1 00:00:00,000 --> 00:00:16,800 Hello everyone, welcome to the magnificence of mathematics. 2 00:00:16,800 --> 00:00:19,520 I'm your host, Eddie Kingston. 3 00:00:19,520 --> 00:00:23,880 If you've ever worked on a farm and had to put up fencing around a particular area, how 4 00:00:23,880 --> 00:00:28,080 could you go about maximizing the area you could enclose with the fencing materials you'd 5 00:00:28,080 --> 00:00:29,160 have on hand? 6 00:00:29,160 --> 00:00:32,960 On the other hand, a lot of you listeners might remember the formula for the area of 7 00:00:32,960 --> 00:00:37,800 a triangle, 1 half base times height, or that of a circle, pi r squared. 8 00:00:37,800 --> 00:00:43,200 But what if you wanted to find the area of, say, a rectangle with a squiggly top, or some 9 00:00:43,200 --> 00:00:44,960 other arbitrary figure? 10 00:00:44,960 --> 00:00:47,760 These two questions can't possibly be related, right? 11 00:00:47,760 --> 00:00:52,920 Well, it turns out that the answers to these two questions turn out to be not only related, 12 00:00:52,920 --> 00:00:55,000 but more or less the opposite of each other. 13 00:00:55,000 --> 00:01:00,080 They have to do with the two subfields of calculus, namely differential calculus and 14 00:01:00,080 --> 00:01:01,840 integral calculus. 15 00:01:01,840 --> 00:01:05,680 Over the course of the next several episodes, we'll answer these two questions using both 16 00:01:05,680 --> 00:01:11,080 differential and integral calculus and see more about how calculus is used a lot in day-to-day 17 00:01:11,080 --> 00:01:12,080 life. 18 00:01:12,080 --> 00:01:17,240 Back in episode 3, I briefly mentioned the idea of a limit in the context of probability. 19 00:01:17,240 --> 00:01:21,520 Knowing what a limit is, is crucial to understanding differential calculus. 20 00:01:21,520 --> 00:01:26,120 The basic idea of a limit is that you want to know how a function behaves near a point 21 00:01:26,120 --> 00:01:28,560 without actually reaching that point. 22 00:01:28,560 --> 00:01:34,300 For example, imagine you draw up two lines, one vertical and one horizontal, intersecting 23 00:01:34,300 --> 00:01:35,600 at one point. 24 00:01:35,600 --> 00:01:38,560 This intersecting point is called the origin. 25 00:01:38,560 --> 00:01:43,640 Now put little marks evenly spaced from each other and label the origin 0, the first mark 26 00:01:43,640 --> 00:01:48,880 to the right of the origin 1, second mark to the right 2, and so on. 27 00:01:48,880 --> 00:01:53,800 Let the first mark to the left be negative 1, the second to the left negative 2, and 28 00:01:53,800 --> 00:01:54,920 so on. 29 00:01:54,920 --> 00:02:00,180 Repeat the same thing for the vertical line, but with right replaced by up and left replaced 30 00:02:00,180 --> 00:02:01,400 with down. 31 00:02:01,400 --> 00:02:06,000 You are now imagining what's called a Cartesian coordinate system, named after 17th century 32 00:02:06,000 --> 00:02:08,240 French philosopher René Descartes. 33 00:02:08,240 --> 00:02:14,160 Now imagine a straight diagonal line going up and to the right, passing through the origin. 34 00:02:14,160 --> 00:02:19,880 You are now imagining the function y equals x, where the x coordinate matches the y coordinate 35 00:02:19,880 --> 00:02:21,280 at every point. 36 00:02:21,280 --> 00:02:25,680 So if you look at where you are on this line at x equals 2, you'll see that you're also 37 00:02:25,680 --> 00:02:28,880 at the y coordinate y equals 2. 