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Calculus

calculus-ai999x

Limits, derivatives, integrals, series, Taylor polynomials, multivariable and vector calculus, special functions and numerical differentiation and integration.

Mathematics & StatisticsQuestion AnsweringToken ClassificationText ClassificationSentence SimilarityFeature Extraction
Hugging Face ↗PyPI ↗GitHub ↗See it on the capability orb
71capabilities
14topics
71formulas typeset
8worked examples below
1.0.0version
About this model

What Calculus does

Calculus (calculus-ai999x) is an offline Python package for calculus. It covers limits, derivatives, integrals, series, Taylor polynomials, multivariable and vector calculus, special functions and numerical differentiation and integration.

It has 71 capabilities — 71 computational procedures — in 14 topics, the largest being Derivatives, Integrals, Special functions, Vector calculus, Multivariable calculus and Series. Ask a question in words with ask(), or call a capability by name with typed inputs with solve(). Every answer carries its value and unit, the steps that produced it and the formula in LaTeX.

When a question is not determined — a value is missing, an input is out of range, or the request is outside the package — it says so and names what it needs, instead of guessing. It runs on an ordinary CPU without a network connection (Python >=3.10), and the same question always gets the same answer.

Install

pip install calculus-ai999x
hf download AI999X/calculus-ai999x --local-dir calculus-ai999x

What it covers

Derivatives14Absolute extrema on a closed interval · Average rate of change · Critical points and their classification …Fourier series1Fourier coefficientsIntegrals14Arc length of a graph · Area between two curves · Area enclosed by two curves …Limits2Limit at a point · Limit at infinityMultiple integrals4Double integral over a rectangle · Double integral over a region · Surface area of a graph …Multivariable calculus6Directional derivative · Gradient · Jacobian determinant …Numerical differentiation1Finite-difference derivativeNumerical integration4Gauss-Legendre quadrature · Midpoint rule · Simpson's rule …Numerical methods1Newton's methodParametric curves2Arc length of a parametric curve · Slope and concavity of a parametric curvePolar curves1Area in polar coordinatesSeries6Partial sum of a geometric series · Ratio test · Sum of an arithmetic series …Special functions8Bessel function of the first kind · Beta function · Chebyshev polynomial …Vector calculus7Curl · Divergence · Green's theorem on a rectangle …
Try it

Worked examples

Each question below was put to calculus-ai999x and the answer shown is the one it returned, with the steps it took.

Limitsm.ask(…)

Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.

limit = 4
Show the steps
  1. Limit at a point
    \lim_{x\to a} f(x)
    with a = 2, f = (x^2 - 4)/(x - 2)
    → limit = 4
Capability: Limit at a point
Derivativesm.ask(…)

Find the absolute maximum and minimum of x^3 - 3x on [0, 2].

absolute_max = 2
absolute_min = -2
critical_points = [1]
Show the steps
  1. Absolute extrema on a closed interval
    \max_{[a,b]} f = \max\{f(a), f(b), f(c) : f'(c) = 0\}
    with a = 0, b = 2, f = x^3 - 3x
    → absolute_max = 2, absolute_min = -2, critical_points = [1]
Capability: Absolute extrema on a closed interval
Seriesm.ask(…)

Find the Taylor polynomial of degree 3 of exp(x) about x = 0 and evaluate it at x = 0.5.

coefficients = [1, 1, 0.5, 0.1666666667]
value = 1.645833333
function_value = 1.648721271
error = 0.002887937367
Show the steps
  1. Taylor polynomial
    P_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k
    with a = 0, f = exp(x), n = 3, x1 = 0.5
    → coefficients = [1, 1, 0.5, 0.1666666667], value = 1.645833333, function_value = 1.648721271, error = 0.002887937367
Capability: Taylor polynomial
Vector calculusm.ask(…)

Find the line integral of F = (-y, x) along x = cos(t), y = sin(t) from t = 0 to t = 2pi.

line_integral = 6.283185307
Show the steps
  1. Line integral of a vector field
    \int_C \mathbf F\cdot d\mathbf r = \int_{t_0}^{t_1} \left(P\,x'(t) + Q\,y'(t)\right)dt
    with P = -y, Q = x, t0 = 0, t1 = 6.283185307, xt = cos(t), yt = sin(t)
    → line_integral = 6.283185307
Capability: Line integral of a vector field
Integralsm.solve(…)
m.solve("definite_integral", f="x^2*exp(-x)", a=0, b=2)
integral = 0.6466471676
Show the steps
  1. Definite integral
    \int_a^b f(x)\,dx
    with f = x^2*exp(-x), a = 0, b = 2
    → integral = 0.6466471676
Capability: Definite integral
Fourier seriesm.ask(…)

Find the Fourier coefficients of order 2 of f(x) = x on [-pi, pi].

a0 = 0
a_k = 0
b_k = -1
Show the steps
  1. Fourier coefficients
    a_k = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{k\pi x}{L}dx,\quad b_k = \frac{1}{L}\int_{-L}^{L} f(x)\sin\frac{k\pi x}{L}dx
    with L = 3.141592654, f = x, k = 2
    → a0 = 0, a_k = 0, b_k = -1
Capability: Fourier coefficients
Multiple integralsm.ask(…)

Find the double integral of x*y over [0, 2] x [0, 3].

integral = 9
Show the steps
  1. Double integral over a rectangle
    \int_a^b\!\int_c^d f(x,y)\,dy\,dx
    with a = 0, b = 2, c = 0, d = 3, f = x*y
    → integral = 9
Capability: Double integral over a rectangle
Multivariable calculusm.ask(…)

Find the directional derivative of x^2*y at (1, 2) in the direction (3, 4).

directional_derivative = 3.2
Show the steps
  1. Directional derivative
    D_{\mathbf u} f = \nabla f\cdot\frac{\mathbf u}{|\mathbf u|}
    with f = x^2*y, p = [1, 2], u = [3, 4]
    → directional_derivative = 3.2
Capability: Directional derivative
Use it

From Python or the command line

Every snippet was run as shown; the output under it is what it printed.

import calculus_ai999x as m

a = m.ask("Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.")
print(a.outcome)
print(a.values)
for step in a.steps:
    print(step["name"])
Output
SOLVED
{'limit': 4.0}
Limit at a point
Figures

Computed, not drawn by hand

Every value in these figures was computed by the package, each point a run of solve(); in a notebook, plot() and plot_relation() draw a capability's curves the same way.

Taylor polynomials closing in on sin x

P_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k
Taylor polynomials closing in on sin xTaylor polynomials of sin x about 0 of degree 1, 3, 5 and 7 against sin x itself, each evaluated by the package at 121 points; marked: x = 2.5, degree-7 value = 0.588534 (sin 2.5 = 0.598472, error 0.00993823).-505-2-10120.588534sin xdegree 1degree 3degree 5degree 7xy
Taylor polynomials of sin x about 0 of degree 1, 3, 5 and 7 against sin x itself, each evaluated by the package at 121 points; marked: x = 2.5, degree-7 value = 0.588534 (sin 2.5 = 0.598472, error 0.00993823).

