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Tag: Linear Approximation

Differentials and Linear Approximation

by theorangedog on Jan.11, 2008, under Skills

Q. Find the linear change in volume dV if the sides of a cube change from 10 to 10.1.

A. We can use the definition of small dy = (dy/dx)dx to determine the estimated linear change. In this problem, we start with the cubic volume equation of small V = x^3 where small x is the length of one side of the cube. The derivative is then small f^{prime}(V) = 3x^2. Filling in for linear approximation we get small dV = 3x^2dx. This results in small dV = 3(10^2)(.1) = 30. We can see that the linear approximation is not entirely accurate, hence the name approximation, as small Delta V = 10.1^3 - 10^3 = 30.301.

Related equations and relationships:
If small df/dx > 0 then small f(x) is increasing. If small df/dx < 0 then small f(x) is decreasing.

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