1 00:00:00,610 --> 00:00:03,800 We have many videos on the mean value theorem, 2 00:00:03,800 --> 00:00:06,260 but I'm going to, I'm going to review it a 3 00:00:06,260 --> 00:00:09,270 little bit so that we can see how this connects 4 00:00:09,270 --> 00:00:11,800 the mean value theorem that we learned in Differential Calculus. 5 00:00:11,800 --> 00:00:13,430 How that connects to what we learned about 6 00:00:13,430 --> 00:00:17,090 the average value of a function using definite integrals. 7 00:00:17,090 --> 00:00:22,951 So the mean value theorem tells us that if I have some function f, that is 8 00:00:22,951 --> 00:00:27,837 continuous, continuous on the closed interval, so itâs 9 00:00:27,837 --> 00:00:30,884 including the end points from a to b. 10 00:00:30,884 --> 00:00:34,304 And it is differentiable, it is differentiable, 11 00:00:34,304 --> 00:00:36,610 so the derivative is defined on the 12 00:00:36,610 --> 00:00:38,517 open interval from a to b, so 13 00:00:38,517 --> 00:00:42,620 not necessary, doesn't necessarily to be defined. 14 00:00:42,620 --> 00:00:44,520 Differentiable at the boundaries as long 15 00:00:44,520 --> 00:00:46,970 as it's differentiable between the boundaries. 16 00:00:46,970 --> 00:00:48,240 Then we know. 17 00:00:48,240 --> 00:00:52,900 Then we know that there exists, there 18 00:00:52,900 --> 00:00:58,880 exists some value, or some number I guess, some number, 19 00:00:58,880 --> 00:01:04,932 some number c such 20 00:01:04,932 --> 00:01:10,900 that c is between the two end points of our interval. 21 00:01:10,900 --> 00:01:17,600 So let me, such that a is less than c is less than b, so c is in this interval. 22 00:01:17,600 --> 00:01:22,940 And, and, and this is kind of the, the meat of it. 23 00:01:22,940 --> 00:01:24,600 The meat of it is that the derivative 24 00:01:24,600 --> 00:01:27,300 of, the derivative of our function at that point, 25 00:01:27,300 --> 00:01:28,820 the derivative of the function at that point you 26 00:01:28,820 --> 00:01:31,130 could use as the slope of the tangent line. 27 00:01:31,130 --> 00:01:34,778 At that point is equal to, essentially the 28 00:01:34,778 --> 00:01:38,980 average, the average rate of change over the interval. 29 00:01:38,980 --> 00:01:42,420 Or you can even think about it as the slope between the two endpoints. 30 00:01:42,420 --> 00:01:44,570 So the, the slope between the two endpoints is 31 00:01:44,570 --> 00:01:47,130 gonna be your change in y which is going to 32 00:01:47,130 --> 00:01:56,610 be your change in your function value so f of b minus f of a over, over b minus a. 33 00:01:56,610 --> 00:01:58,750 And once again we do this, we go into 34 00:01:58,750 --> 00:02:00,870 much more depth in this when we covered this the 35 00:02:00,870 --> 00:02:03,070 first time in Differential Calculus but just to give you 36 00:02:03,070 --> 00:02:06,190 a visualization of it because I think it's always handy. 37 00:02:06,190 --> 00:02:08,039 The mean value theorem that we learned 38 00:02:08,039 --> 00:02:10,650 in Differential Calculus just tells us, hey look, 39 00:02:10,650 --> 00:02:13,400 you know, if this is a, this is 40 00:02:13,400 --> 00:02:18,590 b, I've got my function doing something interesting. 41 00:02:18,590 --> 00:02:21,630 So this is f of a, this is f of b. 42 00:02:22,790 --> 00:02:24,890 So this quantity right over here. 43 00:02:24,890 --> 00:02:27,740 Where you're taking the change in the value of our function divided by. 44 00:02:27,740 --> 00:02:30,420 So this right over here is f of b. 45 00:02:30,420 --> 00:02:34,480 F of b minus f of a is this change in the value of our function, 46 00:02:34,480 --> 00:02:38,650 divided by the change in our x axis, it's our change in y over change in x. 47 00:02:38,650 --> 00:02:42,810 That gives us the slope, this right over here gives us the slope of this line. 48 00:02:42,810 --> 00:02:47,600 This, this, the slope of the line that connects, that connects these two points. 49 00:02:47,600 --> 00:02:50,230 That's this quantity, and this, the mean value theorem 50 00:02:50,230 --> 00:02:52,290 tells us that there's some c in between a 51 00:02:52,290 --> 00:02:56,730 and b where you're going to have the same slope, so it might be at least one place. 52 00:02:56,730 --> 00:03:00,414 So it might be right over there, where you have the exact same slope. 