Question

Given the function f(x) = x^2-6x+4, determine the average rate of change of the function over the interval 1≤x≤4.

Answer 1

The average rate of change of f(x) over the interval 1 ≤ x ≤ 4 is -1.

Explanation:

We are given the function:

`f(x)=x^2-6x+4`

And we want to determine its average rate of change over the interval:

`1\leq x\leq 4`

The average rate of change is synonymous with the slope. So, to find the average rate of change, we simply need to find the slope between the two endpoints.

Evaluate each endpoint:

`f(1)=(1)^2-6(1)+4=-1`

`f(4)=(4)^2-6(4)+4=-4`

So, our endpoints are (1, -1) and (4, -4).

The slope between them is:

`\displaystyle m=((-4)-(-1))/((4)-(1))=(-3)/(3)=-1`

So, the average rate of change of f(x) over the interval 1 ≤ x ≤ 4 is -1.