The function fx x3 describes the volume of a cube fx in cub

The function f(x) = x3 describes the volume of a cube, f(x), in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 1 inches to 1.1 inches.

Solution

V = f(x) = x³

x changes from 1 inches to 1.1 inches ==> dx = 1.1 - 1 = 0.1 inches

==> dv = 3 x^3-1 = 3x² dx          since d/dx xⁿ = n x^n-1 , d/dx f(g(x)) = f \'(g(x)) g \'(x)

==> dv = 3x² dx

==> dv = 3(1)² (0.1)

==> dv = 0.3

Hence change in volume = 0.3 cubic units