[speaker001:] Right.... I'll just have a look through what you've been doing
[Graham:] Yeah er there's the syllabus I've got.
[John:] Oh that's
[Graham:] I managed to arrange that.
[John:] that is brilliant that's great. So your objectives that... and you should be able to do these. Right.
[Graham:] Now what's happened here is we we've done some of each of these there's bits that we've missed out though. But anything that we covered basically I've got in them notes there.
[John:] Okay so the problem that you were having was differentiation and integration.
[Graham:] Mhm.
[John:] And of those two which do you think is harder?
[Graham:] Erm that one's a bit of an odd question to be honest with you because I'm finding because I've had an exam in this, that this is the hardest.
[John:] What differentiation?
[Graham:] Because I've spent more time trying it.
[John:] Okay.
[Graham:] But the thing is I've got an integration exam in a few weeks popping up so I'll be looking more towards that now and [LAUGHTER] end up thinking that's the harder. []
[John:] Right well that's that's fine. Most people myself included find that integration is more difficult than
[Graham:] Mhm.
[John:] differentiation. Which do you find easier multiplication or division?
[Graham:] Multiplication.
[John:] Yeah and addition subtraction?
[Graham:] Addition.
[John:] It's it's the natural thing if you like
[Graham:] Mhm.
[John:] it's the going forward way.
[Graham:] Yeah.
[John:] The other way the inverse is well we got no where did we come from?
[Graham:] [LAUGHTER]
[John:] and you have to back track all the way and work out so integration's very very similar now
[Graham:] Mhm.
[John:] is you started off at school with division
[Graham:] Mhm.
[John:] and you'd never done multiplication you'd find it very hard. Erm if you're familiar with multiplication if you know your tables
[Graham:] Yeah.
[John:] erm assume you do so if I if you know what three fours are
[Graham:] Mhm.
[John:] then when I say to you well how many threes are there in twelve you can
[Graham:] immediately spot.
[John:] four threes' d be twelve or three fours' d be twelve and your not actually working it out I mean
[Graham:] Mhm.
[John:] you're you're going back and saying what would I have done to get
[Graham:] Yeah.
[John:] here? Now integration that's probably the best way to look at it.
[Graham:] Mhm.
[John:] Someone gives you with something to integrate and they say I started off I differentiated something
[Graham:] Mhm.
[John:] and that was what I got.
[Graham:] Mhm.
[John:] Yeah how did I what did I start out from? Okay so although you say you've finished your differentiation exam
[Graham:] Yeah.
[John:] erm I'd still like you to do more work on differentiation.
[Graham:] Mhm.
[John:] Because that's the that's the key once you've got that er you look at some for example if, I say integrate X to the sixth.
[Graham:] Mhm.
[John:] Okay what would make of that? What what were your first tell me your first thoughts on it I don't want the answer right away I want your thoughts on it. X to the sixth.
[Graham:] X to the sixth.
[John:] What's your first thought about that?
[Graham:] It's differentiate.
[John:] To integrate it.
[Graham:] Oh to integrate it.
[John:] Right what are your first thoughts about it?
[Graham:] Right now for integration I always raise the power by one and then
[John:] Right.
[Graham:] divide by the new power so
[John:] Okay you do that straight off.
[Graham:] it'd be... that.
[John:] Right and that's it straight off good.
[Graham:] Mhm.
[John:] a lot of people would think, Ah oh X to the sixth well he must have started of with X to the seventh. Well if he had X to the seventh that would have given him seven X to the sixth
[Graham:] Mhm.
[John:] so we better start off with one seventh of that. Erm how about if I say integrate cos X?
[Graham:] Ee
[John:] [LAUGHTER]
[Graham:] Er
[John:] Are you going to write the answer straight down or are you going to tell me what
[Graham:] Cos
[John:] you're thinking about? Well what are you thinking about?
[Graham:] I thinking a table how to [LAUGHTER] I am thinking about the table that is
[John:] Have you ever done anything that would help with this now?
[Graham:] Erm... we've covered it... I've done it before I'm just trying to remember what the answer was. [LAUGHTER]
[John:] That was the answer think of it that way. Think that you were think that you were differentiating something.
