1 00:00:00,350 --> 00:00:04,350 hello and welcome to my secret math 2 00:00:02,730 --> 00:00:05,910 tutor this video will work on 3 00:00:04,350 --> 00:00:07,680 approximating the instantaneous rate of 4 00:00:05,910 --> 00:00:09,719 change of a function a single point 5 00:00:07,680 --> 00:00:11,730 this time however we'll use our ti-83 or 6 00:00:09,719 --> 00:00:14,879 84 calculator to help us out in that 7 00:00:11,730 --> 00:00:16,230 process so to find something like the 8 00:00:14,880 --> 00:00:17,880 instantaneous rate of change of a 9 00:00:16,230 --> 00:00:19,680 function what you can do is actually 10 00:00:17,880 --> 00:00:21,330 pick two points and then find the 11 00:00:19,680 --> 00:00:23,100 average rate of change between those two 12 00:00:21,330 --> 00:00:25,380 points or you know what we call the 13 00:00:23,100 --> 00:00:27,119 slope now that won't give us the 14 00:00:25,380 --> 00:00:28,590 instantaneous rate of change but if we 15 00:00:27,119 --> 00:00:30,420 move those points closer and closer 16 00:00:28,590 --> 00:00:32,000 together and repeat the process it'll 17 00:00:30,420 --> 00:00:34,140 give us a good idea of what that 18 00:00:32,000 --> 00:00:37,440 instantaneous rate of change actually is 19 00:00:34,140 --> 00:00:39,000 so we will use the calculator to help us 20 00:00:37,440 --> 00:00:41,070 out in this process is we can actually 21 00:00:39,000 --> 00:00:43,200 store our function in we can store in 22 00:00:41,070 --> 00:00:44,879 the slope formula and see what happens 23 00:00:43,200 --> 00:00:47,219 as we're moving one point closer and 24 00:00:44,879 --> 00:00:48,690 closer to the other one this will help 25 00:00:47,219 --> 00:00:50,489 us determine what our limiting value is 26 00:00:48,690 --> 00:00:55,110 and give us a good approximation to that 27 00:00:50,489 --> 00:00:56,430 instantaneous change of a function makes 28 00:00:55,110 --> 00:00:57,719 this a little bit more concrete let's go 29 00:00:56,430 --> 00:01:00,210 ahead and do it with this example 30 00:00:57,719 --> 00:01:02,850 problem so here I have the function x 31 00:01:00,210 --> 00:01:04,439 squared minus 3x minus 4 and I'm curious 32 00:01:02,850 --> 00:01:08,580 what is the instantaneous rate of change 33 00:01:04,438 --> 00:01:11,009 at the point 1 negative 6 so let's go 34 00:01:08,580 --> 00:01:13,009 ahead and grab those calculators and get 35 00:01:11,010 --> 00:01:15,210 started 36 00:01:13,010 --> 00:01:17,100 all right the very first part of this is 37 00:01:15,210 --> 00:01:19,439 we need to get our function into the 38 00:01:17,100 --> 00:01:21,240 calculator itself so go ahead and press 39 00:01:19,439 --> 00:01:23,820 your y equals button it's near the top 40 00:01:21,240 --> 00:01:28,699 here and let's go ahead and type it in 41 00:01:23,820 --> 00:01:33,089 so I have x squared minus 3x minus 4 42 00:01:28,700 --> 00:01:35,600 enter all right so now our function is 43 00:01:33,090 --> 00:01:37,770 stored into the calculator itself and 44 00:01:35,600 --> 00:01:40,110 just for our naming conventions you'll 45 00:01:37,770 --> 00:01:42,060 notice that it stores it under y1 that 46 00:01:40,110 --> 00:01:44,310 will be important for later when we're 47 00:01:42,060 --> 00:01:49,530 making our slope formula so this is 48 00:01:44,310 --> 00:01:50,670 really our y1 function renamed now the 49 00:01:49,530 --> 00:01:53,610 other part of this is we need to 50 00:01:50,670 --> 00:01:58,320 calculate the slope between this point 51 00:01:53,610 --> 00:01:59,909 and some other mystery point you can 52 00:01:58,320 --> 00:02:01,169 consider the slope formula like the 53 00:01:59,909 --> 00:02:03,869 difference between the Y values and the 54 00:02:01,170 --> 00:02:05,159 difference between the X values so I'm 55 00:02:03,869 --> 00:02:06,899 going to use the notation of the 56 00:02:05,159 --> 00:02:10,200 calculator to get a better idea of 57 00:02:06,899 --> 00:02:12,790 what's going on here so wire function is 