Math calculators

Numerical calculus

Derivative of f(x) at a point, definite integrals and roots of equations, with method and error estimate.

Inputs

f(x)For example x^3 - 2x - 5, sin(x)/x or exp(-x^2); angles in radians

From aTo b

Results

f(x)

Methodology

Conventions

Functions of x are written like the scientific calculator's expressions, for example x^2 * sin(x). Angles are in radians.
The derivative uses Ridders' extrapolation of central differences; the integral uses adaptive Simpson's rule; the root uses Brent's method on a bracket where f changes sign.
Each result carries an error estimate. Numerical methods can fail on discontinuities and singularities.

References

Press, Teukolsky, Vetterling and Flannery, Numerical Recipes, 3rd ed. (2007), §5.7, §4.7 and §9.3
Brent, Algorithms for Minimization without Derivatives (1973)

Formulas

Central difference: f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}
Simpson: \int_a^b f \approx \frac{b - a}{6}\left[f(a) + 4f\left(\tfrac{a + b}{2}\right) + f(b)\right]
Bracket: f(a) \cdot f(b) < 0