Left Riemann sum under y = x²

L_n = h\sum_{k=0}^{n-1} f(a + k h),\quad h = \frac{b-a}{n}
Left Riemann sum under y = x²The area under y = x² on [0, 2] and the left Riemann sum with 8 rectangles (heights f(xᵢ) and the curve evaluated by the package): left sum 2.1875, right sum 3.1875, exact integral 2.66667.00.511.5201234left rectangles, sum 2.1875y = x², area 2.66667xy
The area under y = x² on [0, 2] and the left Riemann sum with 8 rectangles (heights f(xᵢ) and the curve evaluated by the package): left sum 2.1875, right sum 3.1875, exact integral 2.66667.

Newton's method converging to √2

x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}
Newton's method converging to √2Newton's method on x² − 2 from x₀ = 3: each tangent (through the curve point computed by the package) meets the axis at the next iterate — 1.83333, 1.46212, 1.415, 1.41421; marked: x₄ = 1.41421, f(x₄) = 6.1568e-07.1230510x₄ = 1.41421y = x² − 2tangent stepsiteratesxy
Newton's method on x² − 2 from x₀ = 3: each tangent (through the curve point computed by the package) meets the axis at the next iterate — 1.83333, 1.46212, 1.415, 1.41421; marked: x₄ = 1.41421, f(x₄) = 6.1568e-07.

Partial sums of a geometric series converging

S_n = a\,\frac{1 - r^n}{1 - r}
Partial sums of a geometric series convergingPartial sums of the geometric series 1 + ½ + ¼ + …, each computed by the package: S₁ = 1, S₄ = 1.875, S₁₂ = 1.99951; the gap to the sum of the series, 2, halves with every term.2.557.51012.500.511.52sum of the series = 211.51.751.875Sₙ = 1 + ½ + … + (½)ⁿ⁻¹n — number of termspartial sum Sₙ
Partial sums of the geometric series 1 + ½ + ¼ + …, each computed by the package: S₁ = 1, S₄ = 1.875, S₁₂ = 1.99951; the gap to the sum of the series, 2, halves with every term.

A solid of revolution and its disks

V = \pi\int_a^b f(x)^2\,dx
A solid of revolution and its disksThe solid made by turning y = √x (0 ≤ x ≤ 4) about the x-axis, with circular cross-sections drawn at x = 1, 2, 3, 4 (profile and radii computed by the package); the package's volume by disks, the integral of π(√x)² from 0 to 4, is 25.1327.024-2-1012solid swept by y = √x, volume 25.1327disks at x = 1, 2, 3, 4y = √xx (axis of revolution)y
The solid made by turning y = √x (0 ≤ x ≤ 4) about the x-axis, with circular cross-sections drawn at x = 1, 2, 3, 4 (profile and radii computed by the package); the package's volume by disks, the integral of π(√x)² from 0 to 4, is 25.1327.
Formulas

The mathematics inside

12 of its 71 formulas; every one is in the capability browser below.

Limit at a point
\lim_{x\to a} f(x)
Absolute extrema on a closed interval
\max_{[a,b]} f = \max\{f(a), f(b), f(c) : f'(c) = 0\}
Taylor polynomial
P_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k
Line integral of a vector field
\int_C \mathbf F\cdot d\mathbf r = \int_{t_0}^{t_1} \left(P\,x'(t) + Q\,y'(t)\right)dt
Definite integral
\int_a^b f(x)\,dx
Fourier coefficients
a_k = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{k\pi x}{L}dx,\quad b_k = \frac{1}{L}\int_{-L}^{L} f(x)\sin\frac{k\pi x}{L}dx
Double integral over a rectangle
\int_a^b\!\int_c^d f(x,y)\,dy\,dx
Directional derivative
D_{\mathbf u} f = \nabla f\cdot\frac{\mathbf u}{|\mathbf u|}
Average rate of change
\frac{\Delta f}{\Delta x} = \frac{f(b)-f(a)}{b-a}
Arc length of a graph
L = \int_a^b \sqrt{1 + f'(x)^2}\,dx
Limit at infinity
\lim_{x\to\infty} f(x)
Double integral over a region
\int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx
Capabilities

Every capability, with an example

Search by name, formula, input or topic. Each capability opens to show what it takes, what it returns, when it applies, and a question it answered.

71 capabilities

Derivatives 14

Absolute extrema on a closed intervalextrema_on_interval

The absolute maximum and minimum values of a polynomial on [a, b] by the closed-interval method, with the critical points inside (a, b).

\max_{[a,b]} f = \max\{f(a), f(b), f(c) : f'(c) = 0\}

Inputs

  • f the polynomial f(x) text
  • a left end
  • b right end

Returns

  • absolute_max the largest value of f on [a, b]
  • absolute_min the smallest value
  • critical_points critical points in (a, b), ascending

Applies when

  • a < b
  • f is a polynomial with rational coefficients

Example

m.ask("Find the absolute maximum and minimum of x^3 - 3x on [0, 2].")
→ absolute_max = 2, absolute_min = -2, critical_points = [1.0]
Average rate of changeaverage_rate_of_change

The average rate of change (f(b) - f(a))/(b - a) of f over [a, b]: the slope of the secant line.

\frac{\Delta f}{\Delta x} = \frac{f(b)-f(a)}{b-a}

Inputs

  • f the function f(x) text
  • a start of the interval
  • b end of the interval

Returns

  • average_rate (f(b) - f(a))/(b - a)
  • fa f(a)
  • fb f(b)

Applies when

  • a != b
  • f defined at a and b

Example

m.ask("Find the average rate of change of f(x) = x^2 from x = 1 to x = 3.")
→ average_rate = 4, fa = 1, fb = 9
Critical points and their classificationcritical_points_classify

All critical points of a polynomial f(x) and which are local maxima or minima (first-derivative test).

f'(c) = 0,\ f' \text{ changes } + \to - \Rightarrow \text{local max}

Inputs

  • f the polynomial f(x) text

Returns

  • critical_points real solutions of f'(x) = 0, ascending
  • local_maxima critical points that are local maxima
  • local_minima critical points that are local minima

Applies when

  • f is a polynomial of degree >= 2 with rational coefficients

Example

m.ask("Find and classify the critical points of x^3 - 3x.")
→ critical_points = [-1.0, 1.0], local_maxima = [-1.0], local_minima = [1.0]
Curvature of a graphcurvature

The curvature kappa = |f''| / (1 + f'^2)^(3/2) of y = f(x) at x = a, and the radius of curvature 1/kappa.

