WEBVTT 1 00:00:00.000 --> 00:00:06.480 Well hello everyone and welcome to our fourth lecture on Laplace transforms. Today we're going 2 00:00:06.480 --> 00:00:12.800 to introduce a special type of function called a unit step function. Now mathematically modelling 3 00:00:12.800 --> 00:00:18.960 real world scenarios often requires us to use non-standard functions in our equations, 4 00:00:19.520 --> 00:00:26.480 equations like differential equations for example. One such function is the unit step function. Now 5 00:00:26.560 --> 00:00:32.720 in coming lectures we'll see how to use Laplace transforms to solve differential equations 6 00:00:33.280 --> 00:00:42.000 that involve the step function. But today we're going to introduce the step function and look at 7 00:00:42.000 --> 00:00:48.880 plots of the step function when it's involved in linear combinations. We're looking at products 8 00:00:48.880 --> 00:00:55.520 involving step functions and finally we'll have a look at determining the equations of functions 9 00:00:55.520 --> 00:01:03.280 from the graphs. Now in our last lecture we saw how Laplace transforms can be applied to solve 10 00:01:03.280 --> 00:01:08.160 an initial value problem where the differential equation has a standard 11 00:01:08.160 --> 00:01:15.760 portion function on the right hand side. Now when we come to mathematically modelling real 12 00:01:15.760 --> 00:01:21.520 world systems however the type of portion function on the right hand side of the differential equation 13 00:01:22.080 --> 00:01:28.880 can make it difficult to solve an initial value problem using traditional time domain 14 00:01:29.600 --> 00:01:34.960 methods such as the method of undetermined coefficients that we saw earlier on in this 15 00:01:34.960 --> 00:01:43.360 course. Now Laplace transforms can simplify the process considerably so this becomes easier 16 00:01:43.360 --> 00:01:51.360 quicker to obtain a solution of a given differential equation. And one function as I've said 17 00:01:51.600 --> 00:01:57.040 that's often used to model real world quantities is the unit step function. 18 00:01:58.880 --> 00:02:05.440 Now in engineering applications we often come across functions whose values change abruptly at 19 00:02:05.440 --> 00:02:13.440 specified values of time. And one common example would be when a voltage is switched on or off 20 00:02:13.440 --> 00:02:21.280 in an electrical circuit at a given value of time. The switching process can be described 21 00:02:21.280 --> 00:02:27.360 mathematically by the unit step function. Now the unit step function is also known as the 22 00:02:27.360 --> 00:02:36.800 heavy side function after the mathematician Oliver Heaviside who introduced it. So what does the 23 00:02:36.800 --> 00:02:43.120 step function look like? The unit step function what does it look like? Well it's defined mathematically 24 00:02:43.120 --> 00:02:53.600 like so u of t minus capital T it's equal to zero when t is less than capital T and it's equal to one 25 00:02:54.480 --> 00:03:01.440 when t is greater than capital T. So if we're to plot this unit step function this is what it looks 26 00:03:01.440 --> 00:03:08.560 like here. We're only defined it for positive values of time so it's zero all the way down here for 27 00:03:08.560 --> 00:03:17.920 negative values of t. So here's our function here u of t minus capital T and up until the value 28 00:03:18.560 --> 00:03:24.640 which we'll see later is called the critical value up until this critical value this function this 29 00:03:24.640 --> 00:03:30.560 unit step function takes the value zero and then it suddenly jumps to take the value one 30 00:03:31.280 --> 00:03:38.080 at this critical value t. So it's zero all the way up to the critical value and then it suddenly 31 00:03:38.080 --> 00:03:45.680 jumps to take the value of one and it stays at one forever and ever and that's what this states. 32 00:03:45.680 --> 00:03:53.680 For t less than capital T the unit step function takes the value zero as you can see here and for t 33 00:03:53.680 --> 00:04:00.880 greater than capital T it takes the value one as you can see in this diagram here. So you can think 34 00:04:00.880 --> 00:04:08.320 of it as a function that's been switched on when t is equal to capital T. I've included some 35 00:04:08.320 --> 00:04:14.000 alternative notation for the unit step function at the foot of the slide here. 36 00:04:16.720 --> 00:04:23.600 Okay so as I mentioned this value when the unit step function is switched on 37 00:04:25.120 --> 00:04:30.320 this value of capital T is called the critical value of the step function and is where 38 00:04:30.320 --> 00:04:39.600 the step function changes value, changes from zero to one. Now as you can see in this definition 39 00:04:39.600 --> 00:04:46.320 here the value of the step function when t is equal to capital T is not defined. Some books will 40 00:04:46.320 --> 00:04:51.440 give it the value one, others books will give it the value zero and some books will even give it 41 00:04:51.440 --> 00:04:58.160 the value a half when t is equal to t. That's not really important for us. All we need to remember 42 00:04:58.160 --> 00:05:05.120 is that up until the critical value this unit step function is equal to zero and once we hit that 43 00:05:05.120 --> 00:05:12.640 critical value t equals capital T think of it as like a switch being flicked on and the unit step 44 00:05:12.640 --> 00:05:21.360 function jumps up to have a value of one which it holds forever and ever. So let's have a look 45 00:05:21.360 --> 00:05:27.840 at specific examples. So when our critical value is taken to be t is equal to three in other words 46 00:05:27.840 --> 00:05:33.520 when capital T is equal to three this is what the function looks like in zero all the way up to three 47 00:05:33.520 --> 00:05:39.360 then it suddenly gets switched on and it jumps up to take the value one and it'll hold that value 48 00:05:39.360 --> 00:05:46.800 forever and ever and mathematically we would write it like this u of t minus three is equal to zero 49 00:05:46.800 --> 00:05:52.560 for t strictly less than three and it's equal to one for t strictly greater than three and that's 50 00:05:52.560 --> 00:06:01.680 what you can see in the diagram there. Now if we multiply our unit step function by a constant 51 00:06:03.360 --> 00:06:10.400 the constant just represents the amplitude of our step function. Obviously when it's a unit step 52 00:06:10.400 --> 00:06:17.280 function it's got amplitude one and we'll just refer to it as a step function when it's got some other 53 00:06:17.360 --> 00:06:24.960 amplitude. So in this case with a critical value of capital T is equal to four the step function 54 00:06:24.960 --> 00:06:30.480 has the value zero all the way up to four and then it's suddenly switched on and it jumps up 55 00:06:30.480 --> 00:06:38.800 to take the value six because there's a six multiplying the unit step function. So that's 56 00:06:38.800 --> 00:06:45.280 what's happening here so it's off all the way up until t is equal to four it suddenly jumps to the 57 00:06:45.280 --> 00:06:55.280 value six at the value t equals four and it stays at six forever and ever. Now a special case might 58 00:06:55.280 --> 00:07:03.760 be when the critical value t is equal to zero so in other words nothing happens it's a unit step 59 00:07:03.760 --> 00:07:10.000 function as the value zero all the way up to time t equals zero and then it suddenly jumps to take 60 00:07:10.000 --> 00:07:18.240 the value one and it stays at one forever and ever. Okay so in terms of step functions that I say 61 00:07:18.240 --> 00:07:24.800 down here u of t minus zero is just equal to one it's just a constant that's equal to one. 62 00:07:25.680 --> 00:07:32.480 So here's an example plot a step voltage of size 12 volts which starts at time t equals four 63 00:07:32.480 --> 00:07:39.280 seconds so in this case we know that our units that will do this with a unit step function we 64 00:07:39.280 --> 00:07:44.640 started with a step function who will take the value zero all the way up to the critical value 65 00:07:44.640 --> 00:07:50.560 of t equals four and then suddenly gets switched on and it jumps up to take the value 12 and it 66 00:07:50.560 --> 00:08:02.480 holds that value for all t after four so and that's what it looks like here. Okay so these are some 67 00:08:03.120 --> 00:08:10.640 plots of step functions of different amplitudes and different critical values so let's have 68 00:08:10.640 --> 00:08:16.800 a look at combining some step functions now so we've got two unit step functions here both of 69 00:08:16.800 --> 00:08:24.320 them have the amplitude one and we're adding them together u of t minus one that's a unit step function 70 00:08:25.120 --> 00:08:33.040 whose critical value is capital t is equal to one in other words it's zero all the way up to the 71 00:08:33.040 --> 00:08:38.400 critical value of t equals one when it suddenly gets switched on and takes the value one because 72 00:08:38.400 --> 00:08:43.440 the amplitude is one so unit step function takes value one forever and ever and to that we're going 73 00:08:43.440 --> 00:08:50.080 to add another unit step function but this time the unit step function has a critical value