Mathematics
# Calculus & Analysis

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
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
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+
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)
compute an indexed sum

sum j^2, j=1 to 100
compute an infinite sum

sum (3/4)^j, j=0..infinity
compute an indexed product

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

product (1-1/n^4), n=2 to infinity
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
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)
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
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
compute the gradient of a function

grad sin(x^2 y)
calculate alternate forms of a vector analysis expression

curl (curl F)
compute a Fourier transform

Fourier transform exp(-x^2)
compute a Laplace transform

LT e^t sin 2t
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)
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)
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