calculus-ai999x — offline, verified calculus solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Mathematics & Statistics · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

calculus-ai999x is an offline, deterministic Python solver that answers calculus questions with verified results from 71 certified capabilities, gives 374 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

calculus-ai999x is a compact, deterministic calculus engine for Python. It reads a question in plain English (or a typed call), picks a certified calculus capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified calculus capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 374 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: limits, derivatives, integrals, series, Taylor polynomials, multivariable and vector calculus, special functions and numerical differentiation and integration.

Model overview

Property	Value
Package	calculus-ai999x 1.0.0 (import calculus_ai999x)
Type	Deterministic neuro-symbolic calculus engine (rule-based reading + certified symbolic capabilities)
Capabilities	71 certified procedures
Knowledge pack	version 1.0.0
Golden answers	374 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install calculus-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/calculus-ai999x/resolve/main/dist/calculus_ai999x-1.0.0-py3-none-any.whl

import calculus_ai999x as calculus

calculus.ask("Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.").show()
# -> SOLVED — limit = 4  (limit_at_point)
calculus.ask("Find the absolute maximum and minimum of x^3 - 3x on [0, 2].").show()
# -> SOLVED — absolute_max = 2, absolute_min = -2, critical_points = [1.0]  (extrema_on_interval)
calculus.ask("Find the Taylor polynomial of degree 3 of exp(x) about x = 0 and evaluate it at x = 0.5.").show()
# -> SOLVED — coefficients = [1.0, 1.0, 0.5, 0.16666666666666666], value = 1.64583, function_value = 1.64872, error = 0.00288794  (taylor_polynomial)
calculus.ask("Find the line integral of F = (-y, x) along x = cos(t), y = sin(t) from t = 0 to t = 2pi.").show()
# -> SOLVED — line_integral = 6.28319  (line_integral_vector)
calculus.solve("definite_integral", f="x^2*exp(-x)", a=0, b=2).show()
# -> SOLVED — integral = 0.646647  (definite_integral)
calculus.ask("Who created you?").message
# -> calculus-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: calculus-ai999x "<question>", calculus-ai999x --capabilities. The outputs above were produced by running this exact code on calculus-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

71 certified procedures. A few of them:

• Procedures: Absolute extrema on a closed interval; Arc length of a graph; Arc length of a parametric curve; Area between two curves; Area enclosed by two curves; Area in polar coordinates; Average rate of change; Average value of a function; Bessel function of the first kind; Beta function; Centroid of a region under a curve; Chebyshev polynomial; Critical points and their classification; Curl; Curvature of a graph; Definite integral; Derivative at a point; Derivative of an integral (fundamental theorem); Directional derivative; Divergence; Double integral over a rectangle; Double integral over a region; Error function; Finite-difference derivative …

calculus_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 71 formulas

Show every formula, by topic

Fourier series

Fourier coefficients

$$a_k=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos \frac{k\pi x}{L}dx,b_k=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin \frac{k\pi x}{L}dxa\_k\; =\; \backslash frac\{1\}\{L\}\backslash int\_\{-L\}^\{L\}\; f(x)\backslash cos\backslash frac\{k\backslash pi\; x\}\{L\}dx,\backslash quad\; b\_k\; =\; \backslash frac\{1\}\{L\}\backslash int\_\{-L\}^\{L\}\; f(x)\backslash sin\backslash frac\{k\backslash pi\; x\}\{L\}dx$$

Derivatives

Derivative at a point

$$f^{\mathrm{\prime }}(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}f\text{'}(a)\; =\; \backslash lim\_\{h\backslash to\; 0\}\backslash frac\{f(a+h)-f(a)\}\{h\}$$

Higher derivative at a point

$$f^{(n)}(a)=\frac{d^{n}f}{d{x}^n}{\mid }_{x=a}f^\{(n)\}(a)\; =\; \backslash frac\{d^n\; f\}\{dx^n\}\backslash Big\backslash vert\; \_\{x=a\}$$

Implicit differentiation

$$\frac{dy}{dx}=-\frac{F_x}{F_y}\backslash frac\{dy\}\{dx\}\; =\; -\backslash frac\{F\_x\}\{F\_y\}$$

Mean value theorem

$$f^{\mathrm{\prime }}(c)=\frac{f(b)-f(a)}{b-a}f\text{'}(c)\; =\; \backslash frac\{f(b)\; -\; f(a)\}\{b\; -\; a\}$$