[Graham:] So it'd be
[John:] And when you differentiated it you got cos X.
[Graham:] Mhm.
[John:] Think of it that way. So that was the answer and what was the question?
[Graham:] Probably sine X.
[John:] It would be sine X okay erm so differentiate sine X what do you get?
[Graham:] Cos X.
[John:] Okay so that's it. So integral of sine X with respect to X cos X
[Graham:] Mhm.
[John:] with respect to X is going to be in X. Now
[Graham:] Yeah.
[John:] suppose I give you sine X and say integrate that.
[Graham:] Cos X.
[John:] Okay you think that's your first thought
[Graham:] Mhm.
[John:] that's it's cos X. Okay write down cos X and differentiate it and see what you get.
[Graham:] Sine X.
[John:] Do you get sine X or...
[Graham:] Oh it's actually minus sine X.
[John:] Right it's
[Graham:] Yeah.
[John:] minus sine X. Okay so if you started off with cos X differentiated it it would give you minus sine X
[Graham:] Mhm.
[John:] Ah you don't want to finish up with minus sine X you want to finish up with sine X so what are you going to have to start off with?
[Graham:] Minus cos X.
[John:] Right and this is the way most people get into integration.
[Graham:] Mhm.
[John:] there's a lot of backtracking it's not a question of learn every integral there is... and you have them all neatly laid out in your head in a
[Graham:] Mhm.
[John:] big table and you say Oh yes it'll be that erm let's say integrate erm... cos well say cos squared... three X.
[Graham:] Right
[John:] Erm
[Graham:] cos
[John:] [whispering] cos squared three X yeah okay. []... ow don't don't try it straight off as an integral what would you sort of have a split your page and work over here sort of playing
[Graham:] Mhm.
[John:] about differentiating thing what sort of thing did you start off with to finish up with a cos squared three X?
[Graham:] Well for starters you have to have minus sine... for the coses... now the square would be erm cubed over three.
[John:] Ah right well it wouldn't so
[Graham:] Well I'm trying to I'm still thinking integration that's all.
[John:] Don't don't worry yeah don't worry about this erm you're still thinking about integration. Now er there are lots of things going on here.
[Graham:] Mhm.
[John:] People throw all these trig functions at you now that's not a very complicated trig function so far cos squared
[Graham:] Mhm.
[John:] but people chuck all this stuff at you and you, Ooh what's going on? now if you're not very very sure of what you're working with in the first place
[Graham:] Mhm.
[John:] you've got very little chance, so what does cos squared three X mean well let's let's forget about the three X could you tell me what cos squared X means?
[Graham:] Erm I'm not really with you. Erm
[John:] I've never I've never heard about this thing called cos squared X, you explain it to me what do what's it all about? Sounds a load of rubbish to me cos squared X what's it mean?
[Graham:] All that is just a function of X whatever number X is X is for.
[John:] Mm.
[Graham:] So...
[John:] Okay I've I've heard of cos X
[Graham:] Mhm.
[John:] but I don't know what this cos squared X is all about what's that how does that tie up with cos X?
[Graham:] Erm... No.
[John:] Mm okay.
[Graham:] I couldn't say.
[John:] Okay that's fine erm... so you're already... even when you're differentiating
[Graham:] Mhm.
[John:] say differentiate cos squared X you're already trying to do something and you don't know what it is
[Graham:] Mhm.
[John:] you're working with. Now this is very common you don't know what you're working with so you can't really be expected even if you get the answer if
[Graham:] Mhm.
[John:] it's someone's given you a list look it's that learn it.
[Graham:] That tends to be the er
[John:] You still can't understand what you're doing and if you
[Graham:] Mhm.
[John:] don't understand it's very hard to learn. It's very hard to retain it lot's of work we've done on this and if you don't remember if you don't understand you don't remember.
[Graham:] Mhm.
[John:] Erm simple as that if I give you a list of telephone numbers you're not likely to remember them for long, you might be able to
[Graham:] Mhm.
[John:] remember them for a day or two but they'd go cos they don't mean anything.
[Graham:] Yeah.