38 00:02:28,880 --> 00:02:33,280 Now imagine you move along this line from this spot and get closer and closer to the 39 00:02:33,280 --> 00:02:38,320 point x equals 1, and you'll see that you're approaching the point y equals 1 as well. 40 00:02:38,320 --> 00:02:43,240 In particular, you are said to be approaching y equals 1 from the right, meaning you start 41 00:02:43,240 --> 00:02:48,560 it at the right hand side of the line and work your way from the right to the left. 42 00:02:48,560 --> 00:02:53,600 Now if you start from the left side and work your way up and to the right of the line to 43 00:02:53,600 --> 00:02:59,360 the point x equals 1, you are said to be approaching the point y equals 1 from the left, since 44 00:02:59,360 --> 00:03:03,120 you're starting from the left hand side and working your way to the right. 45 00:03:03,120 --> 00:03:08,840 Since this is true for any real number a, you can then and only then say that the limit 46 00:03:08,840 --> 00:03:14,400 as x approaches a of this function x is simply that number a. 47 00:03:14,400 --> 00:03:18,720 If we were working with a more complicated function, it might be the case that the limit 48 00:03:18,720 --> 00:03:24,600 as x approaches a number a from the right is a different value than the limit as x approaches 49 00:03:24,600 --> 00:03:26,200 a from the left. 50 00:03:26,200 --> 00:03:29,480 In that case, we say that the limit doesn't exist. 51 00:03:29,480 --> 00:03:33,520 For example, consider the function y equals 1 over x. 52 00:03:33,520 --> 00:03:39,160 Since 1 divided by 1 equals 1, the value of y at x equals 1 is simply 1. 53 00:03:39,160 --> 00:03:46,360 If you consider x equals 1 half, the value of y is 2, since 1 divided by 1 half is 2, 54 00:03:46,360 --> 00:03:51,600 because 2 times 1 half is 1, and multiplication and division are inverses, or opposites of 55 00:03:51,600 --> 00:03:52,600 each other. 56 00:03:52,600 --> 00:03:57,360 Now if you consider x equals 1 third, the value of y is 3 for the same reason. 57 00:03:57,360 --> 00:04:02,280 You can see that as x gets smaller, y gets bigger. 58 00:04:02,280 --> 00:04:08,200 In particular, as we get closer and closer to x equals 0, y just grows and grows without 59 00:04:08,200 --> 00:04:09,200 bound. 60 00:04:09,200 --> 00:04:15,120 We say that the limit as x approaches 0 from the right of 1 over x is infinity, positive 61 00:04:15,120 --> 00:04:16,880 infinity in particular. 62 00:04:16,880 --> 00:04:22,560 Similarly, if you consider x equals negative 1 half, y equals negative 2. 63 00:04:22,560 --> 00:04:26,520 If x equals negative 1 third, y equals negative 3. 64 00:04:26,520 --> 00:04:33,120 So as x gets closer and closer to 0 from the negative, i.e. left side, y gets increasingly 65 00:04:33,120 --> 00:04:34,120 negative. 66 00:04:34,120 --> 00:04:39,920 Hence the limit as x approaches 0 from the left of 1 over x is negative infinity. 67 00:04:39,920 --> 00:04:45,040 So we see that as x gets closer and closer to 0 from these two directions, this function 68 00:04:45,040 --> 00:04:47,180 approaches two different values. 69 00:04:47,180 --> 00:04:52,520 So the limit as x approaches 0 of 1 over x does not exist. 70 00:04:52,520 --> 00:04:56,560 Another way limits have the ability to not exist is if they don't approach anything 71 00:04:56,560 --> 00:04:57,840 in particular. 72 00:04:57,840 --> 00:05:03,280 Consider a function that just gradually oscillates back and forth between negative 1 and 1, touching 73 00:05:03,280 --> 00:05:05,620 every real number along the way. 74 00:05:05,620 --> 00:05:09,760 So at one interval, the function starts at negative 1, and as you move right, it goes 75 00:05:09,760 --> 00:05:14,200 up up up up all the way to 1, and then it goes down down down down down all the way 76 00:05:14,200 --> 00:05:15,520 to negative 1. 