\kappa = \frac{|f''(a)|}{\left(1 + f'(a)^2\right)^{3/2}}

Inputs

  • f the function f(x) text
  • a the point x = a

Returns

  • curvature kappa at x = a
  • radius_of_curvature 1/kappa (absent where kappa = 0)

Applies when

  • f twice differentiable at a

Example

m.ask("Find the curvature of y = x^2 at x = 0.")
→ curvature = 2, radius_of_curvature = 0.5
Derivative at a pointderivative_at_point

The derivative f'(a) of a function f(x) at x = a (the slope of its graph there).

f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

Inputs

  • f the function f(x) text
  • a the point x = a

Returns

  • derivative f'(a)
  • derivative_expression f'(x) as an expression

Applies when

  • f is defined and differentiable at x = a

Example

m.ask("Find the derivative of f(x) = x^3 - 2x at x = 2.")
→ derivative = 10, derivative_expression = 3*x^2 - 2
Higher derivative at a pointnth_derivative_at_point

The n-th derivative f^(n)(a) of f(x) at x = a.

f^{(n)}(a) = \frac{d^n f}{dx^n}\Big|_{x=a}

Inputs

  • f the function f(x) text
  • n the order of the derivative integer
  • a the point x = a

Returns

  • nth_derivative f^(n)(a)
  • nth_derivative_expression f^(n)(x)

Applies when

  • 1 <= n <= 12
  • f is n times differentiable at x = a

Example

m.ask("Find the 4th derivative of sin(2x) at x = 1.")
→ nth_derivative = 14.5488, nth_derivative_expression = 16*sin(2*x)
Implicit differentiationimplicit_derivative

The slope dy/dx of a curve given implicitly by F(x, y) = c, at a point on the curve.

\frac{dy}{dx} = -\frac{F_x}{F_y}

Inputs

  • F the curve as an equation in x and y text
  • p the point (x, y) on the curve vector

Returns

  • dy_dx dy/dx at the point

Applies when

  • the point lies on the curve
  • F_y != 0 there (no vertical tangent)

Example

m.ask("Find dy/dx for x^2 + y^2 = 25 at the point (3, 4).")
→ dy_dx = -0.75
Inflection pointsinflection_points

The points where a polynomial changes concavity (f'' changes sign).

f''(c) = 0 \text{ and } f'' \text{ changes sign at } c

Inputs

  • f the polynomial f(x) text

Returns

  • inflection_points x-values of the inflection points, ascending

Applies when

  • f is a polynomial of degree >= 2 with rational coefficients

Example

m.ask("Find the inflection points of x^3 - 3x^2.")
→ inflection_points = [1.0]
Linear approximationlinear_approximation

Estimate f(x1) by the tangent line at x = a: L(x1) = f(a) + f'(a)(x1 - a); with the actual value and error.

L(x) = f(a) + f'(a)\,(x-a)

Inputs

  • f the function f(x) text
  • a the base point a
  • x1 the point to estimate

Returns

  • approximation L(x1)
  • actual f(x1)
  • error f(x1) - L(x1)
  • slope f'(a)

Applies when

  • f differentiable at a

Example

m.ask("Use the linear approximation of f(x) = sqrt(x) at x = 4 to estimate f(4.1).")
→ approximation = 2.025, actual = 2.02485, error = -0.000154327, slope = 0.25
Mean value theoremmean_value_theorem

The secant slope of a polynomial on [a, b] and every c in (a, b) where f'(c) equals it.

f'(c) = \frac{f(b) - f(a)}{b - a}

Inputs

  • f the polynomial f(x) text
  • a left end
  • b right end

Returns

  • secant_slope (f(b) - f(a))/(b - a)
  • c_values all c in (a, b) with f'(c) = the slope

Applies when

  • a < b
  • f a non-linear polynomial with rational coefficients

Example

m.ask("Find the value of c guaranteed by the mean value theorem for x^3 on [0, 3].")
→ secant_slope = 9, c_values = [1.7320508075688772]
Normal linenormal_line

The normal line (perpendicular to the tangent) to y = f(x) at x = a: slope -1/f'(a).

y = f(a) - \frac{1}{f'(a)}\,(x - a)

Inputs

  • f the function f(x) text
  • a the point x = a

Returns

  • slope slope -1/f'(a)
  • intercept y-intercept
  • y0 f(a)

Applies when

  • f'(a) != 0 (otherwise the normal is vertical)

Example

m.ask("Find the normal line to y = x^2 at x = 1.")
→ slope = -0.5, intercept = 1.5, y0 = 1
Second derivative at a pointsecond_derivative_at_point

The second derivative f''(a) of f(x) at x = a (concavity: positive means concave up).

f''(a) = \frac{d^2 f}{dx^2}\Big|_{x=a}

Inputs

  • f the function f(x) text
  • a the point x = a

Returns

  • second_derivative f''(a)
  • second_derivative_expression f''(x)

Applies when

  • f is twice differentiable at x = a

Example

m.ask("Find the second derivative of x^4 - 3x^2 at x = 1.")
→ second_derivative = 6, second_derivative_expression = 6*(2*x^2 - 1)
Tangent linetangent_line

The tangent line y = m*x + b to y = f(x) at x = a: slope m = f'(a), intercept b = f(a) - a*f'(a).

y = f(a) + f'(a)\,(x - a)

Inputs

  • f the function f(x) text
  • a the point of tangency x = a

Returns

  • slope slope m = f'(a)
  • intercept y-intercept b
  • y0 f(a)
  • equation the line as y = m*x + b

Applies when

  • f is differentiable at x = a

Example

m.ask("Find the equation of the tangent line to y = x^2 + 1 at x = 2.")
→ slope = 4, intercept = -3, y0 = 5, equation = y = 4*x - 3
Velocity and acceleration from positionvelocity_acceleration

Velocity v = s'(t) and acceleration a = s''(t) at time t0 from a position function s(t).

v = \frac{ds}{dt},\quad a = \frac{d^2 s}{dt^2}

Inputs

  • s the position s(t) text
  • t0 the time t0

Returns

  • velocity v(t0) = s'(t0)
  • acceleration a(t0) = s''(t0)
  • speed |v(t0)|
  • position s(t0)

Applies when

  • s twice differentiable at t0

Example

m.ask("A particle has position s(t) = t^3 - 6t^2 + 9t. Find its velocity and acceleration at t = 2.")
→ velocity = -3, acceleration = 0, speed = 3, position = 2

Fourier series 1

Fourier coefficientsfourier_coefficients

The Fourier coefficients a_0, a_k and b_k of f on [-L, L] (series a_0/2 + sum a_k cos(k pi x/L) + b_k sin(k pi x/L)).

a_k = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{k\pi x}{L}dx,\quad b_k = \frac{1}{L}\int_{-L}^{L} f(x)\sin\frac{k\pi x}{L}dx

Inputs

  • f the function f(x) text
  • L the half-period L (> 0)
  • k the order k integer

Returns

  • a0 a_0 = (1/L) integral of f
  • a_k a_k
  • b_k b_k

Applies when

  • L > 0
  • f continuous on [-L, L]
  • 1 <= k <= 50

Example

m.ask("Find the Fourier coefficients of order 2 of f(x) = x on [-pi, pi].")
→ a0 = 0, a_k = 0, b_k = -1

Integrals 14

Arc length of a grapharc_length

The length of the curve y = f(x) from x = a to x = b.