[John:] So let's have a little look at what cos squared X means erm... that cos squared three X, cam I have given you X times cos squared three X to integrate do you think that would have been easier or harder?
[Graham:] Erm Probably a little bit easier, certainly
[John:] Mm.
[Graham:] easier to explain. [LAUGHTER]
[John:] Okay let's have a look at cos X erm cos X what does that mean?...
[Graham:] It's just a a cosine function of... whatever figure that is whatever figure X represents.
[John:] Erm what does X represent when you're taking the cosine of it?
[Graham:] It's a point on the X axis of a graph....
[John:] Erm okay... so you're drawing the function and X goes from nought to what? to fourteen
[Graham:] Erm
[John:] point two or what?
[Graham:] Do you mean on a co on erm a cosine graph? Well X goes up to three sixty doesn't it.
[John:] So if you're working in degrees you'd go zero degrees to three sixty degrees.
[Graham:] Mhm.
[John:] Right it would... from minus infinity to plus
[Graham:] Mm minus yeah.
[John:] infinity you can measure in degrees or in radians.
[Graham:] Mhm.
[John:] Doesn't matter. So X is an angle.
[Graham:] Yeah.
[John:] Right X is an angle and what's the cosine of the angle so if we we look at it this way... erm... it's supposed to go through there through the origin.
[Graham:] Mhm....
[John:] And that's the angle X have it in degrees if you like
[Graham:] Mhm.
[John:] and let's say it's erm... this particular example we've chosen happens to be sixty degrees.... And this is any point X Y this is the Y that's the X... how would you find the cosine of X on that?...
[Graham:] Well you could work it using X and Y.
[John:] Let's let's get rid of this is this is where a lot of the confusion is
[Graham:] Mhm.
[John:] so let's get rid of some of it and let's make the angle theta.
[Graham:] Mhm.
[John:] Okay and that's the X distance along there
[Graham:] Yeah.
[John:] and that's the Y. So could you find cos theta?
[Graham:] Erm... now let me get m...
[John:] Write it down if you're not sure....
[Graham:] It's going to back this really isn't it.... the cosine is erm... awkward.
[John:] Okay which is which is the easiest one of this lot to work out? We're trying to work out which is the hypotenuse which is
[Graham:] the ninety degree angle.
[John:] So the one opposite the ninety is the hypot
[Graham:] Is the hypotenuse.
[John:] Right which is the next easiest one to work out?
[Graham:] The adjacent.
[John:] Mm maybe okay loads of people find that the one opposite the angle we're interested in
[Graham:] Mhm.
[John:] is the... that's that's the opposite
[Graham:] Mhm.
[John:] especially if you're using theta for the angle... looks a bit like theta. So that's the opposite and this one is the adjacent. Doesn't matter if you remember the adjacent first that's fine. If you can remember
[Graham:] Mhm.
[John:] it doesn't matter how you remember it. So adjacent over hypotenuse.... Adjacent is X... the hypotenuse is?... root X squared plus Y squared. Right that's cos theta.... And we plot along here if we plot Y against theta
[Graham:] Mhm.
[John:] we plot this this horrible lot against theta and we get the usual wiggly
[Graham:] Yeah.
[John:] nice little sine wave cos wave.
[Graham:] Mhm.
[John:] It's a number... sine or cos
[Graham:] Yeah.
[John:] it's a ratio of two lengths you can measure them in centimetres miles whatever you like but it'll
[Graham:] Mhm.
[John:] still come down to a a ratio that has no units it's just a a number. That's what cos is now what does... that's cos X cos theta cos
[Graham:] Mhm.
[John:] Z whatever you like.
[Graham:] Yeah.
[John:] What would cos squared Z... what does that mean? What's the notation in other words?...
[Graham:] It's that ratio
[John:] Mm.
[Graham:] squared.
[John:] Right. What's that
[Graham:] The ratio times itself.
[John:] Right what is the ratio?... The ratio is cos Z.
[Graham:] Mhm.
[John:] That's that's the ratio.
[Graham:] Yeah.
[John:] That's the ratio squared.
[Graham:] Mhm. same thing but
[John:] And that's what it means. Now people start getting very involved with this cos squared and integrating it to being cos cubed.
