Lec 2 MIT 18.01 Single Variable Calculus, Fall 2007
පරිච්ඡේද
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5:00පරිච්ඡේදය 2: Two typical examples. 299s · Speaker 2
Two typical examples. And I want to illustrate the second example in a little bit more detail. Because I think it's important to have some visceral sense of this notion of instantaneous speed. And I get to use the example of this very build…
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10:02
So that's using the fact that d by dt of 80 is equal to 0. And d by dt of t squared is equal to 2t. The special case, well, I'm cheating here, but there's a special case that's obvious. I didn't throw it in over here. The case n equals 2 is…
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13:09පරිච්ඡේදය 4: So let me explain this. 301s · Speaker 2
So let me explain this. I don't want to belabor it because I just am doing this in order to introduce you to the ideas on your problem set, which are the first. So on problem set one, you have an example which is based on a simplified model…
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18:11
So now what I'd like to talk about is limits and continuity. And this is a warm -up for deriving all the rest of the formulas. All the rest of the formulas that I'm going to need to differentiate every function you know. Remember, that's ou…
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23:11පරිච්ඡේදය 6: OK, so I want to give an example of this. 303s · Speaker 2
OK, so I want to give an example of this. And also an example of how you're going to think about these sorts of problems. which has two different definitions. Say it's x plus 1 when x is bigger than 0, and minus x plus 2 when x is less than…
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28:14පරිච්ඡේදය 7: here. exists. 302s · Speaker 1
here. exists. And what that means is that there's an honest limiting value both from the left and right. And they also have to be the same. So that's what's going on here. And the second property is that f is defined. So I can't be in one o…
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33:17පරිච්ඡේදය 8: I want to give you is a little bit more subtle. 300s · Speaker 1
I want to give you is a little bit more subtle. It's what's known as a removable discontinuity. And so what this means is that the limit from left and right are equal. So a picture of that would be you have a function which is coming along …
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38:17පරිච්ඡේදය 9: That's this function here. 227s · Speaker 1
That's this function here. But now I'd like to draw also the other branch of the hyperbola down here and allow myself to consider negative values of x. So here's the graph of 1 over x. And the convenience here of distinguishing the left and…
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42:04පරිච්ඡේදය 10: Namely, the limit as x goes to 0 of , that's going to be equal to minus infinity. 304s · Speaker 2
Namely, the limit as x goes to 0 of , that's going to be equal to minus infinity. x going to 0 minus. So both have this property. Finally, let me just make one last comment about these two graphs. This function here is an odd function. And …
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47:08පරිච්ඡේදය 11: So I'll rewrite the limit as x goes to x0. 300s · Speaker 1
So I'll rewrite the limit as x goes to x0. of f minus f divided by x minus x0 times x minus x0. So I wrote down the same expression that I had here. This is just the same limit. But I multiplied and divided by x minus x0. And now, when I ta…
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52:08පරිච්ඡේදය 12: Last question. 35s · Speaker 1
Last question. How do we get the zero from this? So the claim that's being made, so the claim is why is this tending to that? So for example, I'm going to have to erase something to explain that. So the claim is that the limit as x goes to …