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}

0lz try yourself sorry for not answering

proportions are fractions that equal to each other. so if two fractions are the same, they are proportional.

let's check the answers to see if they are proportional.

question a is not proportional because you can't turn 1.5/3 to 3/9. if you try to multiply 1.5/3 by 2 or 3, you wont get 3/9. it's impossible to make the fractions the same.

question b is not proportional because you can't turn 22/33 to 4/6. 33 can't go into 6.

question c is proportional because when you multiply 2 and 3 by 71, you would get 142/213, and those fractions equal to each other

question d is proportional because when you divide 14 and 7 by 2, you would end up with 7/3.5, and those fractions equal to each other.