L = \int_a^b \sqrt{1 + f'(x)^2}\,dx

Inputs

  • f the curve f(x) text
  • a left end
  • b right end

Returns

  • arc_length the length of the curve

Applies when

  • a < b
  • f' continuous on [a, b]

Example

m.ask("Find the arc length of y = x^2 from x = 0 to x = 1.")
→ arc_length = 1.47894
Area between two curvesarea_between_curves

The area between the graphs of two polynomials y = f(x) and y = g(x) over [a, b], counting every piece as positive (the curves may cross).

A = \int_a^b \left|f(x) - g(x)\right|\,dx

Inputs

  • f the first curve f(x) (a polynomial) text
  • g the second curve g(x) (a polynomial) text
  • a left end
  • b right end

Returns

  • area the area between the curves
  • net_integral integral of f - g (signed)
  • crossings x where the curves cross inside (a, b)

Applies when

  • a < b
  • f and g polynomials with rational coefficients

Example

m.ask("Find the area between y = x^2 and y = x from x = 0 to x = 2.")
→ area = 1, net_integral = 0.666667, crossings = [1.0]
Area enclosed by two curvesarea_enclosed_curves

The area of the region enclosed by two polynomial curves y = f(x) and y = g(x), between their first and last intersection.

A = \int_{x_1}^{x_n} \left|f(x) - g(x)\right|\,dx

Inputs

  • f the first curve f(x) (a polynomial) text
  • g the second curve g(x) (a polynomial) text

Returns

  • area the enclosed area
  • x_left first intersection
  • x_right last intersection
  • intersections all intersection x-values

Applies when

  • f and g polynomials with rational coefficients meeting in at least two points

Example

m.ask("Find the area enclosed by y = x^2 and y = x.")
→ area = 0.166667, x_left = 0, x_right = 1, intersections = [0.0, 1.0]
Average value of a functionaverage_value

The average value (1/(b - a)) * integral of f from a to b of a function on [a, b].

f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx

Inputs

  • f the function f(x) text
  • a start of the interval
  • b end of the interval

Returns

  • average_value the average value of f on [a, b]
  • integral the integral of f over [a, b]

Applies when

  • a < b
  • f continuous on [a, b]

Example

m.ask("Find the average value of f(x) = x^2 on [0, 3].")
→ average_value = 3, integral = 9
Centroid of a region under a curvecentroid_region

The area and centroid (x_bar, y_bar) of the region between y = f(x) >= 0 (a polynomial) and the x-axis for a <= x <= b.

\bar x = \frac{1}{A}\int_a^b x f(x)\,dx,\quad \bar y = \frac{1}{2A}\int_a^b f(x)^2\,dx

Inputs

  • f the curve f(x) (a polynomial) text
  • a left end
  • b right end

Returns

  • area area of the region
  • x_bar x-coordinate of the centroid
  • y_bar y-coordinate of the centroid

Applies when

  • a < b
  • f(x) >= 0 on [a, b] (checked exactly)

Example

m.ask("Find the centroid of the region under y = x^2 from x = 0 to x = 1.")
→ area = 0.333333, x_bar = 0.75, y_bar = 0.3
Definite integraldefinite_integral

The definite integral of f(x) from x = a to x = b (net signed area).

\int_a^b f(x)\,dx

Inputs

  • f the integrand f(x) text
  • a lower limit
  • b upper limit

Returns

  • integral the value of the integral

Applies when

  • f real and continuous on the closed interval between a and b

Example

m.ask("Evaluate the integral of x^2 from 0 to 3.")
→ integral = 9
Derivative of an integral (fundamental theorem)ftc_derivative

d/dx of the integral of f(t) from a to g(x), evaluated at x = x0 (first fundamental theorem with the chain rule).

\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)

Inputs

  • f the integrand f(t) text
  • a the constant lower limit
  • g the upper limit g(x) text
  • x0 the point x = x0

Returns

  • derivative d/dx of the integral at x0

Applies when

  • f continuous between a and g(x0)
  • g differentiable at x0

Example

m.ask("Find the derivative of the integral from 0 to x^2 of sin(t) dt at x = 1.")
→ derivative = 1.68294
Improper integral to infinityimproper_integral

The improper integral of f(x) from x = a to infinity, when it converges.

\int_a^\infty f(x)\,dx = \lim_{R\to\infty}\int_a^R f(x)\,dx

Inputs

  • f the integrand f(x) text
  • a lower limit

Returns

  • integral the value of the improper integral

Applies when

  • f continuous on [a, infinity)
  • x^1.01 * f(x) has a finite limit (comparison test), so the integral converges absolutely

Example

m.ask("Evaluate the improper integral of exp(-x) from 0 to infinity.")
→ integral = 1
Left Riemann sumleft_riemann_sum

The left-endpoint Riemann sum of f on [a, b] with n equal subintervals.

L_n = h\sum_{k=0}^{n-1} f(a + k h),\quad h = \frac{b-a}{n}

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of equal subintervals integer

Returns

  • approximation the left Riemann sum

Applies when

  • a < b
  • 1 <= n <= 100000
  • f defined at the sample points

Example

m.ask("Find the left Riemann sum of x^2 on [0, 2] with n = 4.")
→ approximation = 1.75
Right Riemann sumright_riemann_sum

The right-endpoint Riemann sum of f on [a, b] with n equal subintervals.

R_n = h\sum_{k=1}^{n} f(a + k h),\quad h = \frac{b-a}{n}

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of equal subintervals integer

Returns

  • approximation the right Riemann sum

Applies when

  • a < b
  • 1 <= n <= 100000
  • f defined at the sample points

Example

m.ask("Find the right Riemann sum of x^2 on [0, 2] with n = 4.")
→ approximation = 3.75
Surface area of revolutionsurface_area_revolution

The area of the surface formed by rotating y = f(x), a <= x <= b, about the x-axis.

S = 2\pi\int_a^b |f(x)|\sqrt{1 + f'(x)^2}\,dx

Inputs

  • f the curve f(x) text
  • a left end
  • b right end

Returns

  • surface_area the area of the surface of revolution

Applies when

  • a < b
  • f' continuous on [a, b]

Example

m.ask("Find the surface area when y = x from x = 0 to x = 1 is rotated about the x-axis.")
→ surface_area = 4.44288
Volume of revolution (disk method)disk_volume

The volume of the solid formed by rotating y = f(x), a <= x <= b, about the x-axis.

