Calculate the average rate of change of the function f(x) =...

Calculate the average rate of change of the function $f(x) = x^3 - 3x^2$ between the given values. a. Find the average rate of change on the interval $[-3, -2]$. b. Find the average rate of change on the interval $[-2, -1.9]$. c. Find the average rate of change on the interval $[-2, -1.99]$. d. Find the average rate of change on the interval $[-2, -1.999]$. e. Based on your previous answers, estimate the instantaneous rate of change at $x = -2$.

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00:01 We have the function f of x is equal to x cubed.

00:03 And given the average rate of change formula, f of x2 minus f of x1, divided by x2 minus x1, we're going to determine the average rate of change for our function on some different intervals.

00:13 The first interval we have is from negative 2 to negative 1.

00:18 In this case, x1 will be negative 2 and x2 will be negative 1.

00:23 Sounds kind of backwards.

00:24 However, the smaller number should always be your x1, and negative 2 is smaller than negative 1.

00:30 So let's plug in what we have.

00:32 All right.

00:33 So f of x2, that is x cubed with minus 1 plugged in for x.

00:40 That's minus 1 cubed.

00:43 And for the very subtract, minus 2 cubed.

00:46 And divide that by minus 1 minus minus 2.

00:51 So many minuses.

00:53 Okay.

00:54 Let's rearrange this.

00:55 So these numbers are raised to odd powers, meaning you can pull the negative sign out and then just cube them as usual.

01:03 So we'll have negative 1 cubed, which is just 1, minus negative, so plus, and then 2 cubed, which is 8.

01:12 We divide that by minus 1, minus negative 2, or minus 1 plus 2, which is just positive 1.

01:19 And so then negative 1 plus 8, that's positive 7, divided by 1.

01:23 So just 7 is our final result for that interval...

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