see the average rate of change between…
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Horizontal tangent points calculator
The curve x^3 - 3x flattens out at (-1, 2) and (1, -2), where its derivative 3x^2 - 3 is zero. Enter the coefficients of any polynomial and get every real point…
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Two-Point Gauss-Legendre Quadrature Calculator
Two well-chosen points are enough for any cubic: integrating x² from 0 to 1 with two-point Gauss-Legendre gives exactly one third, matching the exact integral…
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Improper integral of exponential calculator
The area under 1·e^(−2x) from 0 to infinity is exactly 0.5, because the integral of a·e^(−kx) over that range equals a/k whenever k is positive.
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Indefinite Integral of Polynomial Calculator
Integrate 3x³ − 4x + 7 in one step and get (3/4)x⁴ − 2x² + 7x + C: each coefficient is divided by its new exponent, with zeros keeping missing powers in place.
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Polynomial even/odd symmetry checker
Enter 1, 0, -3, 0, 2 and you learn that x^4 - 3x^2 + 2 is even, because every nonzero term sits on an even power.
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Simpson's three-eighths rule calculator for definite integrals
The integral of x^3 + 2*x from 0 to 3 comes out at 29.25 with 6 subintervals of 0.5. Enter a function, two bounds and a count divisible by three to apply…
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Accumulation Function Value Calculator
Accumulating 3x² − 2 from 1 to 3 gives exactly 22: the antiderivative is 21 at the top and −1 at the bottom.
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Center of Mass from Density Calculator
A rod from 0 to 3 with density 2x + 1 has a mass of 12, a first moment of 22.5 and balances at 1.875.
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Centroid X-Coordinate of a Region Calculator
The region under y = x from 0 to 2 has an area of 2 and balances at x = four thirds, right of the midpoint.
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Finite geometric series partial sum
Sum a geometric progression without adding term by term: first term 2, ratio 3 and five terms give 2 + 6 + 18 + 54 + 162 = 242, with 162 shown as the last term.
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Moment of a Region About the X-Axis Calculator
For the region under f(x) = x from 0 to 2, the moment about the x-axis is 4/3: the calculator squares the polynomial, integrates it exactly and halves the…
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Net Signed Area Calculator for Polynomials
Net signed area can cancel to nothing: x² − 3 from 0 to 3 gives exactly 0, because the part below the x-axis balances the part above.
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Radius of convergence from coefficients calculator
When each coefficient is half of the one before it, every ratio between neighbours is 2, so the radius of convergence is 2.
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Midpoint Riemann sum calculator
With four equal strips, the midpoint sum for x^2 from 0 to 1 is 0.328125, each strip 0.25 wide. Enter the interval, the number of strips and the polynomial's…
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Taylor Coefficients of Natural Log Calculator
Up to degree 6, ln(1+x) expands as x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6. Choose any degree up to 1000 and get each coefficient as an exact fraction and a…
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Euler Method ODE Solver Calculator
Euler's method in a readable table: y' = x + y from y(0) = 1 with step size 0.1 reaches about 1.5282 after four steps.
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P-series convergence checker
The sum of 1/n² converges because its exponent 2 is greater than 1, while the harmonic series 1/n, with exponent 1, diverges.
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Series Partial Sum Calculator
Adding the first 5 terms of 1/k^2 gives 1.463611111111. Type any term formula in k, say how many terms to add and where to start, and every addend is listed for…
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Sum of the first n cubes calculator
The first 10 cubes add up to 3025, the square of the triangular number 55. Enter any whole number n up to 13776 and get the exact total, the triangular number…
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Damped oscillator position calculator
With amplitude 1, decay rate 0.2, angular frequency 2 and phase 0, the position at time 1 is −0.340712212877 inside an envelope of 0.818730753078.
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ODE half-life from decay constant calculator
A decay constant of 0.05 per unit of time gives a half-life of 13.8629436112 units, because t½ = ln(2)/λ for dQ/dt = −λQ. The starting amount is not needed.
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Runge-Kutta 4 single-step calculator
From y(0) = 1, one step of 0.1 on y' = x + y gives 1.11034166666667, built from the slopes 1, 1.1, 1.105 and 1.2105.
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Separable Differential Equation Solver Calculator
For dy/dx = ky with a rate of 0.5 and y(0) = 2, the exact value at 3 is 8.96337814. Enter the constant, the known starting point and where you need the value…
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Series sum by partial fractions
The sum of 1/(k(k+1)) from 1 to 10 is exactly 10/11. Enter the upper index and the shifts p and q; the tool splits the term into partial fractions, cancels the…
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Euler Method for a 2D System Calculator
Follow two linked equations step by step: with dy/dx = y + z and dz/dx = x − y, starting from y(0) = 1 and z(0) = 0, four steps of 0.1 take y to 1.4 and z to…
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Fixed-point iteration single-step calculator
One fixed-point step, exactly as a homework table needs it: with g(x) = 0.5x + 1 and a current estimate of 0, the next estimate is 1 and the change is 1.
