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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:

asymptotes (2x^3 + 4x^2 - 9)/(3 - x^2)

asymptotes of erf(x)

Cusps & Corners

Compute and visualize cusps and corners of a function.

Find cusps on the graph of a function:

cusps of sqrt |x-2| - cbrt |x+2|

Find corners on the graph of a function:

corners of x + 2 |sin x|

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:

maximize x(1-x)e^x

Minimize or maximize a function of several variables:

maximize 5 + 3x - 4y - x^2 + x y - y^2

Minimize or maximize a function subject to a constraint:

maximize e^x sin y on x^2+y^2=1

Surfaces & Solids of Revolution

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

Compute properties of a surface of revolution:

revolve f(x)=sqrt(4-x^2), x = -1 to 1, around the x-axis

Compute properties of a solid of revolution:

revolve region between y=x^2 and y=x, 0<x<1, about the y-axis

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:

concavity of the curve xe^(-x)

Determine the concavity of a function at a specific point:

Is x^4 concave up at x=0?

Determine the concavity of a function on an interval:

cos[x] + x^3/48 concavity on 1<x<1.8

Tangents & Normals

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:

tangent to x e^-x^2 at x=1/3

Find the normal to a curve specified by an equation:

normal to 3x^2-2xy+y^2=1 at x=0, y=1

Stationary Points

Compute and visualize stationary points of a function.

Locate stationary points of a function:

stationary points of (x^5+x^9-x-1)^3

Find stationary points of a function of several variables:

stationary points (3x+1)y^3 + x^2 y

Area between Curves

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

Compute the area bounded by two curves:

compute the area between y=|x| and y=x^2-6

Specify limits on a variable:

area between y=sinc(x) and the x-axis from x=-4pi to 4pi

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

Compute the curvature of a plane curve:

curvature of sin(x)

Compute the curvature of a space curve at a point:

what is the curvature of (s, sin s, cos s) at s=2

Compute and visualize saddle points of a function.

Locate saddle points of a function:

saddle points of x^3 - y^3 - 2xy + 6

Find the saddle point nearest to a specified point:

saddle point of sin(x y)-tan x near (5,6)

GO FURTHER

Step-by-Step Solutions for Applications of Calculus

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Inflection Points

Compute and visualize inflection points of a function.

Locate inflection points of a function:

inflection points of x+sin(x)

Find inflection points in a specified domain:

inflection points of e^sin(x^3)-1 with -2<=x<=2

Compute arc lengths in various coordinate systems and dimensions.

Compute the arc length of a curve:

arc length of y=x^2 from x=0 to 1

length of the curve {x=2cos(t), y=2sin(t), z=t} from 0 to 7

Monotonicity

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

Explore regions where a curve is increasing, decreasing or flat:

where is 300 x - x^3 increasing?

decreasing intervals of abs(-x^2/3+4)

Determine whether a function is increasing at a specific point:

Is x^4 concave up at x=0?

Determine the concavity of a function on an interval:

monotonicity of the curve sin(x^2) between -pi and pi