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These are the user uploaded subtitles that are being translated: 1 00:00:00,480 --> 00:00:02,740 Today we're going to talk about how to find objects. 2 00:00:03,030 --> 00:00:08,800 Given the function second derivative and a couple of initial conditions we complete this problem will 3 00:00:08,810 --> 00:00:13,770 think the integral of second derivative to find the first derivative then take the integral of the first 4 00:00:13,770 --> 00:00:18,360 derivative find the original function and then plug in our initial conditions. 5 00:00:18,360 --> 00:00:24,720 In this particular problem we've been asked to find F X if double prime of x is equal to 12 x squared 6 00:00:24,750 --> 00:00:26,650 plus 6 X minus 4. 7 00:00:26,760 --> 00:00:33,430 And given the initial conditions that EF 0 0 0 is equal to 4 and 1 is equal to 1. 8 00:00:33,450 --> 00:00:39,540 So our first step is to start with F double prime IVAX and start picking antiderivatives right. 9 00:00:39,540 --> 00:00:47,310 We know that f of x is our essentially original function f prime is its first derivative and of double 10 00:00:47,310 --> 00:00:52,980 prime is its second derivative which means therefore starting with F double prime of X will need to 11 00:00:52,980 --> 00:00:59,010 take the integral of double prime of X to get two f prime of X the first derivative and then take the 12 00:00:59,010 --> 00:01:03,950 integral again to get from prime of X back to f of x. 13 00:01:03,990 --> 00:01:12,480 So let's start with the second derivative f double prime of X and say that that is equal to 12 x squared 14 00:01:13,560 --> 00:01:16,350 plus 6 X minus 4. 15 00:01:16,350 --> 00:01:25,380 Now to find f prime of X the first derivative we need to take the integral of f a prime of x which we 16 00:01:25,380 --> 00:01:30,530 know to be 12 x squared plus 6 x minus 4. 17 00:01:30,840 --> 00:01:34,590 And as with any integral we need that DX notation there. 18 00:01:34,650 --> 00:01:37,150 So is this a simple polynomial integral. 19 00:01:37,170 --> 00:01:42,270 Remember that when we're taking the integral or the antiderivative of this function here we're just 20 00:01:42,270 --> 00:01:43,710 going to use a power rule. 21 00:01:43,710 --> 00:01:46,810 So you take the integral we have x squared. 22 00:01:46,810 --> 00:01:49,750 Here we're going to be adding one to the exponent. 23 00:01:49,830 --> 00:01:57,690 So two plus one in the exponent gives us three and then we divide the coefficient 12 by the new exponent 24 00:01:58,250 --> 00:01:59,390 3. 25 00:01:59,400 --> 00:02:01,460 Same thing here with the 6 x ray. 26 00:02:01,590 --> 00:02:04,060 So we have basically X to the first power. 27 00:02:04,080 --> 00:02:04,340 Right. 28 00:02:04,350 --> 00:02:05,780 X to the first power. 29 00:02:05,790 --> 00:02:09,340 So we'll add 1 to the exponent one plus one gives us two. 30 00:02:09,450 --> 00:02:17,090 So we'll get X squared and then we'll divide the coefficient 6 by the new exponent which is 2. 31 00:02:17,520 --> 00:02:24,180 Finally we get to form and basically here we have four times x to the zero power. 32 00:02:24,210 --> 00:02:24,830 Right. 33 00:02:24,840 --> 00:02:31,200 X to the zero or anything to the zero for that matter is equal to 1 4 times 1 is just for this term 34 00:02:31,200 --> 00:02:34,820 as essentially you can think of it as 4 x to the 0. 35 00:02:34,890 --> 00:02:38,720 So it becomes the same then as the last two terms that we've done. 36 00:02:38,820 --> 00:02:40,400 So we have X to the zero. 37 00:02:40,590 --> 00:02:44,950 Again we'll add 1 to the exponent zero plus 1 gives us 1. 38 00:02:45,150 --> 00:02:51,480 So there's our exponent and then we'll take the coefficient for and divide it by the new exponent which 39 00:02:51,480 --> 00:02:52,620 is 1. 40 00:02:52,740 --> 00:02:58,620 And as with any integral whenever you take the integral you have to go ahead and add what we call C 41 00:02:58,650 --> 00:03:04,890 which is the constant of integration to account for a constant that may have been in f prime of X that 42 00:03:04,890 --> 00:03:08,760 got lost when we took the derivative to get at a prime of x. 43 00:03:09,210 --> 00:03:13,980 So now that we have a prime of x we just need to simplify 12 divided by 3. 44 00:03:13,980 --> 00:03:15,140 Here is 4. 45 00:03:15,240 --> 00:03:21,440 So we get 4 x cubed 6 over 2 will give us 3 x squared. 46 00:03:21,470 --> 00:03:29,190 And obviously this here just becomes for X and then we keep our C for the constant integration so that 47 00:03:29,250 --> 00:03:30,410 f prime of x. 48 00:03:30,420 --> 00:03:34,890 Now we need to go ahead and get to the original function f of x. 49 00:03:34,980 --> 00:03:43,560 So to get f of x we need to take the integral again of the first derivative function as prime of x so 50 00:03:43,560 --> 00:03:57,210 we'll be taking the integral of 4 x cubed plus 3 x squared minus 4 x plus C and again at Dx notation. 