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Examples for

Calculus & Analysis

Calculus is the branch of mathematics studying the rate of change of quantities and the length, area and volume of objects. With the ability to answer questions from single and multivariable calculus, Wolfram|Alpha is a great tool for computing limits, derivatives and integrals and their applications, including tangent lines, extrema, arc length and much more.

Compute definite and indefinite integrals of functions. Integrate with respect to one or more variables.

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

Explore the limit behavior of a function as it approaches a single point or asymptotically approaches infinity.

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+

Compute an indexed product by multiplying a finite or infinite number of terms.

Compute an indexed product:

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

Compute an infinite product:

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

Vector Analysis

Apply the curl, the gradient and other differential operators to scalar and vector fields.

Compute the gradient of a function:

grad sin(x^2 y)

Calculate alternate forms of a vector analysis expression:

Find discontinuities and continuous intervals of a function. Also determine whether particular discontinuities are removable or infinite due to an asymptote.

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)

Take the derivative of single or multivariate functions. Compute the partial derivative of expressions with more than one variable.

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

Calculate and examine sequences of integers or other numerical values. Find continuations and formulas for known or unknown 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)

Series Expansions

Find Taylor series, Laurent series and more about any point.

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

Integral Transforms

Calculate the Fourier transform, Laplace transform and other integral transforms of functions.

Compute a Fourier transform:

Fourier transform exp(-x^2)

Compute a Laplace transform:

GO FURTHER

Step-by-Step Solutions for Calculus

Calculus Web App

Free Unlimited Calculus Practice Problems

RELATED EXAMPLES

Complex Analysis

Coordinate Geometry

Differential Equations

Numerical Analysis

RELATED WOLFRAM RESOURCES

Wolfram Community Calculus Group

Wolfram Calculus Guide Page

Compute the value of indexed summations, or sums of sequences of values. Compute infinite sums and find convergence conditions.

Compute an indexed sum:

sum j^2, j=1 to 100

Compute an infinite sum:

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

Applications of Calculus

Use calculus tools, such as integrals and derivatives, to calculate properties of curves, surfaces, solids and planar regions.

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)

Compute the domain and range of real mathematical functions. Plot the domain and range on a number line.

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)