Question

Question #623ea

Answer 1

Average rate change `(Deltay)/(Deltax)` over the interval `1<=x<=4`

Here `y=f(x)=log_(1/2)x`

`x_1=1 and x_2=4`

So
`(Deltay)/(Deltax)=(f(x_2)-f(x_1))/(x_2-x_1)=(log_(1/2)4-log_(1/2)1)/(4-1)`

`=log_(1/2)(1/2)^(-2)/3=-2log_(1/2)(1/2)/3=-2/3`

Answer 2

`-2/3`

Explanation:

The average rate of change is found using this formula:

`(y_2-y_1)/(x_2-x_1)`

Look familiar? It's just the formula for slope! The average change over a period of time is the slope of the line passing through the start and end points.

In this case, when `x=1`:

`y=log_(1/2)(1) = 0`

(since `(1/2)^0=1`)

And when `x=4`:

`y = log_(1/2)(4) = -2`

(since `(1/2)^-2=2^2=4`)

So now we have our start and end points:

`(y_2-y_1)/(x_2-x_1)`

`(-2-0)/(4-1)`

`-2/3`

So the average rate of change is `-2/3`

Final Answer