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FEMALE SPEAKER: 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 MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Today we will discuss the inverse hyperbolic functions and, with this lecture, finish our block on the logarithmic exponential and hyperbolic functions. And what we're going to find is that much of what we have to say today is simply a specific application to a special function of the same theory that we were talking about in general before. Recall that we can always talk about an inverse function if the original function is a one-to-one function. For example, to introduce today's topic, suppose we take the function 'y' equals hyperbolic sine 'x'. 'y' equals 'sinh x'. Well, as we saw last time, the graph of ...

FEMALE SPEAKER: 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 MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Technically speaking, I suppose, if time permitted, we should spend, or shall we say, give equal time to integral calculus, the same time that we gave to differential calculus, starting from scratch, building ideas slowly and the like. But there are two logistic reasons for not doing this, one of which is that the basic definitions would be the same regardless of which branch we started with. The second one is, of course, that time becomes preciously tight, and we have to make the best of what we can here. So what we're going to do now is try to show, in terms of hindsight being better than foresight, a motivation as to how one would have invented ...

FEMALE SPEAKER: 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 MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Our subject matter today concerns that phase of calculus known as the 'indefinite integral', or the 'antiderivative'. In terms of the concepts that we've talked about so far in our course, and keeping in mind that one concept with many applications is educationally more meaningful than many concepts each with one application, I prefer to call today's lesson the inverse derivative, or inverse differentiation. And again, bring into play a very well-known feature of our course, namely, our discussion of inverse functions. Now, you see, the only problem that comes up in this context is a rather simple one and that is that all the ti...

GUEST SPEAKER: 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 view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. HERBERT GROSS: Hi. Our lecture today involves the backbone of Calculus, namely the concept of limit, and perhaps no place in the history of mathematics is there a more subtle topic than this particular one. I'm reminded of the anecdote of the professor who was staring quite intently, philosophically, in his home, and his wife asked, what are you contemplating, and he said, that it's amazing how electricity works. And the wife said, well what's so amazing about that? All you have to do is flick the switch. And this somehow or other is exactly the predicament that the limit concept falls into. From the point of view of computation, it seems that what comes nat...