[Graham:] Mhm.
[John:] Right there's nothing nothing to do with that at all. We we just have this weird notation that we write it like this and we say it like that cos squared Z what we mean
[Graham:] Mhm.
[John:] is cos Z... squared.
[Graham:] Mhm.
[John:] But if you say that especially when you say it quickly it's starting to sound a bit like cos Z squared.
[Graham:] Mhm.
[John:] Right.
[Graham:] Which is a different thing again.
[John:] Which is nothing at all to do with that.
[Graham:] Yeah.
[John:] It's not that.
[Graham:] Cos that'd just be you'd figure for Z squared as opposed to the whole lot.
[John:] Right and then take the cos of it but this
[Graham:] Yeah.
[John:] is take the cos and then square the answer the answer you get.
[Graham:] Mhm.
[John:] So part of trig functions are probably what give most people more trouble that anything else on differentiation and
[Graham:] Mhm.
[John:] integration. Part of it is not really it it's being a bit out of touch with what you're doing. It doesn't sound real. It's a bit like if you like the difference between working with nice constants nice numbers if I'm saying
[Graham:] Mhm.
[John:] what's erm eighteen times three, yeah I can work that out I might
[Graham:] Yeah.
[John:] not know it immediately but if I say, Well what's X plus seven times Y minus five or... it's only the same it's exactly the same rules
[Graham:] Mhm.
[John:] as you'd use for your three times eighteen
[Graham:] Yeah.
[John:] but the concepts aren't familiar and you're starting to feel a little bit sort of not quite sure what you can do with them and what you can't
[Graham:] Mhm.
[John:] the numbers you've had them since you were about this big
[Graham:] Yeah.
[John:] you've grown up with them you know what you can do with numbers
[Graham:] Mhm.
[John:] you can play with them you're you're quite happy, but these other strange objects you don't really know what the rules are. Well they're all numbers they all follow the same rules
[Graham:] Mhm.
[John:] they all behave the same way. And then you move on to functions and they're diff you're quite happy with Y equals X squared three X squared plus two you think
[Graham:] Yeah.
[John:] I can do that no problem. Then they start throwing in things like Y plus cos X ooh and cos X cos squared X. You don't even know what it means.
[Graham:] Mhm.
[John:] So if you're not happy with the the basic objects that you're playing around with, you you've immediately lost most of your confidence
[Graham:] Mhm.
[John:] you don't know what you're doing. You know that you don't know what you're doing.
[Graham:] Yeah.
[John:] Someone's telling you the answers and you're thinking, Well okay it must be right knows what he's doing he's a teacher right okay I'll put it down. But if you know what you're doing
[Graham:] Mhm.
[John:] then you can take it on you can start understanding you can start building up your knowledge base. like
[Graham:] Mhm.
[John:] putting things into little pigeon holes saying, Oh that's just like I did with that other problem. Er I'd use that bit here. Until you get that breakthrough then everything's a one off. Every
[Graham:] Mm.
[John:] problem you do is, Er start right
[Graham:] Mm.
[John:] from scratch I don't know whether I'm allowed to do that, I think I can possibly get away with this. Erm so your confidence goes, your speed at working through it goes, you might find you're going back and doing it again when it was right then first time.
[Graham:] Mhm.
[John:] So you need to know the objects you're dealing with, then differentiate them, then when you've differentiated them when you've differentiated them you can work back and you think, Ah I know how I got that. So let's look at now you've been doing I noticed today you'd been doing differentiating products. Yeah?
[Graham:] Mhm.
[John:] And you've done quotients as well presumably.
[Graham:] Yeah.
[John:] Yeah. Now you said that if I gave you X cos squared three X it might be a bit easier to integrate.
[Graham:] Mm.
[John:] Why did you say that? It's correct.
[Graham:] Because... what I saw there was that... as a single function as a single function.
[John:] Mm....
[Graham:] Now I'm not entirely positive if I'm right but I think that would be easier to do because you could then alter that round to... and then you'd have your three X as a separate thing.
[John:] your times in there.
[Graham:] Yeah.
[John:] What would you do then?
[Graham:] X to the power of one.