Second derivative at a point

$$f^{\mathrm{\prime }\mathrm{\prime }}(a)=\frac{d^{2}f}{d{x}^2}{\mid }_{x=a}f\text{'}\text{'}(a)\; =\; \backslash frac\{d^2\; f\}\{dx^2\}\backslash Big\backslash vert\; \_\{x=a\}$$

Derivatives / concavity

Inflection points

$$f^{\mathrm{\prime }\mathrm{\prime }}(c)=0\text{ and }{f}^{\mathrm{\prime }\mathrm{\prime }}\text{ changes sign at }cf\text{'}\text{'}(c)\; =\; 0\; \backslash text\{\; and\; \}\; f\text{'}\text{'}\; \backslash text\{\; changes\; sign\; at\; \}\; c$$

Derivatives / curvature

Curvature of a graph

$$\kappa =\frac{\mathrm{\mid }{f}^{\mathrm{\prime }\mathrm{\prime }}(a)\mathrm{\mid }}{{(1+f^{\mathrm{\prime }}(a{)}^2)}^{3\mathrm{/}2}}\backslash kappa\; =\; \backslash frac\{\backslash vert\; f\text{'}\text{'}(a)\backslash vert\; \}\{\backslash left(1\; +\; f\text{'}(a)^2\backslash right)^\{3/2\}\}$$

Derivatives / linearization

Linear approximation

$$L(x)=f(a)+f^{\mathrm{\prime }}(a)(x-a)L(x)\; =\; f(a)\; +\; f\text{'}(a)\backslash ,(x-a)$$

Derivatives / motion

Velocity and acceleration from position

$$v=\frac{ds}{dt},a=\frac{d^{2}s}{d{t}^2}v\; =\; \backslash frac\{ds\}\{dt\},\backslash quad\; a\; =\; \backslash frac\{d^2\; s\}\{dt^2\}$$

Derivatives / optimization

Absolute extrema on a closed interval

$$\underset{[a,b]}{\max }f=\max \{f(a),f(b),f(c):f^{\mathrm{\prime }}(c)=0\}\backslash max\_\{[a,b]\}\; f\; =\; \backslash max\backslash \{f(a),\; f(b),\; f(c)\; :\; f\text{'}(c)\; =\; 0\backslash \}$$

Critical points and their classification

$$f^{\mathrm{\prime }}(c)=0,f^{\mathrm{\prime }}\text{ changes }+\to -\Rightarrow \text{local max}f\text{'}(c)\; =\; 0,\backslash \; f\text{'}\; \backslash text\{\; changes\; \}\; +\; \backslash to\; -\; \backslash Rightarrow\; \backslash text\{local\; max\}$$

Derivatives / rates of change

Average rate of change

$$\frac{\mathrm{\Delta }f}{\mathrm{\Delta }x}=\frac{f(b)-f(a)}{b-a}\backslash frac\{\backslash Delta\; f\}\{\backslash Delta\; x\}\; =\; \backslash frac\{f(b)-f(a)\}\{b-a\}$$

Derivatives / tangent and normal lines

Normal line

$$y=f(a)-\frac{1}{f^{\mathrm{\prime }}(a)}(x-a)y\; =\; f(a)\; -\; \backslash frac\{1\}\{f\text{'}(a)\}\backslash ,(x\; -\; a)$$

Tangent line

$$y=f(a)+f^{\mathrm{\prime }}(a)(x-a)y\; =\; f(a)\; +\; f\text{'}(a)\backslash ,(x\; -\; a)$$

Integrals

Definite integral

$${\int }_a^{b}f(x)dx\backslash int\_a^b\; f(x)\backslash ,dx$$

Derivative of an integral (fundamental theorem)

$$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\mathrm{\prime }}(x)\backslash frac\{d\}\{dx\}\backslash int\_a^\{g(x)\}\; f(t)\backslash ,dt\; =\; f(g(x))\backslash ,g\text{'}(x)$$

Integrals / Riemann sums

Left Riemann sum

$$L_n=h\sum _{k=0}^{n-1}f(a+kh),h=\frac{b-a}{n}L\_n\; =\; h\backslash sum\_\{k=0\}^\{n-1\}\; f(a\; +\; k\; h),\backslash quad\; h\; =\; \backslash frac\{b-a\}\{n\}$$