77 00:05:15,520 --> 00:05:18,840 And this cycle repeats forever in both directions. 78 00:05:18,840 --> 00:05:23,800 One example of this function is called the sine function, so y equals sine of x. 79 00:05:23,800 --> 00:05:28,200 Now what is the limit as x approaches infinity of this function? 80 00:05:28,200 --> 00:05:33,120 Since it's constantly hitting negative 1, 1, and everything in between, it just goes 81 00:05:33,120 --> 00:05:36,360 on forever, not approaching anything in particular. 82 00:05:36,360 --> 00:05:40,700 It's not getting closer and closer to any certain value in the long run, and it's not 83 00:05:40,700 --> 00:05:45,240 growing and growing to either positive or negative infinity, so we say in this instance 84 00:05:45,240 --> 00:05:50,200 that the limit as x approaches infinity of sine of x does not exist either. 85 00:05:50,200 --> 00:05:54,120 Now that we know what limits are, let's get into what derivatives are, and the heart of 86 00:05:54,120 --> 00:05:56,140 what differential calculus is about. 87 00:05:56,140 --> 00:05:59,940 If you've taken an algebra class in high school, you might remember finding slopes 88 00:05:59,940 --> 00:06:02,740 of lines, rise over run, and all that. 89 00:06:02,740 --> 00:06:08,800 This is just how much the value of y changes divided by how much the value of x changes. 90 00:06:08,800 --> 00:06:15,840 For example, for the line y equals 2x, consider the points 1,2, and 2,4, where in the first 91 00:06:15,840 --> 00:06:20,720 point the x coordinates 1 and the y coordinates 2, and in the second point the x coordinates 92 00:06:20,720 --> 00:06:24,400 2 and the y coordinates 4, through both of which the line goes. 93 00:06:24,400 --> 00:06:30,800 The change in y is just 4 minus 2 equals 2, and the change in x is just 2 minus 1 equals 94 00:06:30,800 --> 00:06:31,800 1. 95 00:06:31,800 --> 00:06:36,840 So the slope of this line is the change in y, namely 2, divided by the change in x, namely 96 00:06:36,840 --> 00:06:37,840 1. 97 00:06:37,840 --> 00:06:39,720 So the slope of the line is 2. 98 00:06:39,720 --> 00:06:46,800 Similarly, for any function y equals m times x, for some number m, the slope is just that 99 00:06:46,800 --> 00:06:47,800 number m. 100 00:06:47,800 --> 00:06:51,960 We'll later see that the slope is also the derivative of functions like these. 101 00:06:51,960 --> 00:06:57,240 Any function you can think of has a corresponding straight line at every point, such that the 102 00:06:57,240 --> 00:07:00,560 line just barely touches the function at that point. 103 00:07:00,560 --> 00:07:05,160 This is what's known as a tangent line, and the slope of this tangent line is also the 104 00:07:05,160 --> 00:07:07,920 slope of the function at that point. 105 00:07:07,920 --> 00:07:12,880 This slope at that point is called the derivative of the function at that point. 106 00:07:12,880 --> 00:07:17,760 What if you wanted a function that told you the derivative at every point? 107 00:07:17,760 --> 00:07:22,080 Tangent lines can be approximated by what are called secant lines, which are lines that 108 00:07:22,080 --> 00:07:26,000 just barely touch the function at two different points. 109 00:07:26,000 --> 00:07:30,960 Let's call the x coordinate of one of the points x, and the y coordinate f of x the 110 00:07:30,960 --> 00:07:33,440 function evaluated at that point x. 111 00:07:33,440 --> 00:07:39,160 So for example, for the function f of x equals x squared, the point x equals 3 corresponds 112 00:07:39,160 --> 00:07:43,720 to the point f of 3, which is 3 squared, which is 9. 