Math 12 - Computing Average Rate of Change on a TI-83/84 - YouTube

Okay, so in this video we’re going to be learning how to compute the average rate of

change of a function in the graphing calculator over an interval.

In this example the interval will be the interval from 2 to 2 + h.

We’re going to start by entering the function as Y1, and then we’ll enter the average

rate of change of Y1 over the interval from 2 to 2+h as Y2, I’ll show you how to do

that.

And then we’ll estimate the derivative of the function that’s entered as Y1 at the

x-value 2 using a table of values for the average rate of change.

So we’ll be computing the limit of the average rate of change to estimate the derivative.

So those are the steps we are going to go through as we do this video.

Okay, so I’ve brought up the graphing calculator here.

You can see the cursor in the home screen.

So what I’m going to do is first enter the function in the example that we are working

in the notes as Y1.

So I’ll bring up the equation editor as Y=, so I’m going to press the Y= button

here near the top.

And as I press buttons, you’ll see that the button will turn red when I press it,

so you can keep track of what buttons I’m pressing if you can’t see the cursor.

As I start pressing buttons you’ll see them turn red indicating that I’ve just pressed

one.

So if I press Y1, you can see that it turned red.

Okay, so let’s enter the function as Y1.

So for that I’m going to use the minus (–) key down at the bottom here, and I’m going to

press negative (-) 16 and go to my X-variable button here next to the green ALPHA button

and press X, and then the squared key down here.

And then I’m going to add the term 80X, and then I’m going to add 4—and this is

the function we’re working in the notes in this example.

Okay, so I’m going to press the down button; I’ve entered my function as Y1.

So pressing the down button, for Y2 I’m going to enter the average rate of change

of Y1 over the interval in the example, which is going to be 2 to 2 + h.

So the first thing I have to do is I have to call Y1 from Y2, because Y2 is going to

be the average rate of change of Y1 over the interval.

So I’ll begin by starting with the parentheses, because I’m going to have a difference in

the numerator, so I need to close that in parentheses.

So I’ll hit my left parenthesis button.

Now to call Y1 from Y2, this is where you have to pay attention.

So below the down arrow key you can see a button called “VARS”, V – A – R – S,

so I’m going to press that button. And I want to actually highlight the Y–VARS menu

over here. So I’m going to use the right arrow to highlight Y–VARS, and then I’m

going to press 1 or ENTER—I’m just going to type 1 to access the FUNCTION variables,

because Y1 is a function variable. And then I’m going to press 1 to select Y1, or press

ENTER. So I’ll just press 1. And you can see that the function variable Y1 is now copied

in Y2.

Now the interval is 2 to 2+h. Now on the graphing calculator there is only one variable and

that’s X. So in Y2 I have to remember that X now is going to be playing the role of h

in the average rate of change computation. So the X in Y2 is going to actually be h in

the definition of the average rate of change over the interval. So I’m going to put a

parenthesis here and then type 2 plus X, but what I’m really entering is going to be

2 + h. I just can’t use h I have to use X, because that’s the variable on the graphing

calculator. Then I’m going to choose the minus button here, and then I’m going to

go back and then I have to call Y1 and subtract Y1 of 2. So again going back to the VARS button,

hitting the right arrow button to highlight the Y–VARS menu, choosing 1 for FUNCTION,

and then 1 for Y1 again. And then I’ll evaluate that at 2 because that’s the interval we’re

working with, 2 to 2 + h, and then I’ll close the parenthesis in the top and I’ll

divide by X, but this is really h. So what I’ve really entered for Y2 is my function

evaluated at 2 + h minus my function at 2, divided by h—except X is playing the role

of h in Y2. So I’ve entered the average rate of change now as function Y2. The next

thing I want to do is I want to turn the function off as Y1, because I’m not going to be interested

in the function now, I’m actually interested in the values of the average rate of change

when I evaluate the derivative.

So I’m going to go up to Y1—I’m going to use the up arrow button, and then I’m

going to maneuver the cursor over the equal sign, and then I’m going to press ENTER,

and what you’ll see here if I arrow down over here to Y3 and move the cursor out of

the way. You can see that next to Y1 the equal sign doesn’t have a bold around it. It’s

not “boldened”. So you can see Y2 has a bold—the equal sign is marked in bold,

and Y1’s equal sign is not marked in bold. And that tells me that Y1 is not—so when

I do my table Y1 will be turned off, but Y2 will be turned on and that’s what I want.

Now I’m just going to double-check in my table. I’m going to go 2ND and then TABLE

SETUP above the WINDOW. I just want to check one thing. So I’ll do that. And I want the

independent variable to say “Ask”. So if it doesn’t, you just maneuver using the

arrow keys, highlight Ask next to Independent variable. Leave dependent variable, or “Depend”

to “Auto,” but you want independent to have Ask bold, and that’s what I’ve done

here. So if you go up you can see. And so what I’m doing is that allows me to specify

what values of X, and remember that those are really h-values. I’m going to input

those by hand on the calculator. I don’t want the calculator to self-generate the table,

so this is doing that for me.

Okay, now I’m ready to look at my table, so I’m going to hit the 2ND function and

then above the GRAPH key it says “TABLE”, 2ND and then TABLE, and you can see there’s

is nothing here. Now remember that Y2 is the average rate of change. Now remember when

you compute the derivative, you actually are going to approach, you are going to let h,

which is X now, so you are going to let X approach 0, but you want to do this from both

sides of 0, so you want to choose a sequence of values approaching 0 from both the right

and left, and then you want to try to guess the value of the derivative based on the limiting

value. So we’re really just estimating the value of a limit except that the limit now

is the limit of the average rate of change as h, which is X, approaches 0.

So coming in from the right side, I’ll maybe enter 0.1 ENTER, and 0.01, and maybe 0.001,

and maybe 0.0001, and I’ll stop there. So coming in from the right side, and of course

if I type 0 I’ll get an error, because I can’t divide by 0. And then coming in from

the left side, I might come in at -0.1—except I’ll start with the closest value to 0,

so let’s go –0.0001, and then sort of moving left –0.001, –0.01, and maybe –0.1.

Okay, so remember that we’re approaching 0 here, so I’m interested in the values

as I move up the negative side, as I’m moving closer to 0 the values (of Y2) from the left

side the values are getting closer to about 16. You can see the value here just slightly

to the left of 0 at –0.0001 of 16.002, and if I continue up in the table as I approach

0 from the right side as I move down the table, you can see that the Y-values here in the

table are getting again—they seem to be getting pretty much closer to 16. So in this

example, a good estimate of the limit, which would be the value of my derivative when x

= 2, a good value for that derivative when x = 2, would be about 16. That would be the

estimate I would use in this example based on my table of values, and you can certainly

get closer to 0 to get better estimates of the derivative if you like. But that’s all

we wanted to do in this video, and so that completes this video.