18-01F07-L06 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Please use the following citation format: David Jerison, 18.01 Single Variable Calculus, Fall 2007 . (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms
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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus, Fall 2007 Transcript – Lecture 6 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: All right, so let's begin Lecture Six. We're talking today about exponentials and logarithms. And these are the last functions that I need to introduce, the last standard functions that we need to connect with Calculus, that you've learned about. And they're certainly as fundamental, if not more so, than trigonometric functions. So first of all, we'll start out with a number, a, which is positive, which is usually called a base. And then we have these properties that a to the power 0 is always 1. That's how we get started. And a^1 is a. And of course a^2 , not surprisingly, is a times a, etc. And the general rule is that a^(X1 X2) is a^X1 times a^X2 . So this is the basic rule of exponents, and with these two initial properties, that defines the exponential function. And then there's an additional property, which is deduced from these, which is the composition of exponential functions, which is that you take a to the X1 power, to the X2 power. Then that turns out to be a to the X1 times X2. So that's an additional property that we'll take for granted, which you learned in high school. Now, in order to understand what all the values of a^x are, we need to first remember that if you're taking a rational power that it's the ratio of two integers power of a. That's going to be a ^ m, and then we're want to have to take the nth root of that. So that's the definition. And then, when you're defining a ^ x, so a^x is defined for all x by filling in. So I'm gonna use that expression in quotation marks, "filling in" by continuity. This is really what your calculator does when it gives you a to the power x, because you can't even punch in the square root of x. It doesn't really exist on your calculator. There's some decimal expansion. So it takes the decimal expansion to a certain length and spits out a number which is pretty close to the correct answer. But indeed, in theory, there is an a to the power, square root of 2, even though the square root of 2 is irrational. And there's a to the pi and so forth. All right, so that's the exponential function, and let's draw a picture of one. So we'll
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18-01F07-L06 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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