Estimating the value of a polar function with linear approximation

0:00- [Instructor] We're told the table provides values for the function b of theta. Marcus is estimating a value of b of 1 by using the average rate of change of b over the interval from pi over 4 to pi over 2, and it's including the endpoints there 'cause it's less than or equal.

0:19What is the closest estimate for b of 1? So you can pause this video, have a go at it before we do it together. Okay, so b of theta, it's definitely, at least the way it's written because it's a function of theta, it's some kind of trigonometric function. We could graph it in polar coordinates or rectangular coordinates, but I'm gonna think about it a little bit in rectangular coordinates just because it'll be a little bit easier to visualize the average rate of change.

0:47So let me make this axis the b of theta and let me make this axis right over here theta, so that is theta. And let's see, we have pi over 4, let me put pi over 4 right over here, pi over 4 and then pi over 2 is twice that, so it's right around there. Pi over 2, and let's see, we have 2 and 3, so let's say this is 1, 2, and 3.

1:15So we have this point right over there, which is the point... Maybe I'll do this in another color actually just so we can keep track of things. So this is the point pi over 4, comma, 2. And then this point over here is pi over 2, comma, 3. Oh, actually let me be careful, I put that first point in the wrong place.

1:38That one is right over here. If we assume that this is 2 right over here. So this is the point pi over 4, comma, 2, this is 2 right over here. So you see that, and then this other point is pi over 2, comma, 3, pi over 2, comma, 3 is right over there, so pi over 2, comma, 3.

2:10And what we wanna do is approximate b of 1. And we don't know what this function is doing between these two points. It might be some type of a curve, I'm just guessing, maybe it looks something like this. We really have no idea, but they want us to approximate by using the average rate of change.

2:28So this is really about linear estimation. So what we're thinking about is imagine if we approximate when theta is 1, so let's see, I don't know where exactly where it... But let's say 1 is right over there. So we wanna approximate this right over here, it would be the actual b of theta.

2:46We don't know what that is, we don't have the function defined there. But maybe we could approximate it by looking at the average rate of change, which would really be the slope of the line that connects these two points right over here. The average rate of change is the slope of the line that connects those two points.

3:01And if we know that, let me write that there, the slope is equal to the average rate of change. And if we're able to figure out the average rate of change, then the value, the approximated value for b of 1, and actually this would be b of 1 right here, that's the actual one.

3:21The approximated value, so this is going to be 1, comma, approximately b of 1 based on where you might sit on that line when you've gone that far past pi over 4. So another way to think about it, we're gonna, in a second, look at the average rate of change, but if we are starting at a value of 2, let me write it this way, b of theta is going to be approximately equal to, we can start at the value of 2, we're gonna start at the value of 2, and then we're going to add to that how much are we increasing in theta?

4:01We're increasing in theta by 1 minus pi over 4. This is our ending point, this is our starting point. So 1 minus pi over 4 is how much we're increasing in theta, so how much are we increasing in b? Well, it's going to be the slope times that change in theta. So it's gonna be the average rate of change times this 1 minus pi over 4.

4:26So plus average rate of change times 1 minus pi over 4. This is straight out of y equals mx plus b. Figure out the slope, figure out what the value might be at some point that you aren't given. That's essentially where this comes from. But now, let's actually apply it over here.

4:48So what we want to do, and let me be clear that what I just did right over here is b of 1. If I wanted to do b of theta, I would've had a theta right over here. So b of 1 is gonna be approximately this, and so what is our average rate of change? Well, average rate of change is going to be change in b or change in b of theta divided by change in theta.

5:14Our change in b between these two points is going to be from 2 to 3, so we'd have our finishing point minus our starting point. This might look a lot like slope because essentially that's what we're doing if we're thinking about rectangular coordinates. And then our change in theta is gonna be our ending point minus our starting point.

5:31So we have pi over 2 minus pi over 4. So this is equal to 1 over pi over 4, which is equal to 4 over pi. So we replace this with 4 over pi. Let's see, we could simplify this, this is gonna be approximately to 2 plus we have 4 over pi times 1 is the same thing as 4 over 4.

5:55So it's 4 minus pi over 4, and so this is going to be approximately 2 plus these 4s cancel out, 4 minus pi over pi. Let me get a calculator out now, and I am going to get 2 plus 4 minus pi divided by pi, and we get approximately 2.27. So this right over here is going to be 2.27, and we're done.