Examples for

# Applications of Calculus

The tools of calculus can be used in a variety of applications to compute properties and to describe the behavior of functions, curves, surfaces, solids and many other mathematical objects. Harness Wolfram|Alpha's comprehensive computational understanding of limits, derivatives and integrals to find asymptotes, tangents and normals of curves; compute arc length; find the singular and stationary points of functions; explore increasing and decreasing regions of curves; and beyond.

Compute horizontal, vertical or slant asymptotes.

#### Compute asymptotes of a function:

Compute and visualize cusps and corners of a function.

#### Find cusps on the graph of a function:

#### Find corners on the graph of a function:

Find global and local extrema and stationary points of functions or impose a constraint on a function and compute the constrained extrema.

#### Minimize or maximize a function:

#### Minimize or maximize a function of several variables:

#### Minimize or maximize a function subject to a constraint:

Compute the area of a surface of revolution or compute the volume of a solid of revolution.

#### Compute properties of a surface of revolution:

#### Compute properties of a solid of revolution:

### Concavity & Convexity

Find and visualize where a curve is concave up or concave down.

#### Find the intervals where a curve is concave up, concave down and straight:

#### Determine the concavity of a function at a specific point:

#### Determine the concavity of a function on an interval:

Compute a tangent line to a curve or compute a tangent plane or a normal line to a surface.

#### Find the tangent to the graph of a function at a point:

#### Find the normal to a curve specified by an equation:

Compute and visualize stationary points of a function.

#### Locate stationary points of a function:

#### Find stationary points of a function of several variables:

Compute the areas of enclosed regions, bounded regions between intersecting points or regions between specified bounds.

#### Compute the area bounded by two curves:

#### Specify limits on a variable:

Compute the curvature of functions and parameterized curves in various coordinate systems and dimensions.

#### Compute the curvature of a plane curve:

#### Compute the curvature of a space curve at a point:

Compute and visualize saddle points of a function.

#### Locate saddle points of a function:

#### Find the saddle point nearest to a specified point:

### RELATED EXAMPLES

Compute and visualize inflection points of a function.

#### Locate inflection points of a function:

#### Find inflection points in a specified domain:

Compute arc lengths in various coordinate systems and dimensions.

#### Compute the arc length of a curve:

### Monotonicity

See and measure where a curve is monotonically increasing or decreasing.