Lec 1 MIT 18.01 Single Variable Calculus, Fall 2007

Хөгжмийн зохиолч

• Speaker 1 0:36
• Speaker 2 0:19
• Speaker 3 0:00

1. 0:00

   Хэсэг 1: The following content is provided under a Creative Commons license. 299s · Speaker 3

   The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of…

2. 5:00

   Хэсэг 2: And here's some point. 90s · Speaker 1

   And here's some point. Maybe I should draw it just a bit lower so that I don't. So here's a point P. Maybe it's above the point x0. x0, by the way, this was supposed to be an x0. That was some fixed place on the x -axis. And now, in order t…

3. 6:30

   Хэсэг 3: So what did we learn in high school about what a tangent line is? 298s · Speaker 2

   So what did we learn in high school about what a tangent line is? Well, a tangent line has an equation. Any line through point has the equation y -y0 is equal to m the slope times x -x0. So here's the equation for that line. And now there a…

4. 11:35

   Хэсэг 4: Again, this is still a geometric discussion. 179s · Speaker 1

   Again, this is still a geometric discussion. But now we're going to be able to put symbols and formulas to this computation. And we'll be able to work out formulas in any example. So let's do that. So first of all, I'm going to write out th…

5. 14:35

   Хэсэг 5: here. The change in f. 300s · Speaker 3

   here. The change in f. So remember that the point P is the point x0, f . All right, that's what we got from our formula for the point. And in order to compute these distances, and in particular the vertical distance here, I'm going to have …

6. 19:35

   Хэсэг 6: So this is equal to, let's continue on the next. 111s · Speaker 1

   So this is equal to, let's continue on the next. level here. This is equal to 1 over delta x times, now all I'm going to do is put it over a common denominator. So the common denominator is x0 plus delta x times x0. And so in the numerator …

7. 21:26

   Хэсэг 7: So in other words, what I've shown, let me put it up here, is that f prime of x0 is minus 1 over x0 squared. 302s · Speaker 2

   So in other words, what I've shown, let me put it up here, is that f prime of x0 is minus 1 over x0 squared. Now, let's look at the graph just a little bit to check this for plausibility. What's happening here is, first of all, it's negativ…

8. 26:28

   Хэсэг 8: But all the rest of it relies on everything else you learned in mathematics up to this stage, from grade school through high school. 302s · Speaker 2

   But all the rest of it relies on everything else you learned in mathematics up to this stage, from grade school through high school. So that's the complication. So now we're going to do a little bit of calculus made hard by talking about a …

9. 31:31

   Хэсэг 9: how do we do that? 301s · Speaker 2

   how do we do that? To find this point, this has a name. We're going to find the so -called x -intercept. That's the first thing we're going to do. So to do that, what we need to do is to find where this horizontal line meets that diagonal l…

10. 36:33

    Хэсэг 10: And you could do the same thing analogously the other way. 301s · Speaker 1

    And you could do the same thing analogously the other way. All right, so I'm almost done with the geometry problem. And let's finish it off now. Well, let me hold off for one second before I finish it off. What I'd like to say is just make …

11. 41:34

    Хэсэг 11: And these are just other ways that people refer, notations that people use to refer to derivatives. 300s · Speaker 2

    And these are just other ways that people refer, notations that people use to refer to derivatives. And the first one is the following. We already wrote y is equal to f , and so when we write delta y, that means the same thing as delta f. T…

12. 46:35

    Хэсэг 12: the rest x's. 79s · Speaker 1

    the rest x's. So at least one term of the form x to the n minus 1 times delta x. And how many times does that happen? Well, it happens when there's a factor from here, from the next factor, and so on and so on and so on. There's a total of …

13. 47:54

    Хэсэг 13: that we need. 216s · Speaker 2

    that we need. So here we go. We have delta f over delta x, which remember was 1 over delta x times this, which is this. Times, now this is x to the n plus n x to the n minus 1 delta x plus this junk term minus x to the n. So that's what we …