Question

What is the average rate of change of the function `f(x)=2x^2 -3x -1` on the interval [2, 2.1]?

Answer 1

`5.2`

Explanation:

The average rate of change of the function `f(x)` on the interval `[a,b]` is:

`"average rate of change"=(f(b)-f(a))/(b-a)`

Here, this gives us

`"average rate of change"=(f(2.1)-f(2))/(2.1-2)`

For this function, `f(2)=1` and `f(2.1)=1.52`.

`"average rate of change"=(1.52-1)/(0.1)=0.52/0.1=5.2`

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Extension:

The average rate of change on this interval should be approximately equal to the value of the derivative (rate of change) of the function halfway through the interval.

The function's derivative is

`f'(x)=4x-3`

Halfway through the interval is at `x=2.05`, and the value of the derivative is

`f'(2.05)=5.2`

In this case, they are exactly the same.