Right Riemann sum

$$R_n=h\sum _{k=1}^{n}f(a+kh),h=\frac{b-a}{n}R\_n\; =\; h\backslash sum\_\{k=1\}^\{n\}\; f(a\; +\; k\; h),\backslash quad\; h\; =\; \backslash frac\{b-a\}\{n\}$$

Integrals / applications

Average value of a function

$$f_{\text{avg}}=\frac{1}{b-a}{\int }_a^{b}f(x)dxf\_\{\backslash text\{avg\}\}\; =\; \backslash frac\{1\}\{b-a\}\backslash int\_a^b\; f(x)\backslash ,dx$$

Integrals / arc length

Arc length of a graph

$$L={\int }_a^b\sqrt{1+f^{\mathrm{\prime }}(x{)}^2}dxL\; =\; \backslash int\_a^b\; \backslash sqrt\{1\; +\; f\text{'}(x)^2\}\backslash ,dx$$

Integrals / area

Area between two curves

$$A={\int }_a^b\mid f(x)-g(x)\mid dxA\; =\; \backslash int\_a^b\; \backslash left\backslash vert\; f(x)\; -\; g(x)\backslash right\backslash vert\; \backslash ,dx$$

Area enclosed by two curves

$$A={\int }_{x_1}^{x_n}\mid f(x)-g(x)\mid dxA\; =\; \backslash int\_\{x\_1\}^\{x\_n\}\; \backslash left\backslash vert\; f(x)\; -\; g(x)\backslash right\backslash vert\; \backslash ,dx$$

Integrals / centroids

Centroid of a region under a curve

$$\bar{x}=\frac{1}{A}{\int }_a^{b}xf(x)dx,\bar{y}=\frac{1}{2A}{\int }_a^{b}f(x{)}^{2}dx\backslash bar\; x\; =\; \backslash frac\{1\}\{A\}\backslash int\_a^b\; x\; f(x)\backslash ,dx,\backslash quad\; \backslash bar\; y\; =\; \backslash frac\{1\}\{2A\}\backslash int\_a^b\; f(x)^2\backslash ,dx$$

Integrals / improper integrals

Improper integral to infinity

$${\int }_a^{\mathrm{\infty }}f(x)dx=\underset{R\to \mathrm{\infty }}{\lim }{\int }_a^{R}f(x)dx\backslash int\_a^\backslash infty\; f(x)\backslash ,dx\; =\; \backslash lim\_\{R\backslash to\backslash infty\}\backslash int\_a^R\; f(x)\backslash ,dx$$

Integrals / surface area

Surface area of revolution

$$S=2\pi {\int }_a^b\mathrm{\mid }f(x)\mathrm{\mid }\sqrt{1+f^{\mathrm{\prime }}(x{)}^2}dxS\; =\; 2\backslash pi\backslash int\_a^b\; \backslash vert\; f(x)\backslash vert\; \backslash sqrt\{1\; +\; f\text{'}(x)^2\}\backslash ,dx$$

Integrals / volumes of revolution

Volume of revolution (disk method)

$$V=\pi {\int }_a^{b}f(x{)}^{2}dxV\; =\; \backslash pi\backslash int\_a^b\; f(x)^2\backslash ,dx$$

Volume of revolution (shell method)

$$V=2\pi {\int }_a^{b}xf(x)dxV\; =\; 2\backslash pi\backslash int\_a^b\; x\backslash ,f(x)\backslash ,dx$$

Volume of revolution (washer method)

$$V=\pi {\int }_a^b(f(x{)}^2-g(x{)}^2)dxV\; =\; \backslash pi\backslash int\_a^b\; \backslash left(f(x)^2\; -\; g(x)^2\backslash right)dx$$

Limits

Limit at a point

$$\underset{x\to a}{\lim }f(x)\backslash lim\_\{x\backslash to\; a\}\; f(x)$$

Limit at infinity

$$\underset{x\to \mathrm{\infty }}{\lim }f(x)\backslash lim\_\{x\backslash to\backslash infty\}\; f(x)$$

Multiple integrals

Double integral over a rectangle

$${\int }_a^b\text{\hspace{0.17em}⁣}{\int }_c^{d}f(x,y)dydx\backslash int\_a^b\backslash !\backslash int\_c^d\; f(x,y)\backslash ,dy\backslash ,dx$$

