Lec 3 MIT 18.01 Single Variable Calculus, Fall 2007
Capitole
-
0:00
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…
-
2:34
So that's what we'll spend the first part of the lecture on. And at the same time, I hope to get you very used to dealing with trig functions. think of as a gradual process. All right, so in order to calculate these, I'm going to start over…
-
4:06Capitolul 3: No? Oh, OK. 302s · Speaker 4
No? Oh, OK. So which is it? OK. All right, let's take a vote. Is it sine sine or is it sine cosine? OK. So is this going to be cosine? All right. You better remember these formulas, all right? OK, it turns out that it's sine, cosine. All ri…
-
9:09Capitolul 4: equal to cosine, cosine. 305s · Speaker 2
equal to cosine, cosine. No, it's not cosine squared because there are two different quantities here. Cosine a, cosine b minus, here's the minus sign, sine a. All right. So you'll have to be willing to call those forth at will right now. So…
-
14:14Capitolul 5: values of d by dx, sine x, d by dx, cosine x. 184s · Speaker 1
values of d by dx, sine x, d by dx, cosine x. So that's really what this argument is showing us. Is that we just need one rate of change at one place, and then we work out all the rest of them. So that's really the substance of this proof. …
-
17:19Capitolul 6: and that's this. 302s · Speaker 2
and that's this. This little chunk there. So those are the two pieces. Now, in order to persuade you now that the limit is what it's supposed to be, I'm going to extend the picture just a little bit. I'm going to double it, just for my own …
-
22:21Capitolul 7: vertex gets farther and farther away, The curved line gets closer and closer to being a vertical line. 301s · Speaker 2
vertex gets farther and farther away, The curved line gets closer and closer to being a vertical line. That's sort of the flip side by expansion of the zoom in principle. The principle that curves are nearly straight when you zoom in. If yo…
-
27:22Capitolul 8: cosine theta minus 1 minus 1 is the negative of this. 301s · Speaker 1
cosine theta minus 1 minus 1 is the negative of this. And if I show that this goes to 0, it's the same as showing the other one goes to 0. Another question. So the question is, what about this business about arc length? So the word arc leng…
-
32:25Capitolul 9: So we're just a little away. 301s · Speaker 2
So we're just a little away. So that's what this picture down below in Part A is meant to be. It's supposed to be that theta has opened a tiny crack, just a little bit. And the smallest I can draw it on the board where you can visualize it …
-
37:27Capitolul 10: Well, in order to do that, we've got to keep track of the height, the vertical displacement here. 149s · Speaker 1
Well, in order to do that, we've got to keep track of the height, the vertical displacement here. So we're going to draw this right angle here. And this is the position r. is the change in y. So the picture is, we have something moving arou…
-
40:00Capitolul 11: So the key step is this same principle that we already used, which is that short pieces of curves are nearly straight. 302s · Speaker 2
So the key step is this same principle that we already used, which is that short pieces of curves are nearly straight. So that means that this piece of the circular arc here from p to q is practically the same as the straight segment from p…
-
45:02Capitolul 12: starting from p. 290s · Speaker 2
starting from p. And the orange rotates around 90 degrees and becomes this thing here. So this angle here is the same as the other one which I've just drawn. Different origins, different vertices for the angles. OK, well, I didn't say that …