[John:] What would do then?... Er does that mean you've integrated?
[Graham:] I'm about I'm midway through it. [LAUGHTER]
[John:] You you so you have done some part of the integration?
[Graham:] Yeah.
[John:] Ah no. Now have you learnt the formula for integrating a product?... some people do.
[Graham:] Probably.
[John:] Oh I I don't usually recommend it some people do and find it easier that way I I would rather they just stuck with the the formula for differentiating a product and
[Graham:] Mhm.
[John:] differentiating a quotient and then recognizing the picture. So that you can work backwards. Let's let's go away from trig functions for a bit.
[Graham:] Mhm.
[John:] Okay we'll find out some general applications and then see if we can bring the trig functions back into them.
[Graham:] Mm.
[John:] okay. So... let's say... three X to the power seventeen.... Mm that's a bit let's make it more awkward. Right three X to the power seventeen plus X squared all to the power five.
[Graham:] Right.
[John:] Now could you differentiate that?... What's your fir just tell me tell me I'm more interested in sort what you're thinking about it what you're feeling about it er what you're thinking of trying and not trying than than... the answer you get. So what are you thinking about that?...
[Graham:] That's what I'm thinking.
[John:] Okay.
[Graham:] That can you understand where I'm going with that.
[John:] Oh I can understand what you why you're thinking that yeah.
[Graham:] Now that's the way that we've been shown to do
[John:] Okay.
[Graham:] things along those lines.
[John:] So what would your an let's let's pick a simpler one and see what your answer is to this. Erm... okay we've got Y equals that... erm follow that one through and actually differentiate it.... Okay we'll make it even even more less messy looking and have Y equals...
[speaker001:] Right.
[John:] How would you do that?
[Graham:] I'm trying to think of it handled the the squared on the outside.
[John:] Mm.
[Graham:] I'm thinking that it may be a function of a function.
[John:] Right good.
[Graham:] It could possibly be that.
[John:] I mean it definitely is isn't it. The way you test that is, Ooh this looks a bit awkward, could I do... Ooh yeah I could that.
[Graham:] Mhm.
[John:] I could do Y equals X squared I could do Y equals U squared.
[Graham:] That's what I was about to head for.
[John:] Okay so
[Graham:] Change U for four X
[John:] Right so
[Graham:] U
[John:] in that one right.... You carry on from there.... that's great yeah that's excellent. Okay on this one erm
[Graham:] Mhm.
[John:] . Now... back to integration.
[Graham:] Mhm.
[John:] If I gave you this to integrate, something like... four X cubed minus X... two times four X cubed minus X times
[Graham:] Mhm.
[John:] twelve X squared.... What sort of what would you to say about it?
[Graham:] Again it's a function of a function.
[John:] Okay so it it was... it was something you differentiated
[Graham:] Mhm.
[John:] and you treated it as a function of a function. And when you treated it as a function of a function you get that that thing inside there was the function
[Graham:] Mhm.
[John:] and what's this what's this bit this twelve
[Graham:] Mhm.
[John:] X squared that it's multiplied by?
[Graham:] That's another function. It's another of X.
[John:] Yeah that's a partic it's a very particular tie up between that function and this one isn't there? Which is what?...
[Graham:] It's basically that.
[John:] Right so the function you've got in brackets when you differentiate it it
[Graham:] Mhm.
[John:] gives you this thing you're multing i multiplying it by.
[Graham:] Mhm.
[John:] So what we've got here is we've got the integral of some function of X
[Graham:] Mhm.
[John:] times... F dash are you happy with F dash notation?
[Graham:] Yeah yeah fine.
[John:] Times... the differential of that function. As soon
[Graham:] Mhm.
[John:] as you see that... you know it's Th this all all integration is that's the answer what was the question. You think this is
[Graham:] Yeah.
[John:] where we finished up now where did we start from? Well to finish up with something like that as you your first guess which is almost certainly correct on this one is well
[Graham:] Mhm.
[John:] that came from someone doing this sort of thing.
[Graham:] Yeah.
[John:] Okay?
[Graham:] And so you could plough that straight back in
[John:] Yeah right. So you'd say well we must have let let's say that I wasn't being that helpful
[Graham:] Mhm.
