Question

What is the average rate of change of the function `y = x^2+1` over the interval `[5, 5+a]` ?

Answer 1

The average rate of change will be the total change in `y` divided by the total change in `x`, so we just have to look at the values at each end of the given interval:

When `x = 5`

`y = 5^2+1 = 26`.

When `x = 5 + a`:

`y = (5+a)^2 + 1 = 25+10a+a^2+1`

`= 26+10a+a^2`

`Delta y = (26+10a+a^2)-26 = 10a+a^2`

`Delta x = (5+a)-5 = a`

So `(Delta y)/(Delta x) = (10a + a^2)/a = 10+a`