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Chapter 3.9 Exercise 1
Problem
Use a differential to estimate the change in the volume of a cube caused
by an increase h in the length of each side. Interpret geometrically
the error of your estimate Delta V - dV.
Solution
A cube of edge x has volume x**3. By the method of differential
approximation, adding h to the length of each edge increases the volume
by approximately 3*h*x**2. Here, dV accounts only adding a layer of
thickness h to each of three faces of area x**2, but not the three
smaller boxes with dimensions h*h*x, or the small cube of volume h**3.
That is, Delta V - dV = h**3 + 3*x*h**2.