of two 74 00:08:50.080 --> 00:08:57.360 so once we hit time t equals two seconds this function will get switched on and it'll take the 75 00:08:57.360 --> 00:09:05.360 value one forever and ever from from t equals two onwards so what happens when we add them together 76 00:09:05.360 --> 00:09:11.040 well let's just have a look at this for if t is less than one both the functions are off we haven't 77 00:09:11.040 --> 00:09:17.680 switched either of them on yet because look the critical values are capital t equals one and capital 78 00:09:17.680 --> 00:09:23.600 t equals two so up at the t equals one nothing happens they're both switched off so this one will 79 00:09:23.600 --> 00:09:30.240 have the value zero and this one will have the value zero so my overall function f of t would just be 80 00:09:30.240 --> 00:09:37.920 zero plus zero which is zero so f of t is equal to zero the sum of these two unit step functions up 81 00:09:37.920 --> 00:09:46.400 until the first critical value t equals one now once we hit t equals one this step function here 82 00:09:46.480 --> 00:09:54.720 this one switches on and takes the value one whilst this one is still off this one will not come 83 00:09:54.720 --> 00:10:03.040 into play until we hit the critical value t equals two so the overall function f of t will take the 84 00:10:03.040 --> 00:10:10.880 value one here plus but this one will be zero so it will hold the value one all the way up until 85 00:10:10.960 --> 00:10:16.560 the next critical value at t equals two and that's what you can see in the plot here so we said 86 00:10:16.560 --> 00:10:22.960 earlier that both functions are zero up until t equals one so the overall function is zero so it 87 00:10:22.960 --> 00:10:28.640 keeps going at zero all the way up until you hit the first critical value it then suddenly switches 88 00:10:29.280 --> 00:10:35.680 and jumps up to take the value one now it holds that value all the way along until you hit the 89 00:10:35.680 --> 00:10:44.640 next critical value at t equals two and it'll this one will switch on so now you got both functions 90 00:10:44.640 --> 00:10:50.720 switched on both have a value one and you add them together one plus one gives you the value two 91 00:10:51.280 --> 00:10:58.560 so this will now be the value of the function f of t forever and ever it'll be the hold the value 92 00:10:58.560 --> 00:11:03.920 of two forever and ever on the next slide here i've sort of tried to explain this a bit in a bit 93 00:11:03.920 --> 00:11:11.040 more detail so i'll just go quickly through this so as i said we've added together two unit step 94 00:11:11.040 --> 00:11:17.520 functions u of t minus one in red and u of t minus two in blue this one switches on at the critical 95 00:11:17.520 --> 00:11:23.360 value t equals one and this one will switch on at the critical value t equals two so let's have a look 96 00:11:23.360 --> 00:11:32.080 then so here you can see along here between zero and one nothing happens because we haven't 97 00:11:32.080 --> 00:11:39.520 reached either the critical value set both functions are off so in other words this u t minus one the 98 00:11:39.520 --> 00:11:45.760 red one has the value zero and the blue one has the value zero so in here the sum of these functions 99 00:11:45.760 --> 00:11:51.600 giving me f of t will be zero plus zero which is zero so down here i plot that so in this first 100 00:11:51.600 --> 00:11:57.680 section between zero and one the overall function f of t is the sum of the two unit step functions 101 00:11:57.680 --> 00:12:04.560 and that's zero then we reach the first critical value at t equals one and this function gets 102 00:12:04.560 --> 00:12:11.600 switched on and it takes the value one this function hasn't been switched on yet so it's still off 103 00:12:12.400 --> 00:12:19.200 so my overall function f of t will have the value one from the red function plus zero of the blue 104 00:12:19.200 --> 00:12:27.200 function which is one so it'll take the value one and it'll hold it up until the next critical value 105 00:12:27.760 --> 00:12:36.400 the next critical value will occur here at t equals two and this function this blue function 106 00:12:36.400 --> 00:12:42.240 will get switched on and take the value one so that's what i've shown here the blue function 107 00:12:42.240 --> 00:12:48.000 is zero all the way up to t equals two and then it suddenly gets switched on and it also has the 108 00:12:48.000 --> 00:12:54.160 value one the red function we saw got switched on at t equals one and it jumped to take the value one 109 00:12:54.160 --> 00:13:00.000 and it held the value one forever and ever so on this region from two onwards both functions have 110 00:13:00.000 --> 00:13:05.760 the value one and when you add them together you get two and you can see that the overall function 111 00:13:05.760 --> 00:13:12.160 which is sum of the two functions will have the value one plus one from t equals two onwards and 112 00:13:12.160 --> 00:13:19.200 it'll hold that value forever and ever so that's adding two unit step functions together this is 113 00:13:19.280 --> 00:13:27.120 the sort of function that comes out of it okay so here's one that's slightly more complicated now 114 00:13:27.120 --> 00:13:32.240 i'm not going to go into as much detail on this one and hopefully you can see what's involved so 115 00:13:32.240 --> 00:13:39.040 and first of all i'm going to examine my step functions now obviously this is a unit this unit 116 00:13:39.040 --> 00:13:44.800 step function here is multiplied by two so this function will have amplitude two this one will 117 00:13:44.800 --> 00:13:51.760 have amplitude minus three okay it'll have about three but we're subtracting that so we'll see what 118 00:13:51.760 --> 00:13:58.160 effect that has that has shortly and this is just a unit step function so the first thing i want to 119 00:13:58.160 --> 00:14:04.320 look at is i want to find the critical values the critical values are the all important values 120 00:14:04.320 --> 00:14:12.320 where things happen where functions get switched on so the critical values in this case are five 121 00:14:12.320 --> 00:14:19.520 10 and 15 so mark five 10 and 15 on my plot and let's have a look and see what happens well 122 00:14:20.240 --> 00:14:27.120 up until t equals five there's nothing happening all the functions these three step functions 123 00:14:27.120 --> 00:14:34.880 are all switched off none of them have come into play yet so up until up between zero and five 124 00:14:35.680 --> 00:14:41.920 all these functions have the value zero so the overall function which is just a linear 125 00:14:41.920 --> 00:14:47.920 combination of these three functions the two will have the value zero and i've shown that in blue here 126 00:14:49.680 --> 00:14:53.680 when we hit t equals five this step function gets switched on 127 00:14:55.920 --> 00:15:05.120 and it's going to have amplitude two these two though will not get switched on yet this one 128 00:15:05.120 --> 00:15:11.200 gets switched on at t equals 10 and this one gets switched on on t is equal to 15 so so at this 129 00:15:12.000 --> 00:15:17.760 instant t equals five this function gets switched on and takes the value two and that's what you 130 00:15:17.760 --> 00:15:23.200 can see happening here so it jumps from the value zero the overall function jumps from having the 131 00:15:23.200 --> 00:15:30.720 value zero to taking the value two now it's going to hold that value till we hit the next critical 132 00:15:30.720 --> 00:15:38.720 value and we hit the next critical value at t equals 10 so once we hit that value this unit 133 00:15:38.720 --> 00:15:47.360 step function gets switched on and it's multiplied by minus three so what's going to happen here is 134 00:15:48.320 --> 00:15:54.640 we're going to be up here and we're going to drop down three units and it'll take the overall 135 00:15:54.640 --> 00:16:01.440 function f of t will take the value minus one so that's what you can see happening here so the 136 00:16:01.440 --> 00:16:08.560 overall function f of t had the value two all the way through the interval five up to 10 then 137 00:16:09.200 --> 00:16:16.640 this function got switched on and it took the it it forced a drop of three units so you drop three 138 00:16:16.640 --> 00:16:24.480 units from two and you go down to minus one so the overall function f of t will then hold the value 139 00:16:24.480 --> 00:16:30.560 minus one on the next on this interval until we hit the next critical value and the next critical 140 00:16:30.560 --> 00:16:40.720 value is hit at t equals 15 now when t is equal to 15 this function kicks in and it's got the value 141 00:16:40.720 --> 00:16:46.320 one because that's the amplitude unit step function it's got the it's got the amplitude one and so we 142 00:16:46.320 --> 00:16:54.560 add one to f of t to the f of t from from this interval here and that will bring us up to zero 143 00:16:54.560 --> 00:17:02.480 and then the overall function f of t will hold the value zero forever and ever for all values of 144 00:17:02.480 --> 00:17:10.400 time from t equals 15 onwards so that's what we've got that's that's the linear combination of three 145 00:17:11.040 --> 00:17:17.760 step functions and this is