53 00:03:00,414 --> 00:03:04,289 There exists a c, where the slope of the tangent line at that point is going to be 54 00:03:04,289 --> 00:03:06,226 the same, so this would be ac right over 55 00:03:06,226 --> 00:03:09,410 there, and we actually might have a couple of c's. 56 00:03:09,410 --> 00:03:10,360 That's another candidate c. 57 00:03:10,360 --> 00:03:12,830 There's a least one c where the slope of the tangent 58 00:03:12,830 --> 00:03:16,970 line is the same as the average slope across the interval. 59 00:03:16,970 --> 00:03:21,230 And, once again, we have to assume that f is continuous and f is differentiable. 60 00:03:21,230 --> 00:03:24,360 Now, when you see this, it, something might, it might 61 00:03:24,360 --> 00:03:27,080 evoke some similarities with what we saw when we saw 62 00:03:27,080 --> 00:03:29,470 the, the, or how we defined I guess you could 63 00:03:29,470 --> 00:03:31,990 say, or the formula for the average value of a function. 64 00:03:31,990 --> 00:03:37,650 Remember, what we saw for the average value of function, we said the average, 65 00:03:37,650 --> 00:03:42,647 the average value of a function is going to be equal to 1 over b minus a. 66 00:03:42,647 --> 00:03:46,802 Notice, 1 over b minus a, you have a b minus a in the 67 00:03:46,802 --> 00:03:53,380 denominator here, times the definite integral from a to b of f of x dx. 68 00:03:53,380 --> 00:03:59,320 Now this is interesting, cuz here we have a derivative, here we have an integral, 69 00:03:59,320 --> 00:04:03,760 but maybe we could connect these, maybe we could connect these two things. 70 00:04:05,590 --> 00:04:10,766 Well one thing that might jump out at you, is maybe we could rewrite, maybe 71 00:04:10,766 --> 00:04:15,630 we could rewrite this numerator right over here in this form somehow. 72 00:04:15,630 --> 00:04:18,113 And I encourage you to the pause the video and see 73 00:04:18,113 --> 00:04:21,420 if you can, and I'll give you actually quite a huge hint. 74 00:04:21,420 --> 00:04:24,860 Instead of it being an f of x here, what happens if there's an f prime of x there? 75 00:04:24,860 --> 00:04:27,470 So I encourage you to, to re-try to do that. 76 00:04:27,470 --> 00:04:29,410 So, once again, this is, let me re-write all of this. 77 00:04:29,410 --> 00:04:33,730 This is going to be equal to, this over here is the exact same thing as the 78 00:04:33,730 --> 00:04:40,350 definite integral from a to b of f-prime of x dx. 79 00:04:40,350 --> 00:04:41,090 Think about it. 80 00:04:41,090 --> 00:04:44,750 You're gonna take the anti-derivative of f-prime of x, which is gonna be f of x. 81 00:04:44,750 --> 00:04:46,600 And you're going to evaluate it at b, f of b. 82 00:04:46,600 --> 00:04:50,990 And then from that you're going to subtract it evaluated a minus f of a. 83 00:04:50,990 --> 00:04:52,960 These two thing are identical. 84 00:04:52,960 --> 00:04:55,800 And then you can, of course, divide by. 85 00:04:55,800 --> 00:04:59,970 Divide by b minus a. 86 00:04:59,970 --> 00:05:03,000 Now this is starting to get interesting. 87 00:05:03,000 --> 00:05:07,406 One way to think about it is, there must be a, there must be a 88 00:05:07,406 --> 00:05:11,827 c if we, there must be a c that takes on the average value of. 89 00:05:11,827 --> 00:05:15,729 There must be a c that when you, when you evaluate the 90 00:05:15,729 --> 00:05:20,711 derivative at c, it takes on the average value of the derivative. 91 00:05:20,711 --> 00:05:25,688 Or another way to think about it, another way to think about it if we were to 92 00:05:25,688 --> 00:05:31,360 just write, if we wrote it just right let me say g of x is equal to f prime of x. 93 00:05:31,360 --> 00:05:33,720 Then we get very close to what we have over here. 94 00:05:33,720 --> 00:05:39,350 Because this right over here is going to be g of c, remember f prime of c is the 95 00:05:39,350 --> 00:05:46,190 same thing as g of c, is equal to 1 over is equal to 1 over b minus a. 96 00:05:46,190 --> 00:05:53,160 1 over b minus a, so there exists a c, where g of c is equal to 1 over b minus 97 00:05:53,160 --> 00:06:00,545 a, times the definite integral from a to b, of g of x, g of x dx. 98 00:06:00,545 --> 00:06:02,900 F-prime of x is the same thing as g of x. 99 00:06:02,900 --> 00:06:06,120 So, another way of thinking about it, and this is actually another form 100 00:06:06,120 --> 00:06:10,020 of the mean value theorem, is called the mean value theorem for integrals. 