113 00:07:43,720 --> 00:07:47,560 Also an important thing to notice is that f of x is the exact same thing as y in terms 114 00:07:47,560 --> 00:07:49,120 of function notation. 115 00:07:49,120 --> 00:07:52,600 Now consider moving along the graph by a tiny, tiny amount. 116 00:07:52,600 --> 00:07:55,720 Now we're going to call this tiny, tiny amount h. 117 00:07:55,720 --> 00:08:01,720 So we call the x coordinate of our second point x plus h, and the y coordinate f of 118 00:08:01,720 --> 00:08:03,160 x plus h. 119 00:08:03,160 --> 00:08:08,360 So the secant line intersects the function at two points, the first of which we call 120 00:08:08,360 --> 00:08:13,280 the x coordinate just x, and the y coordinate f of x, and the second point where we call 121 00:08:13,280 --> 00:08:18,600 the x coordinate x plus h, and the y coordinate f of x plus h. 122 00:08:18,600 --> 00:08:24,660 The slope of this secant line is the change in the y coordinate f of x plus h minus f 123 00:08:24,660 --> 00:08:32,400 of x, all divided by the change in the x coordinate x plus h minus x, which is just h. 124 00:08:32,400 --> 00:08:38,800 As h gets closer and closer to zero, the secant line at a point just becomes the tangent line 125 00:08:38,800 --> 00:08:39,920 at that point. 126 00:08:39,920 --> 00:08:47,160 That is, the limit as h approaches zero of the quantity f of x plus h minus f of x divided 127 00:08:47,160 --> 00:08:53,400 by h is the slope of the tangent line, and this limit is by definition the derivative 128 00:08:53,400 --> 00:08:55,640 of the entire function f of x. 129 00:08:55,640 --> 00:09:01,520 This is noted by either f prime of x, which is what's called Lagrange's notation, named 130 00:09:01,520 --> 00:09:06,400 after French mathematician Joseph-Louis Lagrange, whose works were inspired by the famous English 131 00:09:06,400 --> 00:09:13,360 physicist Isaac Newton, or by dy divided by dx, read just dy dx, which is what's called 132 00:09:13,360 --> 00:09:18,360 Leibniz notation, named after German mathematician Gottfried Wilhelm Leibniz. 133 00:09:18,360 --> 00:09:22,640 Regardless of which notation you want to use, this tells you the derivative of the function 134 00:09:22,640 --> 00:09:24,400 at any point you want. 135 00:09:24,400 --> 00:09:29,680 For example, let's consider the function f of x equals m times x for some number m. 136 00:09:29,680 --> 00:09:36,680 If we plug in x plus h in this function, we get f of x plus h equals m times the quantity 137 00:09:36,680 --> 00:09:42,200 x plus h, which if we distribute that out is m times x plus m times h. 138 00:09:42,200 --> 00:09:48,720 Then we subtract f of x, which is just m times x, to get m times h in the numerator of our 139 00:09:48,720 --> 00:09:49,720 limit. 140 00:09:49,720 --> 00:09:52,560 Divide this by h to simply get m. 141 00:09:52,560 --> 00:09:58,400 Since this function no longer depends on what h is, the limit as h approaches zero of m 142 00:09:58,400 --> 00:10:02,380 is simply that number m, and this agrees with what we had earlier. 143 00:10:02,380 --> 00:10:08,120 For a slightly more complicated example, consider f of x equals x squared. 144 00:10:08,120 --> 00:10:14,320 Then f of x plus h is simply the quantity x plus h squared, which if you remember FOIL, 145 00:10:14,320 --> 00:10:20,440 first outer and last from your algebra 1 class, that gets us x squared plus x times h plus 146 00:10:20,440 --> 00:10:28,260 x times h plus h squared, which is x squared plus 2x times h plus h squared. 147 00:10:28,260 --> 00:10:32,580 Subtract off x squared to get 2 times x times h plus h squared. 