the plot of it I mean just very quickly we've got three critical values 146 00:17:17.760 --> 00:17:23.920 five 10 and 15 the first function gets switched on at t equals five so up until then there's nothing 147 00:17:23.920 --> 00:17:30.080 happen when t equals five the step function has the value two so it will hold the value two up until 148 00:17:30.080 --> 00:17:36.400 the next critical value which occurs at 10 and this corresponds to a drop because of the minus sign 149 00:17:36.400 --> 00:17:44.000 by three units so we drop down by three units and then we hold that value of minus one over 150 00:17:44.080 --> 00:17:50.480 this interval until you hit the next critical value which you hit when t is equal to 15 and this unit 151 00:17:50.480 --> 00:17:56.160 step function of amplitude one comes into play and that will move us up one unit and then the 152 00:17:56.160 --> 00:18:04.720 overall function f of t will take the value zero from t equals 15 onwards forever and ever so that's 153 00:18:04.720 --> 00:18:13.840 a linear combination of three step functions so we've looked at sums and differences 154 00:18:13.920 --> 00:18:22.960 of some step functions we'll now move on and look at a couple of different types of products that 155 00:18:22.960 --> 00:18:34.560 involve step functions now one reason for using products that involve step functions is that it 156 00:18:34.560 --> 00:18:41.920 enables us to switch functions other functions off and on at given values of time and we're going 157 00:18:42.000 --> 00:18:49.520 to look at two different products in the first one we're going to look at any function some function 158 00:18:49.520 --> 00:18:57.840 f of t been multiplied by a step function whose critical value is ut minus a now if you remember 159 00:18:57.840 --> 00:19:07.840 the definition of a step function the definition of a step function tells us that the function is 160 00:19:07.840 --> 00:19:14.400 equal to zero up until the critical value and it equals to one from the critical value onwards 161 00:19:15.440 --> 00:19:21.840 now what does that mean well if you're multiplying some function something like perhaps t squared or 162 00:19:21.840 --> 00:19:28.400 something like that by a step function whose critical value is at a now this function this step 163 00:19:28.400 --> 00:19:35.120 function this unit step function here has the value zero up until you hit the critical value at t 164 00:19:35.120 --> 00:19:44.080 equals a so if you multiply anything by this unit step function up until t is equal to a then 165 00:19:44.080 --> 00:19:51.840 you're multiplied by zero so the whole function g of t will become zero and then from after values 166 00:19:51.840 --> 00:19:59.440 of the critical value at a this unit step function takes the value one so our g of t will be equal 167 00:19:59.440 --> 00:20:07.120 to f of t from a onwards so I'll just just recap on that up until the critical value this whole 168 00:20:07.120 --> 00:20:17.280 function g of t will be zero and then from the critical value onwards it'll be equal to our function 169 00:20:17.280 --> 00:20:27.200 f of t so in other words in our plot we'll have a plot of f of t zero up until t equals a and then it 170 00:20:27.200 --> 00:20:34.000 starts at t equals a and goes on forever and ever after that the second type of product we look at is 171 00:20:34.000 --> 00:20:43.120 where we've got a function existing between two values of t only between the two critical values 172 00:20:43.120 --> 00:20:49.840 of two unit step functions and that gives us what we're going to call a mathematical window to the left 173 00:20:50.640 --> 00:20:58.160 that's up to a this overall function g will be equal to zero then you'll have this window between a 174 00:20:58.160 --> 00:21:06.320 and b where the function f of t will exist and then to the right from b onwards the function g of t 175 00:21:06.320 --> 00:21:15.040 will be equal to zero so it only exists within a mathematical window between a and b so first of all 176 00:21:15.040 --> 00:21:26.960 we'll look at type one so here's a function we've got a type one function here we've got some function 177 00:21:26.960 --> 00:21:34.240 f of t multiplying a unit step function whose critical value is a and in our example our function 178 00:21:34.240 --> 00:21:41.200 g of t the product of f of t and the step function is given by four t plus two times a step function 179 00:21:41.200 --> 00:21:48.320 whose critical value is one that's u of t minus one and if you remember all that means is that the 180 00:21:48.320 --> 00:21:55.200 step function has the value zero for t less than one and it's got the value one for t greater than one 181 00:21:56.400 --> 00:22:06.640 so we're wanting to plot this function so what's actually going on here well remember what we said 182 00:22:07.200 --> 00:22:14.400 in a type one function what's actually happening is multiplied by the step function switches 183 00:22:14.400 --> 00:22:21.520 everything off up until the critical value of the step function so so up until t equals one 184 00:22:22.320 --> 00:22:29.520 everything will be equal to zero so that's what i've done here so between t equals zero and the 185 00:22:29.520 --> 00:22:35.360 critical value t equals one everything is off because the step function has the value zero 186 00:22:35.440 --> 00:22:41.920 over that interval so zero times four t plus two is just zero and then suddenly a t equals one 187 00:22:42.800 --> 00:22:50.320 it's like a switch inflict the unit step function kicks into action and has the value one so our 188 00:22:50.320 --> 00:23:00.640 function g of t will take the value four t plus two from one onwards so what we need to do is we need 189 00:23:00.640 --> 00:23:09.680 to draw the graph of the function four t plus two so it's a straight line it's a linear function 190 00:23:09.680 --> 00:23:18.160 because it's t to the power one so to to draw a straight line all we need is any two points on the 191 00:23:18.160 --> 00:23:26.080 line and so for example i could first of all take t equal to zero plug it in here and i would find 192 00:23:26.080 --> 00:23:32.800 that f of zero is equal to two so my first point would be zero two i can then say t equal to one 193 00:23:33.360 --> 00:23:40.960 and plug it in here and i get t f of t or f of one is equal to six so my second point would be one 194 00:23:40.960 --> 00:23:48.880 six and i've plotted them on here and that would enable me to draw my line my function four t plus 195 00:23:49.840 --> 00:23:56.800 and as it's a type one function multiplied by a unit step function basically sets that function 196 00:23:56.800 --> 00:24:03.840 to zero up until the critical value the critical value is that t equals one so my function shown 197 00:24:03.840 --> 00:24:11.520 here f of t is equal to four t plus two is going to be zero up the critical value then it's going 198 00:24:11.520 --> 00:24:17.760 to jump as i've said before shown by a dotted line and then it's just going to take the value of the 199 00:24:17.760 --> 00:24:24.320 function four t plus two for all values of t onwards so that's what it looks like i've just 200 00:24:24.320 --> 00:24:31.040 projected the line back then here just to show where it cuts the where it would cut the vertical 201 00:24:31.040 --> 00:24:37.360 axis so this line here this would be the graph of four t plus two but we're not interested in this 202 00:24:37.360 --> 00:24:44.720 part of it because our switch hasn't been flicked on yet it gets flicked on at t equals one so that's 203 00:24:44.720 --> 00:24:53.280 what that graph looks like okay so let's have a look at another so again we're going to do a type one 204 00:24:53.280 --> 00:25:00.160 function we've got two t plus one times a step unit step function is critical value is two so 205 00:25:00.720 --> 00:25:07.360 that's what our step function our unit step function looks like mathematically it's zero 206 00:25:07.440 --> 00:25:15.280 for t less than two and it's one for t greater than two so first of all let's identify the value 207 00:25:15.280 --> 00:25:21.840 of the critical of the unit the critical value of the step function the critical value is capital 208 00:25:21.840 --> 00:25:31.120 t is equal to two it's that value there okay so that means that my function f of t is equal to two 209 00:25:31.120 --> 00:25:38.800 t plus one will be off will be at zero up until the critical value of t equals two and the reason 210 00:25:38.800 --> 00:25:46.000 for that is because the unit step is zero up until t equals two so you multiply anything by zero you 211 00:25:46.000 --> 00:25:55.120 get zero so the function between my g of t function will be zero all the way up to t is equal to two 212 00:25:55.680 --> 00:26:05.520 now as we said the unit step function gets switched on at t equals two and it'll take the value one 213 00:26:06.080 --> 00:26:12.720 for for all t greater than two so you're going to have a function f of t is two t plus one 214 00:26:12.720 --> 00:26:19.920 multiplying one so your overall function g of t will just be the graph of two t plus one and that's 215 00:26:19.920 --> 00:26:27.520 for all t greater than two so what i need to do is i need to draw the graph of two t plus one again 216 00:26:27.520 --> 