Double integral over a region

$${\int }_a^b\text{\hspace{0.17em}⁣}{\int }_{g_1(x)}^{g_2(x)}f(x,y)dydx\backslash int\_a^b\backslash !\backslash int\_\{g\_1(x)\}^\{g\_2(x)\}\; f(x,y)\backslash ,dy\backslash ,dx$$

Surface area of a graph

$$A={\iint }_R\sqrt{1+f_x^2+f_y^2}dAA\; =\; \backslash iint\_R\; \backslash sqrt\{1\; +\; f\_x^2\; +\; f\_y^2\}\backslash ,dA$$

Triple integral over a box

$${\iiint }_{B}f(x,y,z)dV\backslash iiint\_B\; f(x,y,z)\backslash ,dV$$

Multivariable calculus

Directional derivative

$$D_{\mathbf{u}}f=\mathrm{\nabla }f\cdot \frac{\mathbf{u}}{\mathrm{\mid }\mathbf{u}\mathrm{\mid }}D\_\{\backslash mathbf\; u\}\; f\; =\; \backslash nabla\; f\backslash cdot\backslash frac\{\backslash mathbf\; u\}\{\backslash vert\; \backslash mathbf\; u\backslash vert\; \}$$

Gradient

$$\mathrm{\nabla }f=(f_x,f_y,f_z)\backslash nabla\; f\; =\; \backslash left(f\_x,\; f\_y,\; f\_z\backslash right)$$

Partial derivative

$$\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(p)\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}(p)$$

Tangent plane

$$z=f(x_0,y_0)+f_x(x-x_0)+f_y(y-y_0)z\; =\; f(x\_0,y\_0)\; +\; f\_x(x\; -\; x\_0)\; +\; f\_y(y\; -\; y\_0)$$

Multivariable calculus / change of variables

Jacobian determinant

$$\frac{\mathrm{\partial }(x,y)}{\mathrm{\partial }(u,v)}=x_u{y}_v-x_v{y}_u\backslash frac\{\backslash partial(x,y)\}\{\backslash partial(u,v)\}\; =\; x\_u\; y\_v\; -\; x\_v\; y\_u$$

Multivariable calculus / optimization

Second derivative test (two variables)

$$D=f_{xx}{f}_{yy}-f_{xy}^{2}D\; =\; f\_\{xx\}f\_\{yy\}\; -\; f\_\{xy\}^2$$

Numerical differentiation

Finite-difference derivative

$$f^{\mathrm{\prime }}(a)\approx \frac{f(a+h)-f(a-h)}{2h}f\text{'}(a)\; \backslash approx\; \backslash frac\{f(a+h)\; -\; f(a-h)\}\{2h\}$$

Numerical integration

Gauss-Legendre quadrature

$${\int }_a^{b}fdx\approx \frac{b-a}{2}\sum _{i=1}^n{w}_{i}f\text{\hspace{0.17em}⁣}(\frac{a+b}{2}+\frac{b-a}{2}{x}_i)\backslash int\_a^b\; f\backslash ,dx\; \backslash approx\; \backslash frac\{b-a\}\{2\}\backslash sum\_\{i=1\}^n\; w\_i\backslash ,\; f\backslash !\backslash left(\backslash frac\{a+b\}\{2\}\; +\; \backslash frac\{b-a\}\{2\}x\_i\backslash right)$$

Midpoint rule

$$M_n=h\sum _{k=0}^{n-1}f(a+(k+\frac{1}{2})h)M\_n\; =\; h\backslash sum\_\{k=0\}^\{n-1\}\; f\backslash left(a\; +\; (k\; +\; \backslash tfrac12)\; h\backslash right)$$

Simpson's rule

$$S_n=\frac{h}{3}[f(x_0)+4\sum _{\text{odd }k}f(x_k)+2\sum _{\text{even }k}f(x_k)+f(x_n)]S\_n\; =\; \backslash frac\{h\}\{3\}\backslash left[f(x\_0)\; +\; 4\backslash sum\_\{\backslash text\{odd\; \}\; k\}\; f(x\_k)\; +\; 2\backslash sum\_\{\backslash text\{even\; \}\; k\}\; f(x\_k)\; +\; f(x\_n)\backslash right]$$

