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Step-by-Step Applications of Calculus

Unlock the practical applications of calculus through Wolfram|Alpha's step-by-step instructions, where you'll delve into essential concepts such as finding tangent lines, identifying inflection points, optimizing functions, calculating average values, determining arc lengths and measuring areas between curves. Whether you're a student aiming to grasp real-world calculus applications or a professional seeking to utilize calculus in your field, Wolfram|Alpha equips you with the skills needed to tackle these diverse and valuable calculus applications.

Tangent Lines & Planes

Find a line tangent to a curve using the derivative at a point:

tangent line to sin(x) - 1 at x=0
compute the line tangent to (x^3-1)(x+2) at x=1
tangent line of y=x^2 at x=1

Calculate a multivariate tangent plane by examining partial derivatives:

tangent plane to z=x^2+y^2 at x=1,y=1

Optimization

Explore step by step a variety of tests to discover the extrema:

find the extrema of 3x^4 - 4x^3 - 12x^2 + 1
local extrema of x^3 - 3x + 1

Compute the local maxima or minima of a function:

local minimum for y==x^2-x+1
local maximum x^5 - 10x^3 + 30x

Show the steps for finding the global minima or maxima of a function:

global minima for y==x^4-2x^2+1
minimize sin(x)
global maxima for 1/x
maximize x^5-2x^2+1

Discover how to calculate the extrema of a function on a given interval:

maximize x^5-2x^2+1 on -1<x<1
minimize -2Cos[x] on -1<x<1

Arc Length

Calculate the arc length of a curve step by step:

arc length of Sin[x] for x between 0 and 2pi

Specify a curve in polar coordinates:

arc length of r=theta^2 for 0<theta<2pi

Specify a curve parametrically:

arc length of {Sin[x], Cos[x], x, x} for 0<x<3pi

Inflection Points

Calculate inflection points step by step:

inflection points x^3 - 3 x^2 + 1
points of inflection t * cbrt(t^2 - 4)
inflection points of 3sin(2x)
inflection pts sgn(x)|x|^(3/2) + x^3 - 3x^2 + 1 over -2 < x < 3
x^3-3x^2-4x+1 inflection points

Average Value

Find the average value of an equation on an interval:

average value of 1/x for 1<x<5

Look through the steps to find the average value of a derivative on an interval:

average value of the derivative of Sin(x) for x=1 to 5

RELATED EXAMPLES

  • Applications of Calculus
  • Arc Length
  • Area between Curves
  • Inflection Points
  • Optimization
  • Tangents & Normals

    • Area between Curves

    Learn how to compute the area between curves:

    area between 1/2x(x-2)(x+2) and x
    area of the region between y=x and y=x^2 for -1/2<x<1
    area under 1/x for 1<x<5
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