00:26:33.520 it's a linear function straight line function so i just need two points on the line and i can select 217 00:26:33.520 --> 00:26:41.840 any points that i like so for example i've chosen the values t equals two and t equals three now there's 218 00:26:41.840 --> 00:26:47.040 a reason i chose t equals two the reason i chose t equals two is because that's the critical value so 219 00:26:47.040 --> 00:26:55.120 i want to see where the function exactly jumps to at the left hand end if you like at the critical 220 00:26:55.120 --> 00:27:02.080 value when the unit step function gets switched on so if i calculate the value of f of t when t is 221 00:27:02.080 --> 00:27:09.200 equal to two it'll just be two times two plus one which is five and then i can also calculate the 222 00:27:09.200 --> 00:27:17.920 value at t equals three and then f of three is equal to seven two times three plus one so that 223 00:27:17.920 --> 00:27:24.640 means i've got my coordinate pair two points on the line two five and three seven just as i've 224 00:27:24.640 --> 00:27:31.760 thrown just as i've included here in the diagram so that enables me then to draw my line two t plus 225 00:27:31.760 --> 00:27:38.880 one and there it is and i've just projected back to show her cut the vertical axis but as we know 226 00:27:38.880 --> 00:27:46.320 the overall function g of t is the product of f of t and the unit step function and the unit step 227 00:27:46.320 --> 00:27:55.200 function has the ability to kill off the function all the way up to its critical value the critical 228 00:27:55.200 --> 00:28:02.160 value is t equals two so as we said before our function g of t would be zero all the way up until 229 00:28:02.160 --> 00:28:09.760 that critical value it'll then jump to take this value to take the value five when t equals two 230 00:28:09.760 --> 00:28:16.080 and it'll just take the the form of the function two t plus one for all t 231 00:28:18.000 --> 00:28:23.920 greater than two forever and ever so that's what it looks like the red graph here it goes along here 232 00:28:23.920 --> 00:28:30.240 at zero suddenly jumps take the value five and then it just behaves as the function f of t 233 00:28:30.960 --> 00:28:38.960 equals two t plus one okay so and here we are here's just another slightly more detailed plot of that 234 00:28:40.640 --> 00:28:44.720 okay so let's have a look at another of these type one functions 235 00:28:47.760 --> 00:28:56.400 in this case we've got sine t multiplied by a unit step function whose critical value is two pi 236 00:28:56.960 --> 00:29:02.160 now pi is just a number so two pi is just a number so that we need we don't need to let that 237 00:29:02.160 --> 00:29:10.080 concern us so I can draw the graph of sine t just to illustrate what's going on here so there's the 238 00:29:10.080 --> 00:29:18.160 graph of sine t it's you know it's it's periodic and it oscillates between minus one and one 239 00:29:18.160 --> 00:29:26.240 forever and ever so that's the graph of sine t shown in red here now um what do we know about 240 00:29:26.400 --> 00:29:34.320 our step function we know that our step function has the critical value two pi so you multiply sine 241 00:29:35.040 --> 00:29:41.600 t by a unit step function whose critical value is two pi the effect that that has is 242 00:29:41.600 --> 00:29:50.080 orbital two pi this function our unit step function is equal to zero and therefore our whole function 243 00:29:50.080 --> 00:29:57.440 g of t is equal to zero so that's what you can see here in blue the function g of t takes the value 244 00:29:57.440 --> 00:30:05.280 zero all the way up to two pi and then suddenly the unit step function gets switched on and takes the 245 00:30:05.280 --> 00:30:14.320 value one so overall function g of t will just be sine t times one or just sine t so it's basically 246 00:30:14.320 --> 00:30:23.840 the the red graph comes into play then so it's just the graph of sine t from two pi onwards for 247 00:30:23.840 --> 00:30:32.160 all values of t so the graph would keep going to the right so that's really all that's that's to it 248 00:30:32.160 --> 00:30:40.560 that's all that's to it so basically um type one functions as we said a few slides ago I just go back 249 00:30:40.560 --> 00:30:50.240 to find the slides basically it enables us to kill off a function up to some critical value and the 250 00:30:50.240 --> 00:30:57.600 critical value is defined by the unit step function so in other words if we want to set a function f 251 00:30:57.600 --> 00:31:05.440 of t equal to zero up until some specific some specific value we just set the critical value 252 00:31:06.000 --> 00:31:13.520 in their step function equal to that specific value so our function f of t will start at t equals a 253 00:31:13.520 --> 00:31:20.800 and it will continue like that forever and ever we now move on to look at our type two functions 254 00:31:20.800 --> 00:31:28.880 and our type two functions I've referred to them as mathematical window so as I said in this case 255 00:31:28.880 --> 00:31:35.040 we've got a function f of t multiplying the difference of two unit step functions and what 256 00:31:35.040 --> 00:31:45.840 actually happens here is this forms as I've said a window which is zero to the left of a 257 00:31:48.320 --> 00:31:55.280 it'll take the value one between a and b and then to the right of b it'll take the value zero 258 00:31:56.000 --> 00:32:05.440 so the outside function f of t will only exist between a and b if you mark my f of t times one 259 00:32:05.440 --> 00:32:12.160 between a and b so let's have a look at some examples of this so here's our function g of t 260 00:32:12.160 --> 00:32:19.920 and that's a product of a constant function two and this window function here u of t minus four 261 00:32:19.920 --> 00:32:29.040 minus ut minus ut minus eight so if we were going to sketch this what we would do is we would look 262 00:32:29.040 --> 00:32:37.040 at our critical values four and eight so we mark these as four and eight now we know that what's 263 00:32:37.040 --> 00:32:43.120 going to happen here is that everything is zero up until the first critical value and everything is 264 00:32:43.120 --> 00:32:53.040 zero from the second critical value onwards and inside the window will basically have the plot 265 00:32:53.040 --> 00:32:59.600 of whatever the multiplying function is and that in this case the constant function two so inside 266 00:32:59.600 --> 00:33:10.000 the window this function g of t takes the value two first just going to be two times one and that's 267 00:33:10.000 --> 00:33:15.840 what our function looks like as shown in this diagram down here so let's just have a little 268 00:33:15.840 --> 00:33:27.200 bit more explanation of that so let's go let's go back here so here is our g of t as given in this 269 00:33:29.280 --> 00:33:35.840 expression here so i've actually got it here so g of t is given here i've actually opened out the 270 00:33:35.840 --> 00:33:44.400 brackets and i'm going to look at these as two separate functions two ut minus four critical 271 00:33:44.400 --> 00:33:50.960 value four unit it's going to be a step function whose amplitude is two and will hold the value two 272 00:33:51.600 --> 00:34:01.040 forever beyond t equals four similarly this one this one here is a unit step function of critical 273 00:34:01.040 --> 00:34:06.960 value eight and it's been multiplied by two so we're going to get this function here zero all the 274 00:34:06.960 --> 00:34:12.400 way up until t equals eight and then it suddenly gets switched on and takes the value two and holds 275 00:34:12.400 --> 00:34:21.120 that value forever and ever so what we're doing here is we're doing red function minus blue function 276 00:34:21.120 --> 00:34:27.200 so what happens here if we do that well let's have a look the red function between zero and four 277 00:34:27.200 --> 00:34:32.800 is equal to zero so that's going to be zero all the way along there minus zero the blue function 278 00:34:32.800 --> 00:34:39.040 is also zero over that interval so our resulting function our resulting function will also have 279 00:34:39.040 --> 00:34:47.440 the value zero over that interval moving on let's see what happens now as we travel from four 280 00:34:48.640 --> 00:34:53.200 up to eight which is the critical value of the second function well the red function 281 00:34:54.000 --> 00:35:02.800 here my top function has the value uh two my blue function the other hand has the value zero so all 282 00:35:02.800 --> 00:35:10.560 the way between four and eight we're going to have two minus zero which is equal to two so our g of t 283 00:35:11.280 --> 00:35:18.960 our function will have the value two over that interval and then let's see what happens from 284 00:35:18.960 --> 00:35:28.480 eight onwards so from eight onwards both functions have the value two and my overall function g of t 285 00:35:28.480 --> 00:35:34.720 will just be two minus two all the way along here forever and ever and that's just zero 286 00:35:34.720 --> 00:35:41.440 forever and ever so you can see what the function g of t looks like because sometimes called it's a 287 00:35:41.440 --> 00:35:48.000 hat function so it's on a rectangular pulse so it's got the value zero