Find the average rate of change of f(x)= 5x^2 + x between (x,f(x)) and (x+h,f(x+h)).

Answer 1

This looks like an exercise relating to the definition of the derivative.

Each point is given in the standard "(x,y)" format. We can use "y" for f(x)

The average rate of change is the difference in y values, divided by the difference in x values. The difference is x values is simply h.

The difference in y values is :

(5(x+h)^2 + x + h) - (5x^2 + x)

Expand and simplify

5x^2 +10xh + 5h^2 + x + h - 5x^2 - x

10xh + 5h^2 + h

divide by h

10x + 5h + 1

In the definition of the derivative, h goes to zero. This leaves us with 10x + 1, which is in fact the derivative of the original equation.