Integrals

Compute an indefinite integral:

integrate x^2 sin^3 x dx

Compute a definite integral:

integrate sin x dx from x=0 to pi

Compute an improper integral:

int sinx/x dx, x=0..infinity

Derivatives

Calculate the derivative of a function:

derivative of x^4 sin x

Compute higher derivatives:

second derivative of sin(2x)

Compute partial derivatives:

d/dx x^2 y^4, d/dy x^2 y^4

Limits

Compute a limit:

lim (sin x - x)/x^3 as x->0

limit (1+1/n)^n as n->infinity

Compute a one-sided limit:

lim x/|x| as x->0+

Sequences

Analyze a sequence:

sequence of Fibonacci numbers

Compute a possible formula and continuation for a sequence:

1, 4, 9, 16, 25, ...

Solve a recurrence:

g(n+1)=n^2+g(n)

Sums

Compute an indexed sum:

sum j^2, j=1 to 100

Compute an infinite sum:

sum (3/4)^j, j=0..infinity

Products

Compute an indexed product:

product (k+2)/k, k=1..25

Compute an infinite product:

product (1-1/n^4), n=2 to infinity

Series Expansions

Find a Taylor series expansion:

taylor series sin x

Specify the center point and the order of the expansion:

series (sin x)/(x-pi) at x=pi to order 10

Coordinate Geometry

Compute the area bounded by two curves:

compute the area between y=|x| and y=x^2-6

Locate inflection points of a function:

inflection points of x+sin(x)

Differential Equations

Solve a linear ordinary differential equation:

y'' + y = 0

Specify initial values:

y'' + y = 0, y(0)=2, y'(0)=1

Solve a nonlinear equation:

f'(t) = f(t)^2 + 1

Complex Analysis

Compute properties of a function of a complex variable (use the variable ,<i>z</i>,):

e^z

Compute the residue of a function at a point:

residue of 1/(z^2+4)^2 at z=2i

Vector Analysis

Compute the gradient of a function:

grad sin(x^2 y)

Calculate alternate forms of a vector analysis expression:

curl (curl F)

Integral Transforms

Compute a Fourier transform:

Fourier transform exp(-x^2)

Compute a Laplace transform:

LT e^t sin 2t

Domain & Range

Compute the domain of a function:

domain of f(x) = x/(x^2-1)

Compute the range of a function:

range of e^(-x^2)

Continuity

Determine whether a function is continuous:

is sin(x-1.1)/(x-1.1)+heaviside(x) continuous

Locate discontinuities of a function:

discontinuities (x^3+8)/(x^3+3x^2-4x-12)

Numerical Analysis

Find a root of an equation using Newton's method:

using Newton's method solve x cos x = 0

Numerically integrate functions that cannot be integrated symbolically:

integrate sin(cos x) from x=0 to 1