all the way up to four it's 288 00:35:48.000 --> 00:35:57.120 suddenly switches on and it holds the value two up until t equals eight when it then returns to zero 289 00:35:57.120 --> 00:36:07.520 and it keeps the value zero forever and ever okay so let's now have a look at yet another example 290 00:36:07.520 --> 00:36:15.760 of the type two so we've been asked to plot the function three of ut minus four minus ut minus two 291 00:36:15.760 --> 00:36:20.480 minus ut minus four so basically we know this is going to be a mathematical window 292 00:36:22.000 --> 00:36:28.320 and we know it's only defined between two and four so the left and right values of our window are 293 00:36:28.320 --> 00:36:35.920 defined as two and four so our overall function g of t will take the value zero between zero and two 294 00:36:35.920 --> 00:36:42.320 and between four and infinity forever and ever this you can see this could actually stretch all the 295 00:36:42.320 --> 00:36:48.400 way down to minus infinity but we're only considering positive values of t so critical 296 00:36:48.400 --> 00:36:56.720 values are two and four outside of these the function is equal to zero or g of t is equal to zero 297 00:36:57.680 --> 00:37:04.400 inside the window here we've got the text of value one and we're multiplying that by three 298 00:37:05.040 --> 00:37:12.240 so what we get is this constant line here between two and four so that's what our 299 00:37:14.400 --> 00:37:21.600 hat function looks like a window function looks like and mathematically we would write that as g 300 00:37:21.600 --> 00:37:29.520 of t is equal to zero for zero less than t is less than two it's equal to three for t less than 301 00:37:29.520 --> 00:37:35.920 for two is less than t is less than four it's over that interval and it's equal to zero for t 302 00:37:36.560 --> 00:37:43.680 strictly greater than four and that's over the interval between four and infinity so that's what 303 00:37:43.680 --> 00:37:54.880 the function looks like okay so yet another example slightly more difficult this one because it's no 304 00:37:54.880 --> 00:37:59.440 longer a constant function we're multiplying by but this shouldn't be too difficult it's just a linear 305 00:37:59.440 --> 00:38:04.880 function so let's have a look at sketching this so first of all what we want to do is we want to 306 00:38:04.880 --> 00:38:10.960 identify the critical values that will define our mathematical window the critical values are three 307 00:38:10.960 --> 00:38:19.680 and five so I mark them down on my plot here I know that outside of these values that's less than 308 00:38:19.680 --> 00:38:27.600 from zero to three and from five out to infinity the function g of t takes the value zero and that's 309 00:38:27.600 --> 00:38:34.560 what I've done here now what I know is going to happen is that the multiplying function three t minus 310 00:38:34.560 --> 00:38:40.480 one that's going to exist inside the window that's going to exist inside the window because this 311 00:38:40.480 --> 00:38:47.920 put in here in the square brackets that's equal to one inside my window here so it's going to be three 312 00:38:47.920 --> 00:38:53.280 t minus one times one which is just three t minus one so what I need to do is I need to be able to 313 00:38:53.280 --> 00:39:00.960 sketch the graph of three t minus one inside this window and the way I'll choose to do it is I'll 314 00:39:02.160 --> 00:39:07.920 get the coordinates of two points one at the left hand of the left hand end of the window 315 00:39:07.920 --> 00:39:14.240 and the other one at the right hand end of the window so in other words when t is equal to three 316 00:39:14.240 --> 00:39:21.280 this is to get the left hand value when t is equal to three I want f of three where f of t is just 317 00:39:21.920 --> 00:39:29.040 three t minus one so t is equal to three I get three threes and nine minus one is eight so my 318 00:39:29.040 --> 00:39:36.480 first coordinate pair are three eight so that is the left hand end of my window and that's the value 319 00:39:36.480 --> 00:39:45.840 of my function g of t at the right hand end t is equal to five so I substitute that in here into f of t 320 00:39:45.840 --> 00:39:54.080 so I get 15 minus one is 14 and that will give me the coordinate pair of five 14 at the right hand 321 00:39:54.080 --> 00:40:03.040 end of my window and that will give me my straight line segment representing f of t is three t minus 322 00:40:03.040 --> 00:40:10.800 one and therefore I've got the overall graph of g of t to the left of three it's got the value zero 323 00:40:10.800 --> 00:40:18.000 inside the window it's equal to three t minus one and to the right of five it's equal to zero so there 324 00:40:18.000 --> 00:40:27.200 is my windowed function my mathematical window this function three t minus one only exists inside 325 00:40:27.280 --> 00:40:38.240 that window another example very similar we've got a mathematical window multiplied by a linear 326 00:40:38.240 --> 00:40:45.120 function minus two t plus eight the critical values of the window are one and two so I mark them down 327 00:40:45.120 --> 00:40:52.400 on my diagram I know that outside of that window my function g of t is going to take the value zero 328 00:40:52.400 --> 00:40:58.960 so that's there between zero and one it's got the value zero and from two to infinity it's got the value 329 00:40:58.960 --> 00:41:08.480 zero as well so what I need to do now is I need to plot the what's happening inside the window 330 00:41:08.480 --> 00:41:15.440 well we know that this window here this function here ut minus one minus ut minus two has the value 331 00:41:15.440 --> 00:41:21.520 one inside the window so it's basically going to be the graph of minus two t plus eight inside the 332 00:41:21.520 --> 00:41:29.040 window the coefficient of t is negative so I know that the graph is going to be falling over that 333 00:41:29.040 --> 00:41:37.040 window so all I need to do is get two points on this line here to sketch its graph and again 334 00:41:37.040 --> 00:41:44.960 I choose the left and right values right hand side values that define my window so at t equals one 335 00:41:44.960 --> 00:41:52.320 which comes from the critical value here my function f of t will be minus two times one plus 336 00:41:52.320 --> 00:41:59.360 eight or in other words it's equal to six so i've got my coordinate pair one six at the left hand end 337 00:41:59.360 --> 00:42:07.120 of the window at the right hand end of the window the critical value was t equals two so I plug that 338 00:42:07.120 --> 00:42:13.920 in and I calculate the value of f of t it'll be minus two times two plus eight which is just four 339 00:42:13.920 --> 00:42:20.640 and that gives me my second coordinate pair two four and I can draw the straight line segment 340 00:42:21.520 --> 00:42:32.800 like so and that gives me my overall function g of t it's zero between um zero and one that's what 341 00:42:32.800 --> 00:42:39.760 I've got here for zero for t less than one it takes a form of the graph minus two t plus eight in 342 00:42:39.760 --> 00:42:45.520 other words it's just equal to f of t inside the window between one and two and then for t 343 00:42:45.520 --> 00:42:52.720 bigger than two it takes the value zero and you can see all of that given mathematically over here 344 00:42:54.480 --> 00:43:00.880 okay yet another function slightly different looking but we'll see that it actually boils 345 00:43:00.880 --> 00:43:06.880 down to the same thing as we've been doing in the last two or three examples so sketch the graph 346 00:43:06.880 --> 00:43:14.000 of the function g of t is equal to three minus t so that's going to be my f of t and i could one minus 347 00:43:14.000 --> 00:43:22.320 ut minus three now this might not look like a unit step function but we did see this earlier on 348 00:43:22.880 --> 00:43:30.160 we said that uh if we take the critical value of the step function as zero if we take our capital 349 00:43:30.160 --> 00:43:37.040 t is equal to zero what we have we have u of t minus zero so our step function gets switched on 350 00:43:37.040 --> 00:43:43.120 at t equals zero and it takes the value one from then onwards forever and ever and you can see that's 351 00:43:43.120 --> 00:43:47.920 what's happening here we know it's zero all the way up to t equals zero it suddenly jumps 352 00:43:48.720 --> 00:43:54.400 and gets switched on so that's a dotted line there all the way up to one and it just holds the value 353 00:43:54.480 --> 00:44:02.480 of one forever and ever so i can replace the one here with u of t minus zero and that now looks like 354 00:44:02.480 --> 00:44:10.720 my familiar mathematical window u of t minus a minus u of t minus b so i can now identify 355 00:44:10.720 --> 00:44:16.880 the critical values that will define the left and right boundaries of my mathematical window 356 00:44:16.880 --> 00:44:22.960 the first one is the left hand side so my critical value is zero so that'd be the 357 00:44:23.520 --> 00:44:30.800 boundary of my uh window on the left hand side and on the right hand side the boundary of my window 358 00:44:30.800 --> 00:44:39.440 is three so outside of that to the left of zero well my function g of t would be zero 359 