148 00:10:32,580 --> 00:10:37,520 Note that we have h as a common factor in this expression, so we can factor this as 149 00:10:37,520 --> 00:10:42,960 h times the quantity 2x plus h, and divide that by just h. 150 00:10:42,960 --> 00:10:48,540 The h's cancel out, and we have the limit as h approaches zero of 2x plus h. 151 00:10:48,540 --> 00:10:54,820 We can just evaluate this limit term by term, and get 2x plus zero, which is just 2x. 152 00:10:54,820 --> 00:11:00,240 So the slope of the tangent line at any point x of the function f of x equals x squared 153 00:11:00,240 --> 00:11:03,120 is f prime of x equals 2x. 154 00:11:03,120 --> 00:11:08,320 Whew, okay, so having gone through all that, let's go back to what we talked about before 155 00:11:08,320 --> 00:11:10,500 with the whole fence thing. 156 00:11:10,500 --> 00:11:15,540 Suppose you have 40 square feet of fencing material, say wood or whatever, that you want 157 00:11:15,540 --> 00:11:18,440 to enclose in a rectangular area. 158 00:11:18,440 --> 00:11:23,360 What would you want the dimensions of the fence to be such that the area you enclose 159 00:11:23,360 --> 00:11:25,100 is maximized? 160 00:11:25,100 --> 00:11:30,200 Well since you have 40 square feet of wood to use as the perimeter of your enclosure, 161 00:11:30,200 --> 00:11:37,120 you want to find the length L and the width W such that 2L plus 2W equals 40, that is 162 00:11:37,120 --> 00:11:45,840 the perimeter of the rectangle is 40, and such that the area L times W is maximized. 163 00:11:45,840 --> 00:11:52,640 If we solve our first equation for L by subtracting both sides by 2W and dividing everything by 164 00:11:52,640 --> 00:11:59,680 2, we get L equals 20 minus W. We can plug this into the expression for area and get 165 00:11:59,680 --> 00:12:08,760 W times the quantity 20 minus W and distribute this out to get 20 times W minus W squared. 166 00:12:08,760 --> 00:12:13,760 Finding the maximum area of the enclosure amounts to finding the maximum value of the 167 00:12:13,760 --> 00:12:19,240 function f of W equals 20 times W minus W squared. 168 00:12:19,240 --> 00:12:24,400 The first step in finding the maximum value is to find the derivative of this function. 169 00:12:24,400 --> 00:12:29,880 We can do this term by term, and we actually know what this derivative is already. 170 00:12:29,880 --> 00:12:35,580 The derivative of 20 times W with respect to W is just 20, because that's the slope 171 00:12:35,580 --> 00:12:41,320 of the function f of W equals 20 times W, and the derivative of W squared is just 2 172 00:12:41,320 --> 00:12:47,040 times W. So the derivative of the whole function is 20 minus 2W. 173 00:12:47,040 --> 00:12:52,600 Now the next step is to set this derivative equal to 0 and solve for W. So we subtract 174 00:12:52,600 --> 00:12:59,160 20 from both sides and divide by negative 2 to get W equals 10 feet. But since we had 175 00:12:59,160 --> 00:13:08,160 L equals 20 minus W, the corresponding length L is just 20 minus 10 equals 10 feet. So therefore, 176 00:13:08,160 --> 00:13:14,760 the rectangular shape that maximizes the enclosed area is just a square with side length 10 177 00:13:14,760 --> 00:13:16,920 feet, since all sides are equal. 178 00:13:16,920 --> 00:13:21,800 This is true regardless of how much fencing you have. Calculus has already been seen to 179 00:13:21,800 --> 00:13:26,800 be super useful in everyday life, and we've barely begun scratching the surface. Next 180 00:13:26,800 --> 00:13:56,640 time we'll talk about differential calculus's counterpart, integral calculus. See you then! 181 00:13:56,640 --> 00:14:55,920 The Magnificence of Mathematics was brought to you by Algid Productions LLC. Thank you for listening and don't forget to rate and share!