Math 16A 9-9-08 STUDENT: Sections group that ended up not having their section last week because of confusion. PROFESSOR: If turned in your homework, we'll do the right thing. STUDENT: And it is, (inaudible). PROFESSOR: That is the last thing I posted. The best thing to do is, I haven't looked at schedule at Berkeley dot edu, I hope it hasn't changed. I guess that's consensus. We meet in the same place. Any other questions? Sorry about that. So last time we talked about the slopes of lines and some of their basic properties. I want to say a couple of mores words on that before we go on to slopes of curves which is are what calculus is all about. So there's the equation of a line. And there's the slope as you know. And I want to point out that there's a synonym for that that we're going to use to make it easy to understand what the slope of a curve is. We're also going to call it the rate of change. That's a pretty simple idea. Let me draw our favorite straight line again. Y equals M X-plus B-. Take two points on it. I'll call this point X-one comma Y-one. And this points X-two comma Y-two. So let's me draw this liel triangle here. And so that distance interest there to there. That's X-two minus X-one. And that distance,. Q That points to that point vertically is Y-two minus Y-one. And if you take these two equations, and subtract them, Y-two equals M X-two plus B-and subtract, you'll get this which says that the, then divide, (on board). So it says that that side divide by that side is M. And you can call that the rate of change is how much your vertical position changes divided by how much your horizontal changes. And that's a perfectly good idea for lots of other things that don't have to be straight lined. lines. So legality me draw a curve that looks like this, which is not a straight line. Could be any other curve. Call this point X-one comma Y-one. Call that point X-two comma Y-two. I'll draw the same triangle, right, so this distance is just the same as it was before. X-two minus X-one. This vertical distance is just the same as before, Y-two minus Y-one. And so I'm going to use this same formula and say that Y-two minus Y-one divided by X-two minus X-one is the rate of change of this curve, give the came capital C-as you move from X-one to X-two. It's a straight line it's the same idea we had before. Just the slope M. It doesn't have to be a straight line. I can still say that's the rate of change. And so that's what in calculus is going to be about. Much of calculus is going to be about figuring out how to compute those rates of changes in a nice easy way. Especially whether these two points get very close together. I'll get it that later in the lecture. So the other reason to think about rate ofchange as a nice synonym is because a lot of problems in the real world use language like that as opposed to talk about slopes. The, one economic sample where the rate of change turns out -- in may 2003, mad cow disease was discovered and in Canada and it shut down Canada yen we've exports. They're not the only country with this member. And then finally they cured it and at some point and they started exporting again. Roughly let's say September 1st 2003 and the way it's reported, the way people measure it, exports rose at a rate, think rate of change, of some amount, I think it was $42.5 million a month. That was what was reported. That's the way an economist thinks of it. That's the rate of change up there. so let's use that to say in which month did exports return to their original value of $140 million a month. Excuse me, my handwriting is a business mal, A-B-Y-S-MA L. $140 million a month. No sorry. $170 million a month. That's the rate. So let's translate that into the calculus language of the slide of the board right above it. So what I want to do is write do you think the equation of a straight line. I want function E-X-equals export value in month X-after September 1st 2003. I want to write down an equation for that. And so what we know is this going to be a stlait line because of constant rate of change. Straight line because of constants rate of change. It's equal to zero, at zero, that's when we start counting September 1st am there's no exports then. It starts threaten. So our straight line goes to the .00. Straight line goes through the .00. Okay so that's nice easy straight line. And its slope is that rate, that was given if the problem. The slope is $42.5 million per month. So let's make sure we understand the units. So X-equals zero. That's a start. That's September 1st. X-equals one, that means one month later, so that would be October 1st, X-equals two would be two months later and I'm probably going to get the day, but it's about November 1st. Not quite, but close. So that's the idea. So let's try to write down the equation. E-of X-is going to be a slope times X-plus intercept. So we know that it goes through the origin. Zeros through 00, so what does P-equal, B-equals zero. And M is 42.5 million. So I'll just write down, I'm at the bottom of the board anyway, so there's the equation E-of X-is 42 and a half million times X. I left off the dollars. So now I have the equation of a straight line that tells us what the exports are. That was the first part of problem. We wrote it dpun from the math language that we're used to. so final question is when do we get back to our original export level. I can solve the equation. What equation do I have to solve? I want to know when does the export value equal what? 170 million. Okay. So when does 42.5 million times X-equals 170 million we do division. And the millions cancel. And 170 divided by 42.5 I hope I did it right is four am it take four months before we get back to where when were before make money. So September 1st, October 1st being November 1st, December 1st being January 1st, 4 months means by January first exports are back to normal. Okay. So that's an simple straight line economic model just tomake sure we understand what rate of change is. Are there any questions? Before I go on to curves as opposed to straight lines? Maybe I'll leave that there. So on to section 1.2 which is the slope of a curve at a point. So there that curvy picked two points that were far apart and I said how much did the curve change. What was the rate of change. I went …