00:44:39.440 --> 00:44:48.480 and to the right of three my function g of t will also be zero inside the window my function g of t 360 00:44:48.480 --> 00:44:55.040 would just take the value of this function which we i think called f of t earlier on 361 00:44:56.640 --> 00:45:03.520 okay so what does that require us to do well again this is a straight line segment it's a linear 362 00:45:03.520 --> 00:45:09.440 function t to the power one so i need to draw the graph of three minus t inside the window 363 00:45:09.440 --> 00:45:17.200 and once again i'll do that by just selecting the critical values the the the the edges the 364 00:45:17.440 --> 00:45:24.000 boundaries of the window that's t equals zero from here and t equals three from here 365 00:45:25.360 --> 00:45:32.320 and i'll plug these in so when t is equal to zero my function f of t would be three minus zero is 366 00:45:32.320 --> 00:45:40.080 just three so i get my first coordinate pair as zero three there we are here and at the other end 367 00:45:40.080 --> 00:45:47.760 t is equal to three my function f of t will be three minus three which is zero so my uh 368 00:45:47.760 --> 00:45:53.200 coordinate pair will be three zero and there it is so i've got two points on the line 369 00:45:54.240 --> 00:45:58.960 and i could see my line inside the window i know it's going to have a negative gradient because 370 00:45:58.960 --> 00:46:04.960 it's minus one t so it's going to be falling from left to right and there it is that's what it 371 00:46:04.960 --> 00:46:13.440 that's what it looks like so that is the graph of my overall function g of t it's zero all the way 372 00:46:13.440 --> 00:46:22.160 up until zero nothing's happening it suddenly switches on and over the window between zero and 373 00:46:22.160 --> 00:46:30.480 three it takes the value defined by the function three minus t and then it's it then just returns 374 00:46:30.480 --> 00:46:38.720 to zero outside the left hand boundary obtained from the critical value in here the left hand 375 00:46:38.720 --> 00:46:47.360 boundary of my window so that's another example done i've got i've got another one here slightly 376 00:46:47.920 --> 00:46:55.920 slightly trickier maybe what have i got here well i've actually got uh there's my mathematical 377 00:46:55.920 --> 00:47:02.080 window critical values are two and four so i can easily mark my critical values and my overall 378 00:47:02.080 --> 00:47:09.600 function g of t would be zero to the left of two which was my first critical value and be zero to 379 00:47:09.600 --> 00:47:16.080 the right of four which was my second critical value obtained just by inspecting the mathematical 380 00:47:16.080 --> 00:47:24.320 window now the function that i'm going to multiply by is actually product of two linear functions 381 00:47:24.320 --> 00:47:28.640 i'm going to get a quadratic function and that's what i've got here that's what i'm showing you here 382 00:47:29.360 --> 00:47:37.200 now uh we know a little bit about quadratic so we can get we can plot this we can plot this function 383 00:47:37.920 --> 00:47:45.520 um so what happens here then so uh we've got a negative coefficient for the t squared term so 384 00:47:45.520 --> 00:47:55.440 that means i just get my tablet that means that our graph the quadratic function is going to have 385 00:47:56.560 --> 00:48:04.160 this shape some people say it's a sad face graph so that's going to be the shape of our graph now 386 00:48:05.360 --> 00:48:11.600 what else can we find out well we can certainly identify where it cuts the x-axis because or the 387 00:48:11.600 --> 00:48:20.560 t-axis sorry of course we've got two minus t into t minus four so if it cuts the t-axis 388 00:48:21.520 --> 00:48:29.680 then this would be equal to zero so that'll happen when t is equal to two and when t is equal to four 389 00:48:29.680 --> 00:48:36.960 and that's these two values marked down here okay now um 390 00:48:37.120 --> 00:48:46.160 um what else can we determine well we can certainly because we know the quadratic graph we know it's 391 00:48:46.160 --> 00:48:54.240 going to be symmetric about this midpoint here if we draw a vertical line through this peak here 392 00:48:54.240 --> 00:49:04.240 through the this the apex through the the maximum of my quadratic function if i draw a vertical line 393 00:49:04.240 --> 00:49:10.400 through it here i know the graph is going to be symmetric about this vertical line the left hand 394 00:49:10.400 --> 00:49:17.520 boudre here is two the right hand boudre is four so that point in here must be equal to three 395 00:49:18.320 --> 00:49:25.120 and if i know that t is equal to three i can substitute it into my equation so i get two minus 396 00:49:25.120 --> 00:49:34.720 three times three minus four in other words minus one times minus one which is equal to one and that 397 00:49:34.720 --> 00:49:46.960 will give me the amplitude or the other sorry or the the value of the function at its maximum at its 398 00:49:46.960 --> 00:49:56.080 peak at its turning point if you like so i've been able to sketch the graph um and we know it only 399 00:49:56.080 --> 00:50:04.320 exists inside that window so my overall function g of t then it's zero all the way up until the first 400 00:50:04.320 --> 00:50:10.320 critical value of t equals two it's zero from the critical value of four onwards forever and ever 401 00:50:10.320 --> 00:50:22.160 and inside this window it it's a quadratic function and i've been able to plot that by simply i identified 402 00:50:22.160 --> 00:50:29.600 where it cut the t axis by just setting two minus t into t minus four equals zero so i found that it 403 00:50:29.600 --> 00:50:37.040 cut the t axis at t equals two and t equals four i know that the quadratic it'll be it'll be um 404 00:50:37.760 --> 00:50:46.000 it'll be symmetric about the line through the maximum and that's it there that occurs at t equals 405 00:50:46.000 --> 00:50:51.120 three and i was able to calculate the value of the function at t equals three i found it's equal to 406 00:50:51.120 --> 00:50:59.280 one so i was able to just sketch my graph in there so that's what it looks like okay so um we're 407 00:51:00.240 --> 00:51:02.000 we're going to finish off now by 408 00:51:04.640 --> 00:51:09.600 looking at how to determine the equations of functions from the graphs in the previous questions 409 00:51:09.600 --> 00:51:14.000 we've been given the functions and we've been asked to sketch the graphs we're now going to go the 410 00:51:14.000 --> 00:51:20.160 other way we've got the graphs and we want to determine the functions that gave us these graphs so 411 00:51:20.320 --> 00:51:28.960 here's my graph i want the function of that that that represents that graph so um i've got 412 00:51:29.680 --> 00:51:35.200 one two three critical values i can see there's critical values here because things are changing 413 00:51:35.200 --> 00:51:40.640 at the critical values critical values represent where the function changes so i've got five 10 414 00:51:40.640 --> 00:51:49.360 and 15 of these three critical values so between zero and five okay between zero and five this 415 00:51:49.360 --> 00:51:55.840 function is equal to zero that's what it says between five and ten it's got the value two 416 00:51:55.840 --> 00:52:03.280 that's what i'm saying here between 10 and 15 it's got the value minus one that's this says 417 00:52:03.840 --> 00:52:12.960 and from 15 onwards it's got the value zero and that's what that condition says here so as i said 418 00:52:13.040 --> 00:52:19.040 i've identified the critical values five 10 and 15 so that's going to give rise to three 419 00:52:19.040 --> 00:52:29.200 unit step functions ut minus five ut minus 10 and ut minus 15 so obviously between zero and five 420 00:52:29.200 --> 00:52:35.360 nothing happens so my overall function g of t will have the value zero on this interval 421 00:52:35.360 --> 00:52:44.800 and that's what i say up here okay so we switch on at t equals five and the function g of t steps up 422 00:52:45.840 --> 00:52:55.680 two units okay so that's that's what's happening here so to represent that i'll have 423 00:52:56.400 --> 00:53:04.400 two ut minus five now it'll hold that value of two all the way up to the next critical value 424 00:53:04.400 --> 00:53:12.640 and the next critical value occurs at t equals 10 and as you can see at t equals 10 there's a drop 425 00:53:12.640 --> 00:53:22.880 of three units so we must have minus three times ut minus 10 because this is when the next this is 426 00:53:22.880 --> 00:53:30.960 when the next change occurs at t equals 10 so we drop down from two to minus one that's a drop of 427 00:53:31.040 --> 00:53:40.160 three units okay so we have minus three ut minus 10 and then finally when we hit the next critical 428 00:53:40.160 --> 00:53:47.280 value at t equals 15 we can see that our function g of t jumps up one unit and takes the value zero 429 00:53:47.280 --> 00:53:53.760 and it holds that value forever and ever and that's what i've said here g of t is equal to zero for t 430 00:53:53.760 --> 00:54:04.320 greater than 15 so in other words we're going to add the one time we're going to add one times 431 00:54:04.320 --> 00:54:12.960 the the unit's def function whose