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.