51 00:03:57,210 --> 00:04:02,340 So we'll do the same thing we did last time for 4 x cubed here to take the integral again. 52 00:04:02,400 --> 00:04:09,420 Add 1 to the exponent three plus one gives us 4 and we'll divide our coefficient for our new exponent 53 00:04:09,910 --> 00:04:10,950 4. 54 00:04:11,010 --> 00:04:12,220 Same thing here. 55 00:04:12,240 --> 00:04:18,060 You've got x squared well add 1 to the exponent to get execute and then we'll divide the coefficient 56 00:04:18,180 --> 00:04:23,660 by the new exponent for negative for X here we have X to the first power. 57 00:04:23,730 --> 00:04:30,930 So add 1 and we'll get X squared and then we'll divide the coefficient or the new exponent. 58 00:04:30,930 --> 00:04:32,960 See here is just a constant. 59 00:04:32,970 --> 00:04:37,110 This basically takes over the same form that the forehead last time. 60 00:04:37,140 --> 00:04:43,800 This is c times x to the 0 or c times 1 because anything raised to the zero power is just one. 61 00:04:43,800 --> 00:04:50,990 So we add one to the exponent and the x to the first the coefficient is C C is just a constant. 62 00:04:51,060 --> 00:04:55,620 Like any number here just like for was up here in our original second derivative. 63 00:04:55,620 --> 00:04:58,610 So we get C divided by 1. 64 00:04:58,980 --> 00:05:04,730 And because we took integral again what we have to add another constant to distinguish it from the constant 65 00:05:04,730 --> 00:05:08,860 we added before we'll call this one D instead of C. 66 00:05:09,260 --> 00:05:13,110 So now we need to simplify as much as we can for forgiveness 1. 67 00:05:13,130 --> 00:05:17,150 So we'll just get X to the fourth again three over three will just be one. 68 00:05:17,150 --> 00:05:29,750 So we'll get x cubed and then negative four halves gives us a negative 2 x squared plus C X plus the 69 00:05:31,560 --> 00:05:34,660 so this here is our original function f of x. 70 00:05:34,710 --> 00:05:41,470 Now we need to do is use our initial conditions of zero equals four and a half of one equals one to 71 00:05:41,480 --> 00:05:44,210 find C and D are constants. 72 00:05:44,250 --> 00:05:50,400 So the way that we'll do that because these are both the original function f of zero as opposed to prime 73 00:05:50,400 --> 00:05:52,600 of zero or double prime of zero. 74 00:05:52,650 --> 00:05:57,470 We're going to be plugging them into our original function here f of x. 75 00:05:57,510 --> 00:06:03,530 So what this tells us to do is plug in zero for X and set it equal to 4. 76 00:06:03,600 --> 00:06:09,720 So we'll say and we'll be doing it here with this equation that we simplified for objects. 77 00:06:09,750 --> 00:06:13,730 So we'll set it equal to 4 and plug in zero. 78 00:06:13,740 --> 00:06:29,730 So we'll get 0 to the fourth power plus 0 cubed minus two times zero squared plus C times zero plus 79 00:06:29,880 --> 00:06:30,860 the. 80 00:06:30,900 --> 00:06:32,340 And that'll be our first equation. 81 00:06:32,340 --> 00:06:38,940 We can simplify that obviously because we'll get all of these to cancel since they'll all be zero. 82 00:06:38,970 --> 00:06:43,450 And what we're left with here is four equals D. 83 00:06:43,500 --> 00:06:48,390 Now we need to use our second initial condition to solve for C. 84 00:06:48,420 --> 00:06:53,730 So we'll plug in one for X and that the whole thing equal to one. 85 00:06:53,750 --> 00:07:04,620 So you get one equals one to the fourth power plus one to the third power minus two times 1 squared 86 00:07:05,040 --> 00:07:09,630 plus C times 1 plus the. 87 00:07:09,630 --> 00:07:12,630 And what we'll get when we simplify this is one equals. 88 00:07:12,720 --> 00:07:20,950 This will just be one plus one minus two plus C plus the. 89 00:07:21,240 --> 00:07:22,660 So simplify again. 90 00:07:23,040 --> 00:07:26,270 One plus one minus two is two minus two. 91 00:07:26,400 --> 00:07:33,890 So these were all cancell with one another and we're just left with C plus D. 92 00:07:34,140 --> 00:07:35,550 We already solved for D. 93 00:07:35,550 --> 00:07:37,490 Remember that we got the equals four. 94 00:07:37,730 --> 00:07:41,260 We can go ahead and make that substitution for D. 95 00:07:41,460 --> 00:07:47,190 And then to solve for C We'll just subtract C from both sides and we'll see that negative 3 is equal 96 00:07:47,190 --> 00:07:49,020 to see. 97 00:07:49,020 --> 00:07:51,290 So now we have solutions for C and D. 98 00:07:51,330 --> 00:07:56,400 And what we can do is plug them back into our original function to get our final answer. 99 00:07:56,610 --> 00:08:08,160 So our final answer for f objects will be f objects is equal to x to the fourth plus x cubed minus 2 100 00:08:08,160 --> 00:08:17,490 x squared minus 3 because we get negative 3 for C minus 3 x plus 4 for B so plus 4. 101 00:08:17,700 --> 00:08:18,440 And that's it. 102 00:08:18,450 --> 00:08:25,050 That's our original function f of x given the second derivative f double prime of X and these two initial 103 00:08:25,050 --> 00:08:25,710 conditions. 10895