critical value is t equals 15 so you'll have one ut minus 15 and 432 00:54:12.960 --> 00:54:21.040 that is my overall function i should have probably called it g of t down here not uh not f of t and 433 00:54:21.040 --> 00:54:27.440 just correct that that's that's our function g of t i've called it g here and g here yes and g here 434 00:54:27.440 --> 00:54:36.720 that should be of course g of t at the foot of the slide okay so let's uh do another one of these 435 00:54:36.720 --> 00:54:45.680 just slightly uh different where we no longer got constant functions over each interval we've now got 436 00:54:46.320 --> 00:54:52.000 still a linear this is a linear function with a negative gradient so let's see what we can do 437 00:54:52.000 --> 00:54:57.680 with this thing so well we know it's a type two function we know this is a type two function because 438 00:54:58.880 --> 00:55:05.120 we've got we've got two critical values one and four the function is zero outside of these values 439 00:55:05.120 --> 00:55:11.920 and it only exists inside between one and four so we know that this this a type two function 440 00:55:12.240 --> 00:55:21.120 in involving a mathematical window the mathematical window will be between one and four so uh our 441 00:55:21.120 --> 00:55:27.760 overall function g of t will take the form of f of t which will be the function inside the window 442 00:55:28.320 --> 00:55:36.240 times our mathematical window u of t minus a minus u of t minus b where a is the left hand 443 00:55:36.240 --> 00:55:41.600 boundary of the window and b is the right hand boundary of the window now i can read these off 444 00:55:42.160 --> 00:55:47.920 so i can certainly write down that my function g of t is some function f of t 445 00:55:49.360 --> 00:55:56.080 given by this slanted line the sloping line here times my mathematical window who's got 446 00:55:56.080 --> 00:56:03.200 that's got critical values one and four so it's ut minus one minus ut minus four so that just leaves 447 00:56:03.200 --> 00:56:12.880 me to find what my function f of t actually is so as i say here uh what happens is by the 448 00:56:12.880 --> 00:56:20.560 mathematical window we switch on f of t at equals one and we switch it off again at t equals four 449 00:56:20.560 --> 00:56:27.920 so what i now need to do is if i can identify the coordinates of the points at t equals one 450 00:56:28.000 --> 00:56:35.520 t equals four i will then have two points and that'll be enough to allow me to determine 451 00:56:36.080 --> 00:56:42.320 the equation of the straight line segment inside the window so let's have a look from our diagram 452 00:56:42.320 --> 00:56:51.680 we can read that at the left hand end of the window the line passes through the point one four 453 00:56:51.680 --> 00:56:57.760 and at the right hand end of the window the point passes through the line passes through the point 454 00:56:57.920 --> 00:57:05.680 four one okay so i can now go ahead and get the equation of this line once i know two points 455 00:57:05.680 --> 00:57:12.720 on the line there they are given here okay so i'm going to use the standard form here of straight 456 00:57:12.720 --> 00:57:21.040 line equation y equals mt plus c m is the gradient it's the change in the y direction divided by the 457 00:57:21.040 --> 00:57:25.760 change in the t direction change in the vertical direction divided by the change in the horizontal 458 00:57:25.760 --> 00:57:34.960 direction the change in the vertical direction well i would go from one up to four so i do one 459 00:57:34.960 --> 00:57:45.120 minus four and the corresponding change in the t direction would be four minus one okay so you 460 00:57:45.120 --> 00:57:52.320 know if you if you call this y the vertical one you can think of it as you can think of this one 461 00:57:52.320 --> 00:58:03.760 as how it coordinates t1 y1 and t2 y2 and so the gradient will be y2 minus y1 so it's one minus four 462 00:58:03.760 --> 00:58:11.840 divided by t2 minus t1 which is four minus one and if you work that out you find that the gradient 463 00:58:11.840 --> 00:58:18.800 is minus one and that's really exactly what you would expect because for moving along three units 464 00:58:18.800 --> 00:58:28.880 in the horizontal what's happened you've fallen by three units in the vertical direction so the 465 00:58:28.880 --> 00:58:36.640 vertical change is minus three because it's falling divided by the horizontal change which is three 466 00:58:36.640 --> 00:58:42.720 giving you a slope of minus one and then i can use either point in my equation so if i use the 467 00:58:42.720 --> 00:58:51.120 point four one and the equation y equals mt plus c i know what m is we just found it it's minus one 468 00:58:52.240 --> 00:59:00.320 okay i know what t is in this case it's equal to four and what i want to find is the value of c 469 00:59:00.960 --> 00:59:09.280 if i work all of this out i can find that one is equal to minus four plus c or c is equal to five 470 00:59:09.280 --> 00:59:18.400 so the overall equation of my function inside the window is y is equal to f of t remember this is f of t 471 00:59:19.120 --> 00:59:28.240 f of t is equal to minus t plus five and so my overall function like i've written green g of t 472 00:59:29.040 --> 00:59:37.120 zero for t less than one it's minus t plus five for one is less than t is less than four 473 00:59:37.840 --> 00:59:47.360 and it's zero for t greater than four so from the plot of my function i was able to identify 474 00:59:48.080 --> 00:59:56.160 that this required this was a mathematical window and i identified the critical values 475 00:59:56.960 --> 01:00:03.520 appearing in the mathematical window one and four i set my function g of t to zero outside 476 01:00:03.520 --> 01:00:09.600 these values as for all t less than one and all t greater than four and that left me with the task 477 01:00:09.600 --> 01:00:18.400 of calculating the equation of the straight line segment inside the window and to do that i was able 478 01:00:18.400 --> 01:00:26.480 just to read off the coordinate page and use these two points to calculate the equation 479 01:00:26.480 --> 01:00:31.760 of the line inside the windows giving me minus t plus five 480 01:00:34.960 --> 01:00:42.400 i mean you can check of course quite easily because you can check either end of the window my equation 481 01:00:42.400 --> 01:00:49.840 f of t is minus t plus five when t is equal to one well f of t should be minus one plus five 482 01:00:49.840 --> 01:00:56.720 should be equal to four yes it does when t is equal to four f of t should be minus four plus five 483 01:00:56.720 --> 01:01:01.040 which is equal to one and yes it does so everything is fine everything checks out 484 01:01:02.880 --> 01:01:09.600 okay so here's another example again we're gonna what looks like a mathematical window 485 01:01:09.600 --> 01:01:15.440 i'm going to straight line segment inside that window we've got a linear function inside the 486 01:01:15.440 --> 01:01:22.480 window so first thing we do is we identify the critical values which is a mathematical window 487 01:01:22.480 --> 01:01:30.400 so we identify the critical values as t equals one and t equals two so we know that our overall 488 01:01:30.400 --> 01:01:39.200 function is going to be zero outside the window to the left for t less than one and t greater 489 01:01:39.280 --> 01:01:48.800 than two so um what's going on then so uh the function will switch on a t equals one and off a 490 01:01:48.800 --> 01:01:54.960 t equals two so what we need to do is we need to find the equation of this this line here okay so 491 01:01:54.960 --> 01:01:59.840 let's just find the coordinates of two points on the line so the first one is easy from the left 492 01:01:59.840 --> 01:02:06.240 hand end of the window it's t is equal to one and you can read off that there uh 493 01:02:06.880 --> 01:02:14.240 the vector value is zero so one zero is the first coordinate coordinate that we need the other one 494 01:02:14.240 --> 01:02:21.200 at the other end of the window t is equal to two and f the f t or y whatever you want to call this 495 01:02:21.200 --> 01:02:26.640 equal to three so my second pair is two three so what i want to do now is i want to calculate the 496 01:02:26.640 --> 01:02:33.120 slope of this straight line segment it's a positive slope so i should get a positive value here when 497 01:02:33.760 --> 01:02:40.880 i calculate m so it's vertical change divided by horizontal change so i'll just do three minus 498 01:02:40.880 --> 01:02:47.040 zero just using the coordinate pair three minus zero divided by two minus one so just three over one 499 01:02:47.040 --> 01:02:54.320 which is three so uh this one i've just i've just used this different formulation of a straight 500 01:02:54.320 --> 01:03:02.480 line equation y minus b equals m t minus a where a b are the coordinates of either uh either paired 501 01:03:02.480 --> 01:03:07.040 you can choose either pair to use so so let's have a look at this which one i'm going to use the 502 01:03:07.040 --> 01:03:15.120 easier one i'm going to use one zero so i get uh so that's by a that's my b so i'll have y minus zero 503 01:03:15.120 --> 01:03:22.320 is equal to three because that's gradient t minus one because my