Question

How do you find the average rate of change of `1/x` over the interval [1, 4]?

Answer 1

`"average rate of change" = -1/4`

Explanation:

The average rate of change of a function `f(x)` over an interval `[a,b]` is given by:
`"average rate of change" = ("change in y") / ("change in x") = ( f(b)-f(a) )/(b-a)`

So with `f(x)=1/x` over `[1,4]`
`"average rate of change" = ( f(4)-f(1) ) / (4-1)`
`:. "average rate of change" = (1/4-1) / 3`
`:. "average rate of change" = (-3/4) / 3`
`:. "average rate of change" = -1/4`

Answer 2

`-1/4`

Explanation:

The average rate of change of a function over an interval between (a , f(a)) and (b ,f(b)) is the slope of the `color(blue)"secant line"` joining the 2 points.
That is.

`color(red)(bar(ul(|color(white)(2/2)color(black)((f(b)-f(a))/(b-a))color(white)(2/2)|)))`

here a = 1 and b = 4

`rArrf(1)=1" and " f(4)=1/4`

`rArr(1/4-1)/(4-1)=(-3/4)/3=-1/4larr" average rate of change"`