# Examples by Topic

## Calculus & Analysis

Mathematics
Calculus & Analysis

### 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

### Derivatives »

calculate the derivative of a function

#### derivative of x^4 sin x

compute higher derivatives

#### second derivative of sin(2x)

compute partial derivatives

### Limits »

compute a limit

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

compute a one-sided limit

### 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

### Sums »

compute an indexed sum

#### sum j^2, j=1 to 100

compute an infinite sum

### Products »

compute an indexed product

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

compute an infinite product

### Series Expansions »

find a Taylor series expansion

#### taylor series sin x

specify the center point and the order of the expansion

### 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

### 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

### Complex Analysis »

compute properties of a function of a complex variable (use the variable z)

#### e^z

compute the residue of a function at a point

### Vector Analysis »

compute the gradient of a function

#### grad sin(x^2 y)

calculate alternate forms of a vector analysis expression

### Integral Transforms »

compute a Fourier transform

#### Fourier transform exp(-x^2)

compute a Laplace transform

### Domain & Range »

compute the domain of a function

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

compute the range of a function

### Continuity »

determine whether a function is continuous

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

locate discontinuities of a function

### 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