[John:] and I said would you integrate so it didn't stand out sort of too obviously four X cubed minus X times X squared.... So oh sorry that should be minus one because I taking your word for this one here.
[Graham:] Yeah.
[John:] Erm that's a minus one. Erm...
[Graham:] Oh of course yeah it's just the figures that disappear. That's right.
[John:] Er yeah I hadn't finished off this that's probably what the problem was.... Right that's better.... Okay?
[Graham:] No I would've forgotten [LAUGHTER] that anyway. []
[John:] Okay so we'll let U be this thing that's raised to a power.
[Graham:] Mhm.
[John:] U equals four X cubed minus X X differentiate it... with respect to X twelve X cubed minus four
[Graham:] Mhm.
[John:] so that gives us this and so if I said... erm and that's a... all right we'll leave that as twelve X squared minus one.... Put the twelve in there.... write it out again. So if I said, Integrate four X cubed minus X
[Graham:] Mhm.
[John:] times erm... twelve X squared minus one.... look at that... the the giveaway what's the giveaway? That this has got some chance of being F dash
[Graham:] That times that is that.
[John:] Right. Differentiate this
[Graham:] Mhm.
[John:] differentiate that so the big the first thing to go for is the power is one less and then
[Graham:] Yeah.
[John:] this looks a little bit suspicious where did
[Graham:] Mhm.
[John:] it come from, Ah three fours are twelve X to the three minus the power right oh and this one fits too very nicely. Great so that that's F dash of X. So we started off with something like now we're not we we haven't got there yet
[Graham:] Mhm.
[John:] but we've we've broken the back of it.
[Graham:] Yeah.
[John:] we're just about there. What have we got here? Let's try a four X cubed minus X and
[Graham:] Mhm.
[John:] differentiate it and what do we get? We get this.
[Graham:] Yeah.
[John:] Right so we differentiate... four X cubed minus X... we get... how does it tie up with what we've got there? We get two times... four X cubed... minus X... times twelve X squared minus one... Well that's a shame we should be getting we should get twice it.
[Graham:] Mm.
[John:] So we didn't actually start off with this
[Graham:] Mm.
[John:] to finish up with that answer did we what
[Graham:] Mm.
[John:] did we start off with? If we'd have started off with this it would have given us twice what we got.
[Graham:] Yeah.
[John:] So what must we have started out with.
[Graham:] We started off with that one squared the whole lot squared
[John:] that was squared sorry that was
[Graham:] to make it the function of a function.
[John:] that was squared yeah. Sorry that was squared yeah so the differential of that would've given us twice er would have given us twice that.
[Graham:] Oh of course.
[John:] But we didn't finish up with twice we
[Graham:] No.
[John:] just finished up with four X cubed minus X times its differential so what must we have started off... we differentiated something and it was very very close to this very
[Graham:] Mhm.
[John:] similar.... And this time it didn't give us two X it just gave us one times this one times that one. What... what's missing from this let's let's look at it this way erm... differentiate... erm... no let's go straight for the integral... integrate four X squared.
[Graham:] Four X squared.
[John:] Yeah.
[Graham:] That's erm...
[John:] Make it make it twelve X just to be awkward
[Graham:] [LAUGHTER]
[John:] twelve X. It's a little it's also being a bit helpful.... Okay so if we'd have started off I mean we can just take the constant straight out.
[Graham:] Mhm.
[John:] Right away. Twelve times integral of X squared D X well if we'd have started off with X squared we'd could have we would have if we've started off with X cubed we would have finished up with three X cubed
[Graham:] Mhm.
[John:] So we only we only need to start off with a third of that. Now if we'd have started off four X cubed minus X all squared we would have two times this lot but we didn't we just got once this so we must have started off this is only a coefficient only a a factor.
[Graham:] Yeah.
[John:] What must we have started off with here to finish up with with that?
[Graham:] [sigh]
[John:] ... It's very s it's similar
[Graham:] I'm just working back
[John:] Same sort of thing that we were doing here. If I'd have said, Integrate X cubed.
[Graham:] Mhm.
