Let f be defined as follows. y = f(x) = x^2

Let f be defined as follows. y = f(x) = x^2 - 5x (a) Find the average rate of change of y with respect to x in the following intervals. from x = 4 to x = 5 from x = 4 to x = 4.5 from x = 4 to x = 4.1 (b) Find the (instantaneous) rate of change of y at x = 4.

Transcript

00:01 Okay, so we have this function, f of x, which is equal to x squared minus 5x, and we want to find the average rate of change of this function with respect to x in some intervals.

00:11 So the average rate of change is the change in the y value divided by the change in the x value.

00:16 So, for example, from x equals 4 to x equals 5, to find the average rate of change between these points, we have to do f of 5, so f at the final point, minus f of 4, f at the initial point, divided by the change in the x values, which is 5 minus 4.

00:34 So f of 5, if we plug this in, we get 5 squared minus 5 times 5.

00:39 Put this in brackets, f of 4 is 4 squared minus 5 times 4.

00:44 And then we divide by 5 minus 4, which is 1.

00:48 So i won't write this down.

00:49 We divide by 1 here, which doesn't change anything.

00:52 So here we have 5 squared minus 5 squared, which is 0.

00:55 So we have 0 minus 4 squared, which is 16.

01:00 And then plus because the minus is cancelled, plus 20, which is 4.

01:04 So this is our first answer.

01:07 And then we just need to do basically the exact same thing for the other parts.

01:10 For x equals 4 to x equals 4 .5, we do f of the final point, 4 .5 minus f of the initial point 4, divided by the difference in the x values, 4 .5 minus 4.

01:24 So then we need to work this way.

01:25 We have 4 .5 squared minus 5 times 4 .5.

01:31 Put this order in brackets, minus f of 4, which is 4 squared minus 5 times 4.

01:36 And then divide this by 4 .5 minus 4, which is half...

Question

Let f be defined as follows. y = f(x) = x^2 - 5x (a) Find the average rate of change of y with respect to x in the following intervals. from x = 4 to x = 5 from x = 4 to x = 4.5 from x = 4 to x = 4.1 (b) Find the (instantaneous) rate of change of y at x = 4.

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