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}

answer: 1 2 3 6

step-by-step explanation:

1x6=6 2x3=6