[John:] you'd think well he must have Integrate X squared. you'd say, Well he must have started off with X cubed.
[Graham:] With X cubed.
[John:] That would have given three X squared Well that's
[Graham:] Mhm.
[John:] no good it's three times too much so he must have started off with... X cubed over three and that would've given us one third of three X squared So here... it's something very like that I finished up with I've... if I'd started off with that I would have finished up with twice what
[Graham:] Mhm.
[John:] I've actually got here.... Now the other thing about an integral really when it boils down to it it's only a number or a function or a sine or a cos it comes down to you these are just numbers you're playing with erm...
[Graham:] You'd have to take that from a half.
[John:] You'd have to take a half of it.
[Graham:] Mm.
[John:] Okay? Take a half of it. So if we only had started off with a half of it
[Graham:] Mm you'd end up with that.
[John:] What would we get? We've got Y equals a half of... this thing... four X cubed minus X squared, then D Y by D X would come to a half of two times
[Graham:] four X cubed minus
[John:] No it's not it's X times
[Graham:] times
[John:] its differential.
[Graham:] Mhm.
[John:] Right... So... this is what some one gave us
[Graham:] Mhm.
[John:] they didn't express it as a half of twice they just cancelled it through
[Graham:] Yeah they just cancelled it down.
[John:] once and we said, Ah well we can see the sort of thing we should be working... it sh
[Graham:] Yeah.
[John:] it should've been something like this.
[Graham:] Mhm.
[John:] It should have been that thing squared and now we then having worked it out that far,
[Graham:] [cough]
[John:] right we must have had four X cubed minus X in brackets
[Graham:] Yeah.
[John:] all that squared... I don't know I don't know what the coefficient was I don't really
[Graham:] Mhm.
[John:] care because I'm going to differentiate it now because differentiating is a lot easier than integrating. So I'll differentiate it find out what that gives, Oh well that gives twice it well I don't want twice it so I'll
[Graham:] Mhm.
[John:] start off with half of it. If this had given eighteen times it then you would have started off with one eighteenth. Whatever it was. Er are you fairly happy with that?
[Graham:] Yeah yeah I said that was going
[John:] we'll now we'll now have a we'll do one more of those and then we'll apply it to the wonderful cos sine and sec squared and all sorts of wonderful things like that.
[Graham:] I'll just go and make that coffee
[John:] That sounds like an excellent idea.
[Graham:] [LAUGHTER] How many sugars?
[John:] No sugar no sugar thanks
[Graham:] okey-doke.
[John:] So how are you getting on with the rest of your course?
[Graham:] Erm
[John:] Can I just look through this?
[Graham:] Yeah yeah go on.
[John:] Thanks.
[Graham:] Erm I did first science exam a little while ago
[John:] Yeah.
[Graham:] erm that went... about as well as the first maths one. But I have done a maths erm another science exam sorry
[John:] Yeah.
[Graham:] about a week ago and from the looks of things I've passed it quite successfully.
[John:] Good
[Graham:] So I've picked up on one.
[John:] Right.
[Graham:] Erm.
[John:] So how many subjects have you got?
[Graham:] About nine [LAUGHTER]
[John:] About nine okay and they're all spread out little bits various levels and
[Graham:] Yeah that's
[John:] okay. Which one do you feel most happy with most confident?
[Graham:] Erm anything except the maths and [LAUGHTER] science [].
[John:] Okay.
[Graham:] To be perfectly honest with you because they're the shakiest ones.
[John:] Well what are the what are the others then that you're doing that you're happy with?
[Graham:] Just take a look got me little list knocking round.... It's it's just the technology and things really it's erm dealing with cars electrical studies er it's...
[John:] how
[Graham:] There we are.
[John:] Okay.
[Graham:] There's me timetable it's got the different subjects on it.
[John:] Right which which campus are you on?
[Graham:] Er.
[John:] Right.
[Graham:] Down by Prom.
[John:] Right I know it well. Okay erm...... projects... science... Right now what's your... what are you doing on your science? Thanks very much.
[Graham:] Erm science... at the moment we've just covered erm things like angular momentum erm... it just tends to be stuff like that we've done simple harmonic motion er
[John:] Mm are you happy with that?
