1 00:00:00,000 --> 00:00:05,650 This is calculus, the single variable. And I am Robert Ghrist, professor of 2 00:00:05,650 --> 00:00:11,223 mathematics and electrical and systems engineering, of the University of 3 00:00:11,223 --> 00:00:17,655 Pennsylvania. We're about to begin lecture one on functions. Here we'll review 4 00:00:17,655 --> 00:00:23,313 several functions that you've already encountered in mathematics before - 5 00:00:23,313 --> 00:00:29,746 trigonometrics, polynomials, exponentials. And we'll end the lesson with a mysterious 6 00:00:29,746 --> 00:00:35,482 formula. Calculus is really all about functions. You're probably used to 7 00:00:35,482 --> 00:00:41,140 thinking of functions in terms of their graphs, where you plot X versus Fx). of X. 8 00:00:41,400 --> 00:00:48,982 A slightly different perceptive things of X as an input and F of X as an output. And 9 00:00:48,982 --> 00:00:56,565 it's not a bad idea at all to visualize a function as a machine that takes in X, and 10 00:00:56,565 --> 00:01:03,020 returns F of X. From that perspective, certain terminology is concerning 11 00:01:03,020 --> 00:01:09,587 functions. A clear, the collection of all possible inputs is called the domain. The 12 00:01:09,587 --> 00:01:15,424 collection of all possible outputs is called a range. Now this is single 13 00:01:15,424 --> 00:01:21,829 variable calculus, which means that our domains and ranges are specially simple. 14 00:01:21,829 --> 00:01:27,740 The reals, or some subinterval thereof. Certain operations on functions are 15 00:01:27,740 --> 00:01:33,932 critical to understand. Perhaps the most important is that of composition. The 16 00:01:33,932 --> 00:01:43,101 composition of two functions F and G, is defined to be the function that takes as 17 00:01:43,101 --> 00:01:52,270 its input X, and returns as its output G of X fed into F, that is read F of g of X. 18 00:01:52,270 --> 00:01:59,447 One thinks of this as first you do G, and then you plug the output of G into the 19 00:01:59,447 --> 00:02:06,624 input of F. Lets look at a simple example, the square root of one minus X squared, 20 00:02:06,624 --> 00:02:13,264 can be thought of as the composition of two functions F and G. If G is the 21 00:02:13,264 --> 00:02:20,391 function one minus X squared, then what would F be? F would be the function that 22 00:02:20,391 --> 00:02:29,098 takes an input and returns it's square root. Another operation on functions that 23 00:02:29,098 --> 00:02:36,838 is very important is the inverse. The inverse is written F with a superscript 24 00:02:36,838 --> 00:02:44,579 negative one. That does not mean the reciprocal of F. This denotes the inverse. 25 00:02:44,579 --> 00:02:53,022 The inverse is the function that undoes F. By which I mean, if you plug X into F what 26 00:02:53,022 --> 00:03:01,335 output do you get? You get F of X. If you plug F of X in to f inverse what you will 27 00:03:01,335 --> 00:03:09,356 get is X, F inverse undoes F. Now, notice that you can run this machine both ways if 28 00:03:09,356 --> 00:03:18,840 you plug X into F inverse. What you get when you plug that into F is X back. Again 29 00:03:19,340 --> 00:03:25,975 let's look at a simple example if we consider the function X cubed then its 30 00:03:25,975 --> 00:03:33,221 inverse is going to be X to the one third or rather the cube root of X. Why is that? 31 00:03:33,221 --> 00:03:40,815 If we take the cube root of X cubed we get X. If we take the cube of the cube root, 32 00:03:40,815 --> 00:03:48,246 we get X back again. Now, notice the symmetry that you see in the graphs. The 33 00:03:48,246 --> 00:03:55,578 graphs of F and F inverse are always going to be symmetric about the line Yx. equals 34 00:03:55,578 --> 00:04:05,283 X. That is, the line where the input and the output are the same. Certain classes 35 00:04:05,283 --> 00:04:11,311 of functions are critical to calculus. Perhaps the simplest is that of the 36 00:04:11,311 --> 00:04:17,909 polynomials. A polynomial is a function of the form of a constant plus a constant 37 00:04:17,909 --> 00:04:24,426 times X plus a constant times X squared, all the way up to a constant times X to 38 00:04:24,426 --> 00:04:31,862 the for some finite N. That top power is called the degree of the polynomial. We 39 00:04:31,862 --> 00:04:40,431 can also write, a polynomial, using a summation, notation. The sum, K goes from 40 00:04:40,431 --> 00:04:48,738 zero to N of constant. C Sub K, times Monomial term, X to the K. That summation 41 00:04:48,738 --> 00:04:53,844 symbol, is going to be used very frequently in this course. You might as 42 00:04:53,844 --> 00:05:00,753 well get used to it now. Another class of functions