V = \pi\int_a^b f(x)^2\,dx

Inputs

  • f the curve f(x) text
  • a left end
  • b right end

Returns

  • volume the volume of the solid

Applies when

  • a < b
  • f continuous on [a, b]

Example

m.ask("Find the volume when y = x^2 from x = 0 to x = 1 is rotated about the x-axis.")
→ volume = 0.628319
Volume of revolution (shell method)shell_volume

The volume swept when the region under y = f(x) >= 0 (a polynomial), a <= x <= b with a >= 0, is rotated about the y-axis.

V = 2\pi\int_a^b x\,f(x)\,dx

Inputs

  • f the curve f(x) (a polynomial) text
  • a left end (>= 0)
  • b right end

Returns

  • volume the volume of the solid

Applies when

  • 0 <= a < b
  • f(x) >= 0 on [a, b] (checked exactly)

Example

m.ask("Use the shell method to find the volume when y = x^2 on [0, 2] is rotated about the y-axis.")
→ volume = 25.1327
Volume of revolution (washer method)washer_volume

The volume swept when the region between an outer curve y = f(x) and an inner curve y = g(x) (polynomials, f >= g >= 0) on [a, b] is rotated about the x-axis.

V = \pi\int_a^b \left(f(x)^2 - g(x)^2\right)dx

Inputs

  • f outer radius f(x) (a polynomial) text
  • g inner radius g(x) (a polynomial) text
  • a left end
  • b right end

Returns

  • volume the volume of the solid

Applies when

  • a < b
  • f >= g >= 0 on [a, b] (checked exactly)
  • polynomials

Example

m.ask("Find the volume when the region between y = x + 2 and y = 1 on [0, 1] is rotated about the x-axis using washers.")
→ volume = 16.7552

Limits 2

Limit at a pointlimit_at_point

The two-sided limit of f(x) as x approaches a (including removable 0/0 forms such as sin(x)/x).

\lim_{x\to a} f(x)

Inputs

  • f the function f(x) text
  • a the point x = a

Returns

  • limit the limit of f(x) as x -> a

Applies when

  • the left and right limits exist, are finite and agree

Example

m.ask("Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.")
→ limit = 4
Limit at infinitylimit_at_infinity

The limit of f(x) as x grows without bound (horizontal asymptote, end behaviour).

\lim_{x\to\infty} f(x)

Inputs

  • f the function f(x) text

Returns

  • limit the limit of f(x) as x -> infinity

Applies when

  • the limit exists and is finite

Example

m.ask("Find the limit of (3x^2 + 1)/(x^2 - 5) as x approaches infinity.")
→ limit = 3

Multiple integrals 4

Double integral over a rectangledouble_integral_rectangle

The double integral of f(x, y) over the rectangle [a, b] x [c, d].

\int_a^b\!\int_c^d f(x,y)\,dy\,dx

Inputs

  • f the integrand f(x, y) text
  • a x from
  • b x to
  • c y from
  • d y to

Returns

  • integral the value of the double integral

Applies when

  • a < b, c < d
  • f continuous on the rectangle

Example

m.ask("Find the double integral of x*y over [0, 2] x [0, 3].")
→ integral = 9
Double integral over a regiondouble_integral_region

The double integral of f(x, y) over the region a <= x <= b, g1(x) <= y <= g2(x).

\int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx

Inputs

  • f the integrand f(x, y) text
  • a x from
  • b x to
  • g1 the lower curve y = g1(x) text
  • g2 the upper curve y = g2(x) text

Returns

  • integral the value of the double integral

Applies when

  • a < b
  • g1 <= g2 on [a, b]
  • f, g1, g2 continuous

Example

m.ask("Find the double integral of x*y over the region 0 <= x <= 1, x^2 <= y <= x.")
→ integral = 0.0416667
Surface area of a graphsurface_area_graph

The area of the surface z = f(x, y) over the rectangle [a, b] x [c, d].

A = \iint_R \sqrt{1 + f_x^2 + f_y^2}\,dA

Inputs

  • f the surface z = f(x, y) text
  • a x from
  • b x to
  • c y from
  • d y to

Returns

  • surface_area the surface area

Applies when

  • a < b, c < d
  • f continuously differentiable on the rectangle

Example

m.ask("Find the surface area of z = x + y over [0, 1] x [0, 1].")
→ surface_area = 1.73205
Triple integral over a boxtriple_integral_box

The triple integral of f(x, y, z) over the box [x0, x1] x [y0, y1] x [z0, z1].

\iiint_B f(x,y,z)\,dV

Inputs

  • f the integrand f(x, y, z) text
  • x0 x from
  • x1 x to
  • y0 y from
  • y1 y to
  • z0 z from
  • z1 z to

Returns

  • integral the value of the triple integral

Applies when

  • x0 < x1, y0 < y1, z0 < z1
  • f continuous on the box

Example

m.ask("Find the triple integral of x*y*z over [0, 1] x [0, 2] x [0, 3].")
→ integral = 4.5

Multivariable calculus 6

Directional derivativedirectional_derivative

The rate of change of f at a point in the direction of a vector u (u is normalised to unit length).

D_{\mathbf u} f = \nabla f\cdot\frac{\mathbf u}{|\mathbf u|}

Inputs

  • f the function f(x, y) or f(x, y, z) text
  • p the point vector
  • u the direction vector vector

Returns

  • directional_derivative D_u f at the point

Applies when

  • u != 0 with as many components as p
  • f differentiable at p

Example

m.ask("Find the directional derivative of x^2*y at (1, 2) in the direction (3, 4).")
→ directional_derivative = 3.2
Gradientgradient

The gradient vector of f(x, y) or f(x, y, z) at a point and its length (the greatest rate of increase).

\nabla f = \left(f_x, f_y, f_z\right)

Inputs

  • f the function f(x, y) or f(x, y, z) text
  • p the point (x, y) or (x, y, z) vector

Returns

  • gradient the gradient vector
  • magnitude its length

Applies when

  • f differentiable at the point

Example

m.ask("Find the gradient of x^2 + y^2 at (1, 2).")
→ gradient = [2.0, 4.0], magnitude = 4.47214
Jacobian determinantjacobian_determinant

The Jacobian determinant d(x, y)/d(u, v) of a transformation x = x(u, v), y = y(u, v) at a point.

\frac{\partial(x,y)}{\partial(u,v)} = x_u y_v - x_v y_u

Inputs

  • xu x as a function of u and v text
  • yv y as a function of u and v text
  • p the point (u, v) vector

Returns

  • jacobian the Jacobian determinant at the point

Applies when

  • x and y differentiable at the point

Example

m.ask("Find the Jacobian of x = u*cos(v), y = u*sin(v) at (2, 0.5).")
→ jacobian = 2
Partial derivativepartial_derivative

The partial derivative of f(x, y) or f(x, y, z) with respect to one variable, at a point.

\frac{\partial f}{\partial x}(p)

Inputs

  • f the function f(x, y) or f(x, y, z) text
  • var the variable choice
  • p the point (x, y) or (x, y, z) vector

Returns

  • partial the partial derivative at the point

Applies when

  • f differentiable at the point

Example

m.ask("Find the partial derivative of x^2*y + y^3 with respect to x at (1, 2).")
→ partial = 4
Second derivative test (two variables)hessian_classification

Classifies a critical point of f(x, y) as a local minimum, local maximum or saddle point from the Hessian.

