Find the average rate of change on the interval specified for real numbers h. f (x) = - 2 x 3 on [ x, x + h ]

S = - x^{2} - xh - h^{2}

Step-by-step explanation:

In this question, we have f (x) = y = - 2x^{3}.

Given a function y, the average rate of change S of y=f (x) in an interval [x_s, x_f] will be given by the following equation:

S = / frac{f (x_{f}) - f (x_s) }{x_f - x_s}

So, in your problem, f (x) = - 2x^{3}, x_{f} = x+h and x_{s} = x. Applying this to the equation S above, we have:

S = / frac{f (x+h) - f (x) }{x+h - x}

where f (x+h) = - 2 (x+h) ^{3} = - 2 (x^3 + x^{2}h + xh^{2} + h^{3})

Now

S = / frac{-2 (x^3 + x^{2}h + xh^{2} + h^{3}) - 2x^{3}}{h}

S = / frac{-x^{2}h - xh^{2} - h^{3}}{h}

The numerator can be simplified by dividing h. So

S = / frac{h (-x^{2} - xh - h^{2}) }{h}

Simplyfing h, the average rate of change will be

S = - x^{2} - xh - h^{2}