very simple to work with are the 43 00:05:00,753 --> 00:05:07,923 rational functions. Rational functions are functions of the form P of X over Q of X 44 00:05:07,923 --> 00:05:14,316 where each is a polynomial. These are very simple to work with, for example 3X minus 45 00:05:14,316 --> 00:05:20,450 one over X squared plus X minus six. The only thing you have to be careful of with 46 00:05:20,450 --> 00:05:27,012 a rational function is the denominator. When the denominator takes a value of 47 00:05:27,012 --> 00:05:35,103 zero, then the function may not be well defined. Other powers, besides those of 48 00:05:35,103 --> 00:05:41,723 positive integers are useful. Let's review, what is X to the zero? And we all 49 00:05:41,723 --> 00:05:48,306 know what that is, that's equal to one. What is X to the negative one half? We'll 50 00:05:48,306 --> 00:05:55,473 recall a fractional power denotes roots. for example, X to the one half means the 51 00:05:55,473 --> 00:06:02,245 square root of X. The negative sine in the exponent means that we take th e 52 00:06:02,245 --> 00:06:09,573 reciprocal. So the answer is one over root X. What is X to the 22 sevenths? Well, 53 00:06:09,573 --> 00:06:17,090 that means we take X to the twenty-second power, and look at the seventh root of 54 00:06:17,090 --> 00:06:23,891 that. What is X to the pie? When you take an irrational power, it seems as though we 55 00:06:23,891 --> 00:06:29,576 ought to be able to make sense of that by some sort of limit. Well, let's, let's 56 00:06:29,576 --> 00:06:35,480 keep that in the back of our heads for the moment and turn to some other matters. 57 00:06:35,880 --> 00:06:41,960 Let's see, the trigonometric functions are of primary importance in calculus and the 58 00:06:41,960 --> 00:06:47,461 rest of mathematics. You should be familiar with the, the basic trigonometric 59 00:06:47,461 --> 00:06:53,180 functions, sine, and cosine. One of the things you're going to have to keep in 60 00:06:53,180 --> 00:06:58,320 mind, that remember, is that cosine squared plus sine squared equals one. 61 00:06:58,680 --> 00:07:05,539 There are several ways to think about that geometrically. For example, if we look at 62 00:07:05,539 --> 00:07:12,399 a right triangle with hypotenuse one, the, the sine of the angle theta gives you the 63 00:07:12,399 --> 00:07:19,341 opposite side length. The cosine gives you the adjacent side length to that Theta. We 64 00:07:19,341 --> 00:07:25,374 could embed that picture intouh, a, a diagram for the unit circle. Where we see 65 00:07:25,374 --> 00:07:31,735 that cosine of feta and sine of feta returns the X and Y coordinates and the 66 00:07:31,735 --> 00:07:40,856 point on the unit circle with angle feta to the X axis. That explains the nature of 67 00:07:40,856 --> 00:07:45,784 the formula cosine squared plus sine squared equals one. It's simply the 68 00:07:45,784 --> 00:07:52,721 Pythagorean Theorem for the X and Y coordinate that triangle. There are other 69 00:07:52,721 --> 00:07:58,458 trigonometric functions as well. Besides sine and cosine, you should be familiar 70 00:07:58,458 --> 00:08:04,194 with the tangent function, the ratio of the sine with the cosine, as well as the 71 00:08:04,194 --> 00:08:09,858 cotangent function, its reciprocal, cosine over sine. The secant function is the 72 00:08:09,858 --> 00:08:15,740 reciprocal of the cosine, and the cosecant function, is the reciprocal of the sine. 73 00:08:16,260 --> 00:08:24,073 All four of these have vertical asymptotes at regions where the denominator term goes 74 00:08:24,073 --> 00:08:31,523 to zero. The inverse trigonometric functions are likewise very useful. One 75 00:08:31,523 --> 00:08:37,790 must be a little bit careful with these however. One often writes sine negative 76 00:08:37,790 --> 00:08:44,016 one in the exponent. To denote the inverse, but this can cause confusion. 77 00:08:44,016 --> 00:08:50,329 Students might think that th is, is one over the sine, that is the cosecent. 