D = f_{xx}f_{yy} - f_{xy}^2

Inputs

  • f the function f(x, y) text
  • p the critical point (x, y) vector

Returns

  • fxx f_xx
  • fxy f_xy
  • fyy f_yy
  • D D = f_xx f_yy - f_xy^2
  • classification local minimum, local maximum or saddle point

Applies when

  • the gradient is zero at the point
  • D != 0 (D = 0 is inconclusive and refused)

Example

m.ask("Classify the critical point of x^2 + 3y^2 - 2x at (1, 0).")
→ fxx = 2, fxy = 0, fyy = 6, D = 12, classification = local minimum
Tangent planetangent_plane

The tangent plane z = f_x x + f_y y + c to the surface z = f(x, y) at (x0, y0).

z = f(x_0,y_0) + f_x(x - x_0) + f_y(y - y_0)

Inputs

  • f the surface z = f(x, y) text
  • p the point (x0, y0) vector

Returns

  • z0 f(x0, y0)
  • fx f_x(x0, y0)
  • fy f_y(x0, y0)
  • c the constant term c

Applies when

  • f differentiable at (x0, y0)

Example

m.ask("Find the tangent plane to z = x^2 + y^2 at (1, 2).")
→ z0 = 5, fx = 2, fy = 4, c = -5

Numerical differentiation 1

Finite-difference derivativenumerical_derivative

A forward, backward or central difference approximation of f'(a) with step h, with the exact derivative and the error.

f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}

Inputs

  • f the function f(x) text
  • a the point x = a
  • h the step size h > 0
  • method forward, backward or central difference choice

Returns

  • approximation the difference quotient
  • exact the exact f'(a)
  • error approximation - exact

Applies when

  • h > 0
  • f defined at the points used

Example

m.ask("Estimate the derivative of x^2 at x = 1 using a central difference with h = 0.1.")
→ approximation = 2, exact = 2, error = 0

Numerical integration 4

Gauss-Legendre quadraturegauss_legendre

The n-point Gauss-Legendre approximation of the integral of f on [a, b] (exact for polynomials of degree <= 2n - 1), with the exact value and the error.

\int_a^b f\,dx \approx \frac{b-a}{2}\sum_{i=1}^n w_i\, f\!\left(\frac{a+b}{2} + \frac{b-a}{2}x_i\right)

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of nodes integer

Returns

  • approximation n-point Gauss-Legendre value
  • exact the integral
  • error approximation - exact

Applies when

  • a < b
  • 1 <= n <= 64
  • f continuous on [a, b]

Example

m.ask("Use 2-point Gauss-Legendre quadrature to approximate the integral of x^3 + x^2 from 0 to 2.")
→ approximation = 6.66667, exact = 6.66667, error = 0
Midpoint rulemidpoint_rule

The composite midpoint approximation of the integral of f on [a, b] with n subintervals, with the exact value and the error.

M_n = h\sum_{k=0}^{n-1} f\left(a + (k + \tfrac12) h\right)

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of equal subintervals integer

Returns

  • approximation midpoint approximation M_n
  • exact the integral
  • error M_n - exact

Applies when

  • a < b
  • 1 <= n <= 100000
  • f continuous on [a, b]

Example

m.ask("Approximate the integral of x^2 from 0 to 2 using the midpoint rule with n = 4.")
→ approximation = 2.625, exact = 2.66667, error = -0.0416667
Simpson's rulesimpson_rule

The composite Simpson approximation of the integral of f on [a, b] with an even number n of subintervals, with the exact value and the error.

S_n = \frac{h}{3}\left[f(x_0) + 4\sum_{\text{odd } k} f(x_k) + 2\sum_{\text{even } k} f(x_k) + f(x_n)\right]

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of equal subintervals integer

Returns

  • approximation Simpson approximation S_n
  • exact the integral
  • error S_n - exact

Applies when

  • a < b
  • n even, 2 <= n <= 100000
  • f continuous on [a, b]

Example

m.ask("Approximate the integral of x^4 from 0 to 2 using Simpson's rule with 4 subintervals.")
→ approximation = 6.41667, exact = 6.4, error = 0.0166667
Trapezoidal ruletrapezoid_rule

The composite trapezoidal approximation of the integral of f on [a, b] with n subintervals, with the exact value and the error (approximation - exact).

T_n = \frac{h}{2}\left[f(x_0) + 2\sum_{k=1}^{n-1} f(x_k) + f(x_n)\right]

Inputs

  • f the function f(x) text
  • a left end
  • b right end
  • n number of equal subintervals integer

Returns

  • approximation trapezoidal approximation T_n
  • exact the integral
  • error T_n - exact

Applies when

  • a < b
  • 1 <= n <= 100000
  • f continuous on [a, b]

Example

m.ask("Approximate the integral of x^2 from 0 to 2 using the trapezoidal rule with n = 4.")
→ approximation = 2.75, exact = 2.66667, error = 0.0833333

Numerical methods 1

Newton's methodnewton_method

n iterations of Newton's method x_{k+1} = x_k - f(x_k)/f'(x_k) for a root of f, from x_0.

x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}

Inputs

  • f the function f(x) text
  • x0 the starting value x_0
  • n number of iterations integer

Returns

  • x_n the n-th iterate
  • f_x_n f(x_n)
  • iterates x_1 .. x_n

Applies when

  • f'(x_k) != 0 at every iterate
  • 1 <= n <= 50

Example

m.ask("Apply Newton's method to f(x) = x^2 - 2 starting at x0 = 1 for 3 iterations.")
→ x_n = 1.41422, f_x_n = 6.0073e-06, iterates = [1.5, 1.4166666666666667, 1.4142156862745099]

Parametric curves 2

Arc length of a parametric curveparametric_arc_length

The length of the curve x = x(t), y = y(t) for t0 <= t <= t1.

L = \int_{t_0}^{t_1}\sqrt{x'(t)^2 + y'(t)^2}\,dt

Inputs

  • xt x(t) text
  • yt y(t) text
  • t0 start parameter
  • t1 end parameter

Returns

  • arc_length the length of the curve

Applies when

  • t0 < t1
  • x and y continuously differentiable

Example

m.ask("Find the arc length of the parametric curve x = cos(t), y = sin(t) from t = 0 to t = pi.")
→ arc_length = 3.14159
Slope and concavity of a parametric curveparametric_slope

dy/dx and d^2y/dx^2 of the curve x = x(t), y = y(t) at a parameter value t0.