a value is one by choosing this 504 01:03:23.360 --> 01:03:32.320 coordinate pair to use so i can simplify that to find that my equation is three t minus three 505 01:03:32.480 --> 01:03:40.080 okay so that's the function that's going to multiply the mathematical window whose critical values 506 01:03:40.080 --> 01:03:48.240 where t equals one and t equals two and there's my solution at the foot of the slide again you know 507 01:03:48.240 --> 01:03:56.160 i could check if this was correct at the left hand end of the window t is equal to one so y would be 508 01:03:56.160 --> 01:04:04.480 three minus three which is zero that's correct and at the other end uh t is equal to t is equal to 509 01:04:04.480 --> 01:04:11.760 two so i would have three twos or six minus three give me the value three so that's absolutely 510 01:04:11.760 --> 01:04:20.640 correct um so that's fine so that is the graph of my overall function which i've called f of t this time 511 01:04:20.960 --> 01:04:33.200 um and i've called h of t been this function here okay right so uh maybe a couple more so this one 512 01:04:33.200 --> 01:04:39.360 comes from Mobius to Tem the function f of t represent the graph below okay so just looking 513 01:04:39.360 --> 01:04:44.720 at this the critical values are two and three so i know i'm going to have a mathematical window uh 514 01:04:44.720 --> 01:04:54.320 ut minus three minus ut u of t minus two minus u of t minus three so the function inside here 515 01:04:54.320 --> 01:05:00.320 switches on a t equals two and it switches off a t equals three so what we want now is we want the 516 01:05:00.320 --> 01:05:08.960 equation of this straight line segment that lies inside the window okay so what do we do well we 517 01:05:08.960 --> 01:05:18.160 need to read off some coordinate points and i do that by getting the left hand coordinate pair will 518 01:05:18.160 --> 01:05:26.560 be two four and at the right hand end of the window we'll have three minus two so there we are so first 519 01:05:26.560 --> 01:05:33.920 of all let's calculate the gradient so uh change in y divide by change in t so we'll get minus two 520 01:05:33.920 --> 01:05:40.880 minus four divided by three minus two because minus six over one which is minus six and if i look at 521 01:05:40.880 --> 01:05:50.080 my diagram i can see that that is correct because um my horizontal change i go from two to three so 522 01:05:50.080 --> 01:05:56.560 i'm changing one in the horizontal direction there you are and in the vertical direction i've dropped 523 01:05:56.560 --> 01:06:04.640 from four down to two so that's a change of six and it's falling so it's a negative change so we get 524 01:06:05.360 --> 01:06:12.480 minus six over one vertical change or horizontal change give me minus six so you know you can either 525 01:06:12.480 --> 01:06:21.040 do it like this or you could actually get the slope just by reading directly from the graph so here 526 01:06:21.120 --> 01:06:30.960 again i'm going to use my equation y minus b equals mt minus a i know uh i know a i know b and i know 527 01:06:30.960 --> 01:06:37.360 m so it's just a matter of plugging these in well i know a and b i've chosen to be the point two four 528 01:06:38.080 --> 01:06:44.400 it's just easy to work with one with a negative in it so my a is two my b is four and my m is 529 01:06:44.400 --> 01:06:52.880 minus six so i put all of these in i end up with this equation here and that will tell me that my 530 01:06:52.880 --> 01:07:00.320 overall function is given by this equation here there's my mathematical window and it's multiplied 531 01:07:00.320 --> 01:07:09.920 by minus six t plus 16 and that ensures that it only exists inside the window so again we can 532 01:07:10.640 --> 01:07:16.560 check your values at either end of the window to see if this function is correct so on the left 533 01:07:16.560 --> 01:07:22.800 hand side it's equal to two so i minus six times two is minus 12 plus 16 is four that's correct that's 534 01:07:22.800 --> 01:07:28.640 the value there and the right hand end of the window where t is equal to three and i have minus six times 535 01:07:28.640 --> 01:07:35.760 three is minus 18 plus 16 is minus two and that's exactly the value uh the function the diagram 536 01:07:36.720 --> 01:07:42.560 takes at the value t three takes the value minus two so that's all correct we've been able to check 537 01:07:42.560 --> 01:07:50.000 your answer so one more example then so again i want to determine the function f of t represent 538 01:07:50.000 --> 01:07:59.600 this graph below again clearly it's a mathematical window uh for t less than two the function is 539 01:07:59.600 --> 01:08:07.200 equal to zero and for t greater than four it's equal to zero it only exists between two and four 540 01:08:07.200 --> 01:08:13.600 so that tells me that i should write down the equation for the mathematical window as ut minus 541 01:08:13.600 --> 01:08:22.080 two minus ut minus four and again that leaves me the task of deriving the equation of that 542 01:08:22.160 --> 01:08:29.360 straight line segment that exists inside the window it's uh going to have a positive gradient 543 01:08:29.360 --> 01:08:35.360 because it's climbing you could actually just read the gradient off the diagram here so for two 544 01:08:37.280 --> 01:08:46.800 steps two two steps in the t direction i would climb six in the vertical direction i would go from 545 01:08:46.800 --> 01:08:53.360 minus two up to four over that window so the change in the vertical direction is six and the change in 546 01:08:53.360 --> 01:09:00.640 the horizontal direction is two so the gradient should be three and it is and but you can verify 547 01:09:00.640 --> 01:09:08.720 that by identifying the points two minus two at the left hand side and four four at the right hand 548 01:09:08.720 --> 01:09:15.920 side and calculate the gradient in the standard way horizontal vertical change four minus minus two 549 01:09:15.920 --> 01:09:21.760 divided by horizontal change four minus two and it does give you six or two which is three 550 01:09:24.080 --> 01:09:30.640 which i got just by inspection really off the diagram so either way is perfectly okay and again 551 01:09:30.640 --> 01:09:42.080 i use my equation for for for a straight line y minus b equals mt minus a i can choose either point 552 01:09:42.960 --> 01:09:48.080 and i think it's sensible to choose the one without the negative value so i'm going to choose four four 553 01:09:48.080 --> 01:09:54.560 as my point so a is four b is four i know m is three i can substitute all of that in 554 01:09:55.840 --> 01:10:05.440 and i obtain my equation uh three t minus eight for the function that exists inside the window so 555 01:10:06.080 --> 01:10:14.240 my overall function then my overall function i shown the diagram here will be my window ut minus 556 01:10:14.240 --> 01:10:21.920 two minus ut minus four that ensures that everything outside less than two and greater than four is 557 01:10:21.920 --> 01:10:29.920 equal to zero so if my overall function is equal to zero that's exactly what we see here and my it 558 01:10:29.920 --> 01:10:38.160 ensures that the only thing that survives is inside the window where the where f t will take 559 01:10:38.160 --> 01:10:45.440 the value of this function three t minus eight so let's just check that at the left hand end we've got 560 01:10:47.120 --> 01:10:52.640 t is equal to two so you get six minus eight which is minus two yes that's correct two minus two 561 01:10:52.640 --> 01:10:59.120 lies on the line at the right hand end t is equal to four so three times four minus eight 12 minus 562 01:10:59.120 --> 01:11:05.600 eight is equal to four and as you can see the point four four does indeed lie on the line at the 563 01:11:05.600 --> 01:11:12.720 right hand end of the window so that is correct so uh just to summarize what we've been doing so 564 01:11:12.720 --> 01:11:18.560 in this lecture we've introduced the unit step function and we've looked at graphs of functions 565 01:11:18.560 --> 01:11:24.880 that involve step functions we first of all considered functions involving sums and differences 566 01:11:24.880 --> 01:11:31.760 of step functions and then we moved on to look at graphs of functions that involve different types 567 01:11:31.760 --> 01:11:40.720 of products of step functions with other standard functions in our next lecture we're going to take 568 01:11:40.720 --> 01:11:47.360 a look at calculating Laplace transforms of step functions and we're going to introduce the second 569 01:11:47.360 --> 01:11:54.880 shifting theorem and this will enable us to transform time shifted functions now we need 570 01:11:54.880 --> 01:12:02.240 these results later when we come to solving differential equations that involve step functions 571 01:12:02.240 --> 01:12:13.200 to model applied voltages or forces so very quickly now I'll just say what you should be able to do 572 01:12:13.200 --> 01:12:21.040 you should now be able to tackle mobius questions from number one to number 11 and from the notes 573 01:12:21.040 --> 01:12:29.520 you should be able to attempt these tutorial questions so we can just stop at that we can 574 01:12:29.520 --> 01:12:33.840 leave it at that and I'll stop recording