58 00:02:10,199 --> 00:02:14,869 y1 59 00:02:12,790 --> 00:02:17,480 and I'm going to figure out what is the 60 00:02:14,870 --> 00:02:20,480 difference between the output of the Y 61 00:02:17,480 --> 00:02:24,040 values over the X values well I know 62 00:02:20,480 --> 00:02:27,140 what happens with one of my Y values 63 00:02:24,040 --> 00:02:30,950 that happens when I plug in a 1 into the 64 00:02:27,140 --> 00:02:32,480 function itself and I'll leave the other 65 00:02:30,950 --> 00:02:34,310 one as X because that hasn't been 66 00:02:32,480 --> 00:02:37,130 determined yet it'll be my other point 67 00:02:34,310 --> 00:02:41,300 that I end up choosing and down below 68 00:02:37,130 --> 00:02:43,070 we'll have my X values so this slope 69 00:02:41,300 --> 00:02:46,550 formula is what now I want to program 70 00:02:43,070 --> 00:02:49,700 into the calculator and we'll go ahead 71 00:02:46,550 --> 00:02:51,260 and put that under Y to start this off 72 00:02:49,700 --> 00:02:54,500 I'll go ahead and use a pair of 73 00:02:51,260 --> 00:02:59,260 parentheses and then we need to put in 74 00:02:54,500 --> 00:03:02,000 y1 of 1 to do that you'll go to VARs 75 00:02:59,260 --> 00:03:06,049 arrow over until we get to the Y 76 00:03:02,000 --> 00:03:11,720 variables make sure the first one is 77 00:03:06,050 --> 00:03:14,200 selected function and select y1 so you 78 00:03:11,720 --> 00:03:16,730 can see now that I have parentheses y1 79 00:03:14,200 --> 00:03:21,049 will open up another set of parentheses 80 00:03:16,730 --> 00:03:22,700 and put a 1 on the inside there so what 81 00:03:21,050 --> 00:03:25,360 I'm telling the calculator literally is 82 00:03:22,700 --> 00:03:27,380 to take one and plug it into my function 83 00:03:25,360 --> 00:03:32,000 now let's go ahead and repeat that 84 00:03:27,380 --> 00:03:33,950 process - and we'll put a y1 and we'll 85 00:03:32,000 --> 00:03:39,350 feed X into it so again that's under 86 00:03:33,950 --> 00:03:44,089 variables arrow over to Y variables one 87 00:03:39,350 --> 00:03:45,829 for function and y1 only this time we'll 88 00:03:44,090 --> 00:03:49,190 open up the set of parentheses and just 89 00:03:45,830 --> 00:03:51,110 put an X on the inside all right so I 90 00:03:49,190 --> 00:03:52,520 use the set of parentheses when I start 91 00:03:51,110 --> 00:03:54,830 in this process let's go ahead and put 92 00:03:52,520 --> 00:03:56,600 another set to close the process they'll 93 00:03:54,830 --> 00:04:00,440 take care of everything on the top of my 94 00:03:56,600 --> 00:04:01,700 slope formula now divided by let's go 95 00:04:00,440 --> 00:04:04,490 ahead and open up another set of 96 00:04:01,700 --> 00:04:09,649 parentheses for the bottom here we'll 97 00:04:04,490 --> 00:04:11,480 have 1 minus X close parentheses so 98 00:04:09,650 --> 00:04:14,210 we've programmed in this nice slope 99 00:04:11,480 --> 00:04:16,099 formula we've programmed it in as y2 and 100 00:04:14,210 --> 00:04:19,190 now we can feed it in different values 101 00:04:16,100 --> 00:04:20,630 and see what we get all right so for the 102 00:04:19,190 --> 00:04:21,709 feeding in process we need to get back 103 00:04:20,630 --> 00:04:23,870 to the homescreen let's go ahead and 104 00:04:21,709 --> 00:04:25,919 press 2nd quit and go back to that home 105 00:04:23,870 --> 00:04:31,930 screen 106 00:04:25,919 --> 00:04:34,299 perfect all right so the point that we 107 00:04:31,930 --> 00:04:37,479 are interested in is the point 1 108 00:04:34,300 --> 00:04:39,490 negative 6 and I'll choose other X 109 00:04:37,479 --> 00:04:42,340 values a little bit further away to see 110 00:04:39,490 --> 00:04:43,479 what the slope is through those and here 111 00:04:42,340 --> 00:04:45,940 are some of the other X values we'll end 112 00:04:43,479 --> 00:04:47,740 up choosing and see what we get so to 113 00:04:45,940 --> 00:04:51,430 feed n values into my slope formula now 114 00:04:47,740 --> 00:04:56,169 I need to go back into