\frac{dy}{dx} = \frac{y'(t)}{x'(t)},\quad \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\frac{dy}{dx}}{x'(t)}

Inputs

  • xt x(t) text
  • yt y(t) text
  • t0 the parameter value

Returns

  • dy_dx dy/dx at t0
  • d2y_dx2 d^2y/dx^2 at t0

Applies when

  • dx/dt != 0 at t0

Example

m.ask("Find dy/dx for x = t^2, y = t^3 at t = 1.")
→ dy_dx = 1.5, d2y_dx2 = 0.75

Polar curves 1

Area in polar coordinatespolar_area

The area swept by the polar curve r = r(theta) between theta = alpha and theta = beta.

A = \frac{1}{2}\int_{\alpha}^{\beta} r(\theta)^2\,d\theta

Inputs

  • r the polar curve r(theta) text
  • al alpha, the start angle (radians)
  • be beta, the end angle (radians)

Returns

  • area the area

Applies when

  • alpha < beta <= alpha + 2 pi
  • r continuous

Example

m.ask("Find the area enclosed by the polar curve r = 1 + cos(theta) from theta = 0 to theta = 2pi.")
→ area = 4.71239

Series 6

Partial sum of a geometric seriesgeometric_partial_sum

The sum of the first n terms a + a r + ... + a r^(n-1).

S_n = a\,\frac{1 - r^n}{1 - r}

Inputs

  • a the first term
  • r the common ratio
  • n the number of terms integer

Returns

  • sum the sum of the first n terms

Applies when

  • 1 <= n <= 500

Example

m.ask("Find the sum of the first 4 terms of the geometric series with first term 2 and ratio 3.")
→ sum = 80
Ratio testratio_test

The limit L of |a_(n+1)/a_n| and the ratio test's verdict for the series sum a_n (converges if L < 1, diverges if L > 1).

L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|

Inputs

  • a the general term a_n, in n text

Returns

  • ratio_limit L = lim |a_(n+1)/a_n|
  • verdict converges or diverges

Applies when

  • L exists, is finite and is not 1 (L = 1 is inconclusive and refused)

Example

m.ask("Apply the ratio test to the series 2^n/n^3.")
→ ratio_limit = 2, verdict = diverges
Sum of an arithmetic seriesarithmetic_series_sum

The sum of the first n terms of an arithmetic sequence, and its last term.

S_n = \frac{n}{2}\left(2a_1 + (n-1)d\right)

Inputs

  • a1 the first term
  • d the common difference
  • n the number of terms integer

Returns

  • sum the sum of the first n terms
  • last_term the n-th term

Applies when

  • n >= 1

Example

m.ask("Find the sum of the first 10 terms of the arithmetic series with first term 2 and common difference 3.")
→ sum = 155, last_term = 29
Sum of an infinite geometric seriesgeometric_series_sum

The sum a + a r + a r^2 + ... of a convergent geometric series.

\sum_{n=0}^{\infty} a r^n = \frac{a}{1-r}

Inputs

  • a the first term
  • r the common ratio

Returns

  • sum the sum of the series

Applies when

  • |r| < 1

Example

m.ask("Find the sum of the geometric series with first term 3 and ratio 0.5.")
→ sum = 6
Sum of an infinite seriesseries_sum

The exact sum of a convergent series sum_{n=k}^infinity a_n (telescoping, geometric-type, zeta values, alternating harmonic, ...).

\sum_{n=k}^{\infty} a_n

Inputs

  • a the general term a_n, in n text
  • k the starting index integer

Returns

  • sum the sum of the series

Applies when

  • the series converges
  • a closed form exists

Example

m.ask("Find the sum of the series 1/n^2 from n = 1 to infinity.")
→ sum = 1.64493
Taylor polynomialtaylor_polynomial

The degree-n Taylor polynomial of f about x = a (Maclaurin when a = 0): its coefficients, its value at x1, f(x1) and the truncation error.

P_n(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k

Inputs

  • f the function f(x) text
  • a the centre x = a
  • n the degree integer
  • x1 the evaluation point

Returns

  • coefficients c_0 .. c_n with c_k = f^(k)(a)/k!
  • value P_n(x1)
  • function_value f(x1)
  • error f(x1) - P_n(x1)

Applies when

  • f is n times differentiable at a and defined at x1
  • 0 <= n <= 12

Example

m.ask("Find the Taylor polynomial of degree 3 of exp(x) about x = 0 and evaluate it at x = 0.5.")
→ coefficients = [1.0, 1.0, 0.5, 0.16666666666666666], value = 1.64583, function_value = 1.64872, error = 0.00288794

Special functions 8

Bessel function of the first kindbessel_j

The Bessel function J_n(x) of whole-number order n.

J_n(x) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,(m+n)!}\left(\frac{x}{2}\right)^{2m+n}

Inputs

  • n the order n integer
  • x the argument x

Returns

  • J J_n(x)

Applies when

  • 0 <= n <= 50
  • |x| <= 1000

Example

m.ask("Evaluate the Bessel function J_0(1).")
→ J = 0.765198
Beta functionbeta_function

The beta function B(a, b) for positive a and b.

B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}

Inputs

  • a the first argument (> 0)
  • b the second argument (> 0)

Returns

  • beta B(a, b)

Applies when

  • a > 0
  • b > 0

Example

m.ask("Evaluate the beta function B(2, 3).")
→ beta = 0.0833333
Chebyshev polynomialchebyshev_polynomial

The value of the Chebyshev polynomial of the first kind T_n(x) (T_n(cos t) = cos(n t)).

T_{k+1}(x) = 2x\,T_k(x) - T_{k-1}(x)

Inputs

  • n the degree n integer
  • x the argument x

Returns

  • T T_n(x)

Applies when

  • 0 <= n <= 500

Example

m.ask("Evaluate the Chebyshev polynomial T_3(0.5).")
→ T = -1
Error functionerror_function

The error function erf(x) and the complementary error function erfc(x) = 1 - erf(x).

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt

Inputs

  • x the argument x

Returns

  • erf erf(x)
  • erfc erfc(x)

Applies when

  • x real

Example

m.ask("Evaluate erf(0.5).")
→ erf = 0.5205, erfc = 0.4795
Gamma functiongamma_function

The value of the gamma function (Gamma(n) = (n-1)! for whole n).

\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}\,dt

Inputs

  • x the argument x

Returns

  • gamma Gamma(x)

Applies when

  • x is not 0 or a negative whole number
  • x < 171.6

Example

m.ask("Evaluate the gamma function at x = 4.5.")
→ gamma = 11.6317
Legendre polynomiallegendre_polynomial

The value of the Legendre polynomial P_n(x).

(k+1)P_{k+1}(x) = (2k+1)\,x\,P_k(x) - k\,P_{k-1}(x)

Inputs

  • n the degree n integer
  • x the argument x

Returns

  • P P_n(x)

Applies when

  • 0 <= n <= 200

Example

m.ask("Evaluate the Legendre polynomial P_3(0.5).")
→ P = -0.4375
Regularized incomplete gamma functionsregularized_gamma

The regularized lower and upper incomplete gamma functions P(a, x) and Q(a, x) = 1 - P(a, x).