78 00:08:50,329 --> 00:08:56,993 That's a bad idea. I recommend instead, using the terminology arc sine. For the 79 00:08:56,993 --> 00:09:04,202 inverse of the sine function. The arc sine function, takes on values between negative 80 00:09:04,202 --> 00:09:11,262 Pi over two and pi over two, and has a restricted domain going from negative one 81 00:09:11,262 --> 00:09:19,151 to one. The arc cosine function, likewise, has a restricted domain from negative one 82 00:09:19,151 --> 00:09:25,957 to one, but is chosen to take values, between zero and Pi. The arc tangent 83 00:09:25,957 --> 00:09:32,243 function, unlike these two, has an unbounded domain. It is well defined for 84 00:09:32,243 --> 00:09:39,475 all inputs. But it has a restricted range between negative Pi over two and Pi over 85 00:09:39,475 --> 00:09:49,253 two. Our last class of important functions concerns the exponential. You should of 86 00:09:49,253 --> 00:09:56,597 seen the exponential function E to the X in your previous precalculus and calculus 87 00:09:56,597 --> 00:10:03,853 background. You know what it's graph looks like. You know also it's inverse. The log 88 00:10:03,853 --> 00:10:10,312 function or more properly than natural logarithm function. These are logarithm 89 00:10:11,020 --> 00:10:17,222 base E. These two are inverse to one another. That means their graphs are 90 00:10:17,222 --> 00:10:22,977 symmetric about the diagonal line Yx. equals X. You know that for example, E to 91 00:10:22,977 --> 00:10:29,676 the zero equals one, because anything to the zero equals one. And therefore, you 92 00:10:29,676 --> 00:10:36,792 know also that log of one equals zero. Now, you've seen these functions before. 93 00:10:36,792 --> 00:10:43,754 But if I ask you, what exactly do I mean by E when I say log base E, or E to the X. 94 00:10:43,754 --> 00:10:50,803 What is meant by that? You could answer that E is the number for which the natural 95 00:10:50,803 --> 00:10:57,972 log of E equals one. But then, what do we mean by the natural logarithm? Well of 96 00:10:57,972 --> 00:11:05,699 course E is simply a number a particular irrational number location on the real 97 00:11:05,699 --> 00:11:13,329 number line. B why is that number so special? Where did it come from? Why is it 98 00:11:13,329 --> 00:11:22,759 so useful so prevalent in mathematics? Before we answer that question. We need to 99 00:11:22,759 --> 00:11:29,729 go over some properties associated to the exponential function, E to the X. Let's 100 00:11:29,729 --> 00:11:36,699 recall some of the algebraic properties first. For example, E to the X times E to 101 00:11:36,699 --> 00:11:44,066 the Y equals E to the X plus Y. In like manner, E to the X raised to the Yth power 102 00:11:44,066 --> 00:11:50,876 equals E to the X times Y. Both of these properties follow from the, the simple 103 00:11:50,876 --> 00:11:58,248 properties for exponents. E to the X however, has some very unique properties 104 00:11:58,248 --> 00:12:05,345 concerning derivatives and integrals. In your previous exposure to calculus, you've 105 00:12:05,345 --> 00:12:12,285 seen a little bit concerning derivatives and integrals. And you may have seen the 106 00:12:12,285 --> 00:12:18,968 following two important examples. The derivative of E to the X equals E to the 107 00:12:18,968 --> 00:12:24,835 X. The integral of the E to the X DX likewise is E to the X. Oh plus a 108 00:12:24,835 --> 00:12:31,380 constant, don't forget that. We'll be talking more about these ideas and these 109 00:12:31,380 --> 00:12:38,180 properties later, but for the moment keep them in mind, we'll see them again next 110 00:12:38,180 --> 00:12:46,311 lecture. Now there's one formula that ties together all of the functions that we've 111 00:12:46,311 --> 00:12:52,320 looked at in this first lesson trigonometrics, exponentials, and that is 112 00:12:52,320 --> 00:12:59,138 Euler's formula. Euler's formula states that E, e to the IX equals Cosine of X 113 00:12:59,138 --> 00:13:06,683 plus I times of sine of X. You may or may not have seen this before? Let's think for 114 00:13:06,683 --> 00:13:14,047 a moment about what it means. It's a wonderful formula if we can make sense of 115 00:13:14,047 --> 00:13:19,559 it. First of all, a little bit about the, the terminology used. The I in the 116 00:13:19,559 --> 00:13:25,445 exponent is the imaginary number, square root of negative one. It's the number that 117 00:13:25,445 --> 00:13:30,900 has the property that I squared is equal to negative one. This is not a real 118 00:13:30,900 --> 00:13:36,858 number. That doesn't mean that it doesn't exist, it just means it is not on the real 119 00:13:36,858 --> 00:13:46,640 number line. Euler's formula concerns the exponentiation of an imaginary variable. 120 00:13:47,240 --> 00:13:54,466 What exactly does that mean? How is this related to trigonometric functions? The 121 00:13:54,466 --> 00:14:01,832 graphs of these two don't seem to be alike at all. That is something, that we will 122 00:14:01,832 --> 00:14:07,952 answer, in our next lesson. Our introduction to functions is complete. But 123 00:14:07,952 --> 00:14:13,666 the mystery Euler's formula remains. In our next lesson, we'll resolve that 124 00:14:13,666 --> 00:14:18,918 mystery, by coming to a deeper understanding, of what the exponential 125 00:14:18,918 --> 00:14:20,540 function is and does.