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 roots 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