P(a,x) = \frac{1}{\Gamma(a)}\int_0^x t^{a-1}e^{-t}\,dt

Inputs

  • a the shape a (> 0)
  • x the upper limit x (>= 0)

Returns

  • P P(a, x) = gamma(a, x)/Gamma(a)
  • Q Q(a, x) = Gamma(a, x)/Gamma(a)

Applies when

  • a > 0
  • x >= 0

Example

m.ask("Find the regularized incomplete gamma function P(2, 3).")
→ P = 0.800852, Q = 0.199148
Riemann zeta functionriemann_zeta

The Riemann zeta function zeta(s) = sum 1/n^s for s > 1 (the p-series sum).

\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

Inputs

  • s the argument s (> 1)

Returns

  • zeta zeta(s)

Applies when

  • s > 1

Example

m.ask("Find zeta(2).")
→ zeta = 1.64493

Vector calculus 7

Curlcurl

The curl of a vector field F = (P, Q, R) at a point.

\nabla\times\mathbf F = (R_y - Q_z,\ P_z - R_x,\ Q_x - P_y)

Inputs

  • P the x-component P(x, y, z) text
  • Q the y-component Q(x, y, z) text
  • R the z-component R(x, y, z) text
  • p the point (x, y, z) vector

Returns

  • curl curl F at the point

Applies when

  • the components are differentiable at the point

Example

m.ask("Find the curl of F = (x*y, y*z, z*x) at (1, 2, 3).")
→ curl = [-2.0, -3.0, -1.0]
Divergencedivergence

The divergence P_x + Q_y + R_z of a vector field F = (P, Q, R) at a point.

\nabla\cdot\mathbf F = P_x + Q_y + R_z

Inputs

  • P the x-component P(x, y, z) text
  • Q the y-component Q(x, y, z) text
  • R the z-component R(x, y, z) text
  • p the point (x, y, z) vector

Returns

  • divergence div F at the point

Applies when

  • the components are differentiable at the point

Example

m.ask("Find the divergence of F = (x*y, y*z, x*z) at (1, 2, 3).")
→ divergence = 6
Green's theorem on a rectanglegreens_theorem

Both sides of Green's theorem for F = (P, Q) on the rectangle [a, b] x [c, d]: the counter-clockwise circulation around the boundary and the double integral of Q_x - P_y.

\oint_C P\,dx + Q\,dy = \iint_R \left(Q_x - P_y\right)dA

Inputs

  • P the x-component P(x, y) text
  • Q the y-component Q(x, y) text
  • a x from
  • b x to
  • c y from
  • d y to

Returns

  • circulation the boundary line integral (counter-clockwise)
  • double_integral the integral of Q_x - P_y

Applies when

  • a < b, c < d
  • P and Q continuously differentiable on the rectangle

Example

m.ask("Verify Green's theorem for F = (-y, x) on the rectangle [0, 1] x [0, 2].")
→ circulation = 4, double_integral = 4
Laplacianlaplacian

The Laplacian f_xx + f_yy (+ f_zz) of a function at a point.

\nabla^2 f = f_{xx} + f_{yy} + f_{zz}

Inputs

  • f the function f(x, y) or f(x, y, z) text
  • p the point vector

Returns

  • laplacian the Laplacian at the point

Applies when

  • f twice differentiable at the point

Example

m.ask("Find the Laplacian of x^2*y + z^3 at (1, 2, 3).")
→ laplacian = 22
Line integral of a vector fieldline_integral_vector

The line integral (work) of F = (P, Q) along the path x = x(t), y = y(t) from t0 to t1.

\int_C \mathbf F\cdot d\mathbf r = \int_{t_0}^{t_1} \left(P\,x'(t) + Q\,y'(t)\right)dt

Inputs

  • P the x-component P(x, y) text
  • Q the y-component Q(x, y) text
  • xt the path x(t) text
  • yt the path y(t) text
  • t0 start parameter
  • t1 end parameter

Returns

  • line_integral the integral of F . dr along the path

Applies when

  • the integrand is continuous for t between t0 and t1

Example

m.ask("Find the line integral of F = (-y, x) along x = cos(t), y = sin(t) from t = 0 to t = 2pi.")
→ line_integral = 6.28319
Line integral with respect to arc lengthline_integral_scalar

The integral of f(x, y) ds along the path x = x(t), y = y(t), t0 <= t <= t1 (mass of a wire, average along a curve).

\int_C f\,ds = \int_{t_0}^{t_1} f(x(t),y(t))\sqrt{x'(t)^2 + y'(t)^2}\,dt

Inputs

  • f the function f(x, y) text
  • xt the path x(t) text
  • yt the path y(t) text
  • t0 start parameter
  • t1 end parameter

Returns

  • line_integral the integral of f ds along the path

Applies when

  • t0 < t1
  • the integrand continuous on [t0, t1]

Example

m.ask("Find the line integral of x + y ds along x = t, y = t from t = 0 to t = 1.")
→ line_integral = 1.41421
Outward flux through a box (divergence theorem)divergence_theorem_box

The outward flux of F = (P, Q, R) through the surface of the box [x0, x1] x [y0, y1] x [z0, z1].

\oiint_S \mathbf F\cdot\mathbf n\,dS = \iiint_B \nabla\cdot\mathbf F\,dV

Inputs

  • P the x-component text
  • Q the y-component text
  • R the z-component text
  • x0 x from
  • x1 x to
  • y0 y from
  • y1 y to
  • z0 z from
  • z1 z to

Returns

  • flux the outward flux

Applies when

  • the box is non-degenerate
  • F continuously differentiable on the box

Example

m.ask("Find the outward flux of F = (x^2, y^2, z^2) through the box [0, 1] x [0, 1] x [0, 1].")
→ flux = 3
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Releases and repository files

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VersionReleased
1.0.02026-10-25
Questions

Frequently asked

What does calculus-ai999x do?

Calculus answers questions in calculus: limits, derivatives, integrals, series, Taylor polynomials, multivariable and vector calculus, special functions and numerical differentiation and integration. Each answer gives the value with its unit, the steps and the formula in LaTeX.

How do I install calculus-ai999x?

With pip: pip install calculus-ai999x. The same files are on Hugging Face (hf download AI999X/calculus-ai999x --local-dir calculus-ai999x) and GitHub.

Does it run offline?

Yes. It is a Python package that runs on an ordinary CPU without a network connection (Python >=3.10); no GPU and no API key are needed.

How many capabilities does calculus-ai999x have?

71 capabilities in 14 topics, including Derivatives, Integrals, Special functions, Vector calculus, Multivariable calculus.

Which field and tasks does it cover?

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

What does an answer look like?

Asked "Find the limit of (x^2 - 4)/(x - 2) as x approaches 2.", it answers: limit = 4.

What happens when a question cannot be answered?

It says so and names what is missing or out of range, instead of guessing a number.

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