[Graham:] Oh yeah yeah no problems at all with it.
[John:] Good.
[Graham:] Sailed through that.
[John:] Excellent excellent excellent, cos that's you know that's quite a lot of maths in that and you get these sines and coses and omega T omega T and
[Graham:] That's it this is it it's something that's unusual because with stuff like I'll I can see what I'm aiming for.
[John:] Right.
[Graham:] And so I'm bi I'm a no problems with it at all.
[John:] Good good erm... just about everyone learns things practically
[Graham:] Mm.
[John:] It's not you you can't just have an abstract theory that's not tied
[Graham:] Yeah.
[John:] into anything it must come from something. I don't know if you saw a programme a while ago about carbon sixty, a new form of carbon that some professors in America discovered.
[Graham:] I s I heard something about somebody was talking to me about it a little while ago. I can't remember what it was like
[John:] Well three sort of top professors in America they
[Graham:] Mhm.
[John:] they made this big discovery they all went back to their hotel rooms.
[Graham:] Mhm.
[John:] The next morning when they got together everyone had got a model.
[Graham:] Mm.
[John:] One had made one gluing bits of straw together another one had used something else but they'd
[Graham:] Mhm.
[John:] all got a physical model.
[Graham:] Yeah.
[John:] So they were playing with each others models, No that bit has to go there
[Graham:] Mm.
[John:] because of this and that would happ They were making it real concrete something
[Graham:] Mhm.
[John:] you can pick up play with like a bit of Lego
[Graham:] Yeah.
[John:] erm and then you understand. If I say erm you know can I can I have a look at your pen it usually
[Graham:] Mhm.
[John:] means pick it and feel the weight of it and
[Graham:] Yeah.
[John:] see how it works. Not give me a an abstract technical description of it
[Graham:] Yeah.
[John:] or have a quick glance but getting to understand it. Now you're obviously I mean most people are but you can obviously understand better think better if they're more practical
[Graham:] Very much so.
[John:] everyone can.
[Graham:] Mhm.
[John:] Everyone is like no matter how sort of academic they pretend to be
[Graham:] Yeah.
[John:] they always understand things much better if it's it's something concrete so if you've got simple harmonic motion and you can relate it to something physical
[Graham:] Mm
[John:] you're with it. So how are we going to relate this differentiation and integration to something physical? Erm let's go back to differentiation, what does is mean why the why the big deal you know why do we make such a fuss about, Hey wouldn't it be nice to integrate things.
[Graham:] Mhm.
[John:] Why? You know why do we go to this trouble?
[Graham:] We need to integrate to find out... the gradient at a certain point
[John:] To
[Graham:] on a curve
[John:] differentiate yeah
[Graham:] Well that's to
[John:] to find the gradient.
[Graham:] Yeah.
[John:] We differentiate to find the gradient.
[Graham:] Mhm.
[John:] We integrate
[Graham:] We integrate from the gradient
[John:] We yeah integrate for other reasons like you're doing moment of inertia and
[Graham:] Mhm.
[John:] things like that. Erm we're finding basically area under a curve or volume of revolution
[Graham:] Yeah.
[John:] integrating but as you say differentiate find the gradient.
[Graham:] Mhm.
[John:] How steep is that hill? I don't know what hill do you know perhaps you know erm Road or something like that how
[Graham:] Yeah I know where you mean.
[John:] how steep is that hill? Well it's erm one in fifty
[Graham:] Mhm.
[John:] What does that mean? Well for every for every fifty you go along horizontally
[Graham:] Yeah you go one up.
[John:] you go up one. Er that's one way of expressing its
[Graham:] Mhm.
[John:] gradient. Another way would be to tell me what the angle is from
[Graham:] Mhm.
[John:] the horizontal. And you'd find that the tan of that angle
[Graham:] Mhm.
[John:] is the other way of expressing the gradient. Up so many along so many.
[Graham:] Yeah.
[John:] Yeah? So that's that's what gradient is erm that's thinking of something very solid like a hill.
[Graham:] Mhm.
[John:] How does rate of change D Y by D X or D S by D T
[Graham:] Mhm