Trapezoidal rule

$$T_n=\frac{h}{2}[f(x_0)+2\sum _{k=1}^{n-1}f(x_k)+f(x_n)]T\_n\; =\; \backslash frac\{h\}\{2\}\backslash left[f(x\_0)\; +\; 2\backslash sum\_\{k=1\}^\{n-1\}\; f(x\_k)\; +\; f(x\_n)\backslash right]$$

Numerical methods / root finding

Newton's method

$$x_{k+1}=x_k-\frac{f(x_k)}{f^{\mathrm{\prime }}(x_k)}x\_\{k+1\}\; =\; x\_k\; -\; \backslash frac\{f(x\_k)\}\{f\text{'}(x\_k)\}$$

Parametric curves

Arc length of a parametric curve

$$L={\int }_{t_0}^{t_1}\sqrt{x^{\mathrm{\prime }}(t{)}^2+y^{\mathrm{\prime }}(t{)}^2}dtL\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\backslash sqrt\{x\text{'}(t)^2\; +\; y\text{'}(t)^2\}\backslash ,dt$$

Slope and concavity of a parametric curve

$$\frac{dy}{dx}=\frac{y^{\mathrm{\prime }}(t)}{x^{\mathrm{\prime }}(t)},\frac{d^{2}y}{d{x}^2}=\frac{\frac{d}{dt}\frac{dy}{dx}}{x^{\mathrm{\prime }}(t)}\backslash frac\{dy\}\{dx\}\; =\; \backslash frac\{y\text{'}(t)\}\{x\text{'}(t)\},\backslash quad\; \backslash frac\{d^2y\}\{dx^2\}\; =\; \backslash frac\{\backslash frac\{d\}\{dt\}\backslash frac\{dy\}\{dx\}\}\{x\text{'}(t)\}$$

Polar curves

Area in polar coordinates

$$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta A\; =\; \backslash frac\{1\}\{2\}\backslash int\_\{\backslash alpha\}^\{\backslash beta\}\; r(\backslash theta)^2\backslash ,d\backslash theta$$

Series

Partial sum of a geometric series

$$S_n=a\frac{1-r^n}{1-r}S\_n\; =\; a\backslash ,\backslash frac\{1\; -\; r^n\}\{1\; -\; r\}$$

Ratio test

$$L=\underset{n\to \mathrm{\infty }}{\lim }\mid \frac{a_{n+1}}{a_n}\mid L\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left\backslash vert\; \backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right\backslash vert$$

Sum of an arithmetic series

$$S_n=\frac{n}{2}(2a_1+(n-1)d)S\_n\; =\; \backslash frac\{n\}\{2\}\backslash left(2a\_1\; +\; (n-1)d\backslash right)$$

Sum of an infinite geometric series

$$\sum _{n=0}^{\mathrm{\infty }}a{r}^n=\frac{a}{1-r}\backslash sum\_\{n=0\}^\{\backslash infty\}\; a\; r^n\; =\; \backslash frac\{a\}\{1-r\}$$

Sum of an infinite series

$$\sum _{n=k}^{\mathrm{\infty }}{a}_n\backslash sum\_\{n=k\}^\{\backslash infty\}\; a\_n$$

Series / Taylor

Taylor polynomial

$$P_n(x)=\sum _{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^{k}P\_n(x)\; =\; \backslash sum\_\{k=0\}^\{n\}\backslash frac\{f^\{(k)\}(a)\}\{k!\}(x-a)^k$$

Special functions

Bessel function of the first kind

$$J_n(x)=\sum _{m=0}^{\mathrm{\infty }}\frac{(-1{)}^m}{m!(m+n)!}{\left(\frac{x}{2}\right)}^{2m+n}J\_n(x)\; =\; \backslash sum\_\{m=0\}^\{\backslash infty\}\backslash frac\{(-1)^m\}\{m!\backslash ,(m+n)!\}\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\{2m+n\}$$

Beta function

$$B(a,b)={\int }_0^1{t}^{a-1}(1-t{)}^{b-1}dt=\frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(a+b)}B(a,b)\; =\; \backslash int\_0^1\; t^\{a-1\}(1-t)^\{b-1\}dt\; =\; \backslash frac\{\backslash Gamma(a)\backslash Gamma(b)\}\{\backslash Gamma(a+b)\}$$

Error function

$$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt\backslash operatorname\{erf\}(x)\; =\; \backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\backslash int\_0^x\; e^\{-t^2\}\backslash ,dt$$