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Mixing tank concentration calculator
A 100-unit tank fed at rate 5 with brine of concentration 2, starting from pure water, reaches 0.7869386806 after 10 time units, on its way to the inlet value…
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Taylor Series Method ODE Calculator
From y(0) = 1, two order-4 Taylor steps of 0.1 carry y' = x + y to 0.2 and give 1.24280514170139. Choose an order from 1 to 4 and see each derivative term and…
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Natural cubic spline interpolation calculator
Through four points alternating between 0 and 1, the natural cubic spline gives 0.5 when x is 1.5.
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Inverse interpolation calculator
Read a table backwards: with points (2, 4) and (3, 8) in your data, a target of 6 gives x of 2.5 by straight-line interpolation.
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Least squares residuals calculator
See how far your data sit from a line you already have: for four points checked against slope 1 and intercept 1, the residual sum of squares is 0.09 and the…
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Least-squares regression prediction calculator
Fit a straight line and read off a forecast: for the pairs (1, 3), (2, 5), (3, 7) and (4, 9) the least-squares line has slope 2 and intercept 1, so for a new x…
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Newton Backward Difference Interpolation Calculator
For the table (0, 1), (1, 2), (2, 5), (3, 10), Newton's backward formula estimates 7.25 at x of 2.5, adding 10, −2.5, −0.25 and 0 term by term.
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Newton-Raphson Single Step Calculator
One Newton-Raphson step, shown in full: for x² − 2 starting at 1.5, the function value is 0.25, the derivative is 3, the correction is 0.0833333333 and the next…
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Steffensen method root finder calculator
Starting from 1, Steffensen's method finds the root of x^2 - 2 at 1.4142135623731 in 7 updates, with no derivative needed.
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Double Integral Midpoint Rule Calculator
Approximate a double integral over a rectangle by sampling the center of every grid cell: x^2 + y^2 for x from 0 to 2 and y from 0 to 1, on a 2 by 2 grid, gives…
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Gradient magnitude calculator
Turn two partial derivatives into one steepness number: with ∂f/∂x of 3 and ∂f/∂y of 4, the gradient magnitude is 5 and its square is 25.
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Hessian determinant calculator
Classify a critical point in one go: for x^2 + 3xy + 2y^2 at (0, 0) the Hessian determinant is −1, so the point is a saddle.
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Normal line to surface calculator
On the sphere x² + y² + z² at the point (1, 2, 2), the gradient is (2, 4, 4), so the normal line is r(t) = (1, 2, 2) + t·(2, 4, 4) and the unit normal is (1, 2…
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Numerical Partial Derivative with Respect to x Calculator
At the point (2, 3), the function x^2*y + sin(x*y) changes at a rate of about 14.8805108600492 in the x direction.
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Balanced ternary calculator
42 becomes +---0 in balanced ternary, a base-three system whose digits are −1, 0 and +1. The calculator shows each power of three it uses, counts the trits and…
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Integer parity checker
Count the ones in a number's binary form and get its parity: 13 is 1101, with three ones, so its parity is odd and the parity bit is 1.
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Fast modular exponentiation calculator
Get the exact remainder of a huge power without overflow: 2 to the power 10 modulo 1000 is 24, and 3 to the power 5 modulo 13 is 9.
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Edge list to adjacency matrix converter
Turn an edge list into an adjacency matrix with the node order spelled out: the three lines A B, B C and C A give a symmetric 3 by 3 matrix with zeros on the…
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Graph vertex degree calculator
Paste an edge list and get each vertex's degree: in the triangle A→B, B→C, C→A every vertex has in-degree 1, out-degree 1 and degree 2, and the degrees add up…
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Divisibility test
Is 252 divisible by 7? Yes: it fits 36 times with a remainder of 0. Enter any two whole numbers, negatives included, and get a clear yes or no together with the…
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Divisor calculator
28 has exactly six divisors, and you get every one of them in ascending order, from 1 to 28 itself, together with the count.
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Regular dodecagon
A regular dodecagon with a side of 4 has an area of 179.138439 square units. From the side alone you get the area, 3·(2+√3)·s², the perimeter, 12·s, or the…
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Double-angle formula
Enter an angle in degrees or radians, pick sine, cosine or tangent, and get the value for the doubled angle with its formula: 30° with the cosine identity gives…
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Double-angle identities
One angle in radians, all three double-angle values at once: for π/6 (0.5235987756) you get sin(2x) ≈ 0.8660254038, cos(2x) of 0.5 and tan(2x) ≈ 1.7320508076…
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e^x calculator
e squared is 7.38905609893065: type one exponent and get e raised to it, where e ≈ 2.718281828459045.
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