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➗ MathFree
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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➗ MathFree
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…
K
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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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…
F
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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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.
B
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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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…
B
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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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…
A
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➗ MathFree
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…
B
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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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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➗ MathFree
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…
F
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➗ MathFree
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.
K
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➗ MathFree
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…
F
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➗ MathFree
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.
S
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➗ MathFree
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.
K
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➗ MathFree
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…
A
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➗ MathFree
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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🖼️ ImagesFree
RGB to XYZ converter
Get CIE 1931 XYZ values from an sRGB color: orange 255, 128, 0 gives X 48.96429558, Y 36.70460115 and Z 4.50627447 on the 0 to 100 scale.
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🖼️ ImagesFree
LAB to RGB converter
The LAB color lightness 53.2407941413, a 80.0924595964, b 67.2031965159 comes back as rgb(255, 0, 0), pure red #FF0000.
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🖼️ ImagesFree
RGB to LCH converter
Describe an RGB color the way people adjust it: lightness from 0 for black to about 100 for white, chroma as colorfulness and hue as an angle from 0 up to 360°.
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🖼️ ImagesFree
RGB to OKLCH converter
Get the OKLCH coordinates of an RGB color: 255, 87, 51 becomes lightness 0.680358, chroma 0.210012 and hue 33.691637°.
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🖼️ ImagesFree
RGB to YUV converter
Split an RGB color into brightness and two color differences: orange 255, 128, 0 gives Y 0.5936509804, U −0.2921263922 and V 0.3564952157 with BT.601 weights.
K
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🖼️ ImagesFree
YIQ to RGB converter
Turn Y, I and Q into a screen color: 0.5, 0.5 and 0.5 give red 255, green 10 and blue 204, or #ff0acc, and you are told that red was clipped from 1.2885.
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🖼️ ImagesFree
CMYK to HSV converter
Turn a print ink recipe into hue, saturation and value: C0 M50 Y100 K0 becomes RGB 255, 128, 0, an orange with 100 % saturation and 100 % value.
B
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🖼️ ImagesFree
HSV to HSL converter
Move a color from HSV to HSL without guessing: hue 210, saturation 70% and value 80% become lightness 52% on the same hue, with saturation recalculated for HSL.
A
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🖼️ ImagesFree
HWB to HSL converter
Hue 210 with 20 % whiteness and 10 % blackness becomes hsl(210 77.78% 55%), with the hue kept exactly as you entered it.
K
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🖼️ ImagesFree
RGB to LUV converter
Get the CIE 1976 Luv coordinates of an RGB color: lightness L* from 0 for black to about 100 for white, plus u* and v* measured from the D65 white.
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🖼️ ImagesFree
XYZ to xyY converter
Split a color into chromaticity and brightness: X 41.24, Y 21.26 and Z 1.93 become a pair of x and y coordinates for the CIE 1931 diagram, with luminance Y kept…
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🖼️ ImagesFree
Expand shorthand hex color
Turn shorthand into the long form browsers use: #f0a becomes #ff00aa because each digit is doubled, and you also get #FF00AA in capitals and the channels 255, 0…
S
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🖼️ ImagesFree
Hex8 to RGBA converter
Read a transparent hex color channel by channel: #33669980 splits into red 51, green 102, blue 153 and alpha 128, which is 0.501961 on a 0 to 1 opacity scale.
B
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🖼️ ImagesFree
Hex to CMYK converter
Get the process-ink mix behind a screen color: #3498db becomes cyan 76%, magenta 31%, yellow 0% and key 14%, worked out with the standard subtractive formula.
S
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