variables VARs Y 115 00:04:51,430 --> 00:04:58,300 variables one for function and we've 116 00:04:56,169 --> 00:05:00,400 stored that slope formula into y2 so I'm 117 00:04:58,300 --> 00:05:02,550 gonna arrow down make sure y2 is 118 00:05:00,400 --> 00:05:04,599 selected and then press Enter 119 00:05:02,550 --> 00:05:08,830 all right it looks like the first value 120 00:05:04,600 --> 00:05:11,350 I'll feed is to make sure that when you 121 00:05:08,830 --> 00:05:13,539 feed it values that it's also encased 122 00:05:11,350 --> 00:05:16,990 inside parentheses all right so let's 123 00:05:13,539 --> 00:05:21,310 press Enter it looks like we get a value 124 00:05:16,990 --> 00:05:23,020 of zero so what's that really mean it 125 00:05:21,310 --> 00:05:24,760 means if I choose the point at 1 and I 126 00:05:23,020 --> 00:05:26,469 choose the point at x equals 2 and I 127 00:05:24,760 --> 00:05:29,680 find the slope between them the slope is 128 00:05:26,470 --> 00:05:31,750 going to be 0 so now that this is on my 129 00:05:29,680 --> 00:05:34,840 screen let's go ahead and feed in 1.5 130 00:05:31,750 --> 00:05:40,360 repeat those process of variables Y 131 00:05:34,840 --> 00:05:45,219 variables function make sure Y to open 132 00:05:40,360 --> 00:05:47,340 up parentheses 1.5 enter okay so now I'm 133 00:05:45,220 --> 00:05:52,630 getting a different one this is negative 134 00:05:47,340 --> 00:05:55,659 0.5 and you can repeat that process for 135 00:05:52,630 --> 00:05:57,610 the rest of these values now the neat 136 00:05:55,660 --> 00:05:59,740 part is if it's already on the screen 137 00:05:57,610 --> 00:06:03,729 like this I can shortcut the process I 138 00:05:59,740 --> 00:06:06,520 can go 2nd enter and then we can change 139 00:06:03,729 --> 00:06:07,810 this so that I don't have to go through 140 00:06:06,520 --> 00:06:10,448 that variables menu and make a little 141 00:06:07,810 --> 00:06:19,560 bit quicker you know all right this 142 00:06:10,449 --> 00:06:19,560 one's that negative 0.9 second enter 0 1 143 00:06:20,900 --> 00:06:32,830 negative 0.99 and one last time let's 144 00:06:26,840 --> 00:06:36,650 see 0 0 1 close parentheses negative 145 00:06:32,830 --> 00:06:39,050 0.999 so you can see the calculator is 146 00:06:36,650 --> 00:06:40,638 quickly finding the value of the slope 147 00:06:39,050 --> 00:06:42,830 between those two points make it nice 148 00:06:40,639 --> 00:06:44,389 and easy on me and now all I have to do 149 00:06:42,830 --> 00:06:45,680 is look at these slopes and say well 150 00:06:44,389 --> 00:06:47,030 does it look like they're getting close 151 00:06:45,680 --> 00:06:48,470 to anything does it look like they're 152 00:06:47,030 --> 00:06:51,280 getting close to some sort of limiting 153 00:06:48,470 --> 00:06:54,020 value all right so let's take this away 154 00:06:51,280 --> 00:06:56,330 all right for our slopes as we get 155 00:06:54,020 --> 00:06:57,680 closer and closer to 1 the slopes looks 156 00:06:56,330 --> 00:07:01,550 like they're getting closer to negative 157 00:06:57,680 --> 00:07:04,190 1 so I will say that that is my limiting 158 00:07:01,550 --> 00:07:06,919 value and it's my approximation to the 159 00:07:04,190 --> 00:07:10,940 instantaneous change of the function at 160 00:07:06,919 --> 00:07:12,650 that point all right so again the 161 00:07:10,940 --> 00:07:14,000 process for this is really just choosing 162 00:07:12,650 --> 00:07:15,530 two points finding the slope between 163 00:07:14,000 --> 00:07:16,760 those two points and then moving them 164 00:07:15,530 --> 00:07:18,080 closer and closer together and 165 00:07:16,760 --> 00:07:19,490 fortunately with the elbow of the 166 00:07:18,080 --> 00:07:21,169 calculator you can move those points 167 00:07:19,490 --> 00:07:23,630 closer to one another fairly simply 168 00:07:21,169 --> 00:07:24,770 making it nice and quick if you'd like 169 00:07:23,630 --> 00:07:27,400 to 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