Gamma function

$$\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt\backslash Gamma(x)\; =\; \backslash int\_0^\backslash infty\; t^\{x-1\}e^\{-t\}\backslash ,dt$$

Regularized incomplete gamma functions

$$P(a,x)=\frac{1}{\mathrm{\Gamma }(a)}{\int }_0^x{t}^{a-1}{e}^{-t}dtP(a,x)\; =\; \backslash frac\{1\}\{\backslash Gamma(a)\}\backslash int\_0^x\; t^\{a-1\}e^\{-t\}\backslash ,dt$$

Special functions / orthogonal polynomials

Chebyshev polynomial

$$T_{k+1}(x)=2x{T}_k(x)-T_{k-1}(x)T\_\{k+1\}(x)\; =\; 2x\backslash ,T\_k(x)\; -\; T\_\{k-1\}(x)$$

Legendre polynomial

$$(k+1)P_{k+1}(x)=(2k+1)x{P}_k(x)-k{P}_{k-1}(x)(k+1)P\_\{k+1\}(x)\; =\; (2k+1)\backslash ,x\backslash ,P\_k(x)\; -\; k\backslash ,P\_\{k-1\}(x)$$

Special functions / series

Riemann zeta function

$$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\backslash zeta(s)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{n^s\}$$

Vector calculus

Curl

$$\mathrm{\nabla }\times \mathbf{F}=(R_y-Q_z,P_z-R_x,Q_x-P_y)\backslash nabla\backslash times\backslash mathbf\; F\; =\; (R\_y\; -\; Q\_z,\backslash \; P\_z\; -\; R\_x,\backslash \; Q\_x\; -\; P\_y)$$

Divergence

$$\mathrm{\nabla }\cdot \mathbf{F}=P_x+Q_y+R_z\backslash nabla\backslash cdot\backslash mathbf\; F\; =\; P\_x\; +\; Q\_y\; +\; R\_z$$

Green's theorem on a rectangle

$${\oint }_{C}Pdx+Qdy={\iint }_R(Q_x-P_y)dA\backslash oint\_C\; P\backslash ,dx\; +\; Q\backslash ,dy\; =\; \backslash iint\_R\; \backslash left(Q\_x\; -\; P\_y\backslash right)dA$$

Laplacian

$${\mathrm{\nabla }}^{2}f=f_{xx}+f_{yy}+f_{zz}\backslash nabla^2\; f\; =\; f\_\{xx\}\; +\; f\_\{yy\}\; +\; f\_\{zz\}$$

Line integral of a vector field

$${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_{t_0}^{t_1}(P{x}^{\mathrm{\prime }}(t)+Q{y}^{\mathrm{\prime }}(t))dt\backslash int\_C\; \backslash mathbf\; F\backslash cdot\; d\backslash mathbf\; r\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; \backslash left(P\backslash ,x\text{'}(t)\; +\; Q\backslash ,y\text{'}(t)\backslash right)dt$$

Line integral with respect to arc length

$${\int }_{C}fds={\int }_{t_0}^{t_1}f(x(t),y(t))\sqrt{x^{\mathrm{\prime }}(t{)}^2+y^{\mathrm{\prime }}(t{)}^2}dt\backslash int\_C\; f\backslash ,ds\; =\; \backslash int\_\{t\_0\}^\{t\_1\}\; f(x(t),y(t))\backslash sqrt\{x\text{'}(t)^2\; +\; y\text{'}(t)^2\}\backslash ,dt$$

Outward flux through a box (divergence theorem)

$${∯}_S\mathbf{F}\cdot \mathbf{n}dS={\iiint }_B\mathrm{\nabla }\cdot \mathbf{F}dV\backslash oiint\_S\; \backslash mathbf\; F\backslash cdot\backslash mathbf\; n\backslash ,dS\; =\; \backslash iiint\_B\; \backslash nabla\backslash cdot\backslash mathbf\; F\backslash ,dV$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified calculus capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (limits, derivatives, integrals, series, Taylor polynomials, multivariable and vector calculus, special functions and numerical differentiation and integration); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{calculus_ai999x_1_0_0,
  title  = {calculus-ai999x: an offline, verified calculus solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/calculus-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/calculus-ai999x/discussions or visit https://ai999x.com.

Support the project

calculus-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com