101 00:06:10,020 --> 00:06:11,680 Mean value theorem for integrals. 102 00:06:11,680 --> 00:06:15,468 Let me, so this is the mean, I'll just write 103 00:06:15,468 --> 00:06:22,160 the acronym, mean value theorem for, for integrals or integration. 104 00:06:22,160 --> 00:06:24,450 Which essentially and, and to give you in a slightly 105 00:06:24,450 --> 00:06:28,060 more formal sense, is if you have some function g. 106 00:06:28,060 --> 00:06:33,100 So, if g is, let me actually go down a little bit. 107 00:06:33,100 --> 00:06:41,790 Which tells us that g of x is continuous, continuous on this closed interval. 108 00:06:41,790 --> 00:06:45,210 On, going from a to b, then there exists, then there exists, 109 00:06:51,650 --> 00:06:57,480 a c in this interval where g of c is equal to, what is this? 110 00:06:57,480 --> 00:07:00,610 This is the average value of our function. 111 00:07:00,610 --> 00:07:05,715 There exists, exists a c where g of c is equal to the 112 00:07:05,715 --> 00:07:10,960 average value of your function over the integral. 113 00:07:10,960 --> 00:07:14,290 This was our definition of the average value of a function. 114 00:07:14,290 --> 00:07:16,090 So anyway, this is just another way of 115 00:07:16,090 --> 00:07:18,020 saying you might see some of the mean value 116 00:07:18,020 --> 00:07:20,280 theorem of integrals, and that just to show you 117 00:07:20,280 --> 00:07:23,660 that it's really closely tied is using different notation. 118 00:07:23,660 --> 00:07:27,110 But it's usually, it's, it's essentially the same exact ideas the 119 00:07:27,110 --> 00:07:29,490 mean value theorem you learned in 120 00:07:29,490 --> 00:07:32,060 Differential Calculus, but now different notation. 121 00:07:32,060 --> 00:07:34,380 And I guess you could have a slightly different interpretation. 122 00:07:34,380 --> 00:07:36,930 We were thinking about it in, in Differential Calculus. 123 00:07:36,930 --> 00:07:40,710 We're thinking about having a point where the slope of the tangent line 124 00:07:40,710 --> 00:07:43,950 of the function of that point is the same as the average rate. 125 00:07:43,950 --> 00:07:46,090 So that's when we had our kind of differential mode and we were 126 00:07:46,090 --> 00:07:49,440 kind of thinking, thinking in terms of slopes and slopes of tangent lines. 127 00:07:49,440 --> 00:07:51,930 And now when we're in integral mode we're thinking much 128 00:07:51,930 --> 00:07:55,680 more in terms of average value, average value of the function. 129 00:07:55,680 --> 00:07:59,580 So there's some c where g of c, there's, there's some c 130 00:07:59,580 --> 00:08:03,930 where the function evaluated at that point is equal to the average value. 131 00:08:03,930 --> 00:08:05,800 So another way of thinking about it, 132 00:08:08,410 --> 00:08:10,340 if I were to draw, if I were to draw. 133 00:08:10,340 --> 00:08:17,120 G of, if I were to draw a g of x, if I were to draw g of x. 134 00:08:17,120 --> 00:08:20,810 So that's x, that is my y axis. 135 00:08:20,810 --> 00:08:24,470 This is the graph of y is equal to g 136 00:08:24,470 --> 00:08:26,880 of x, which of course was the same thing as f-prime 137 00:08:26,880 --> 00:08:29,540 of x, but we core, I guess, we've just rewritten 138 00:08:29,540 --> 00:08:32,270 it now to be more consistent with our average value formula. 139 00:08:32,270 --> 00:08:37,340 And we're talking about the interval from a to b. 140 00:08:37,340 --> 00:08:39,794 We've already seen how to calculate the average value. 141 00:08:39,794 --> 00:08:45,290 We've already seen how to calculate the average value so may be the average value 142 00:08:45,290 --> 00:08:50,610 is, is that right over there so that is g average. 143 00:08:50,610 --> 00:08:52,270 So our average value is this. 144 00:08:52,270 --> 00:08:54,140 The mean value theorem for integrals just tells there's 145 00:08:54,140 --> 00:08:57,500 some c where our function must take on, where our 146 00:08:57,500 --> 00:09:00,560 function must take on that value at c, whereas that 147 00:09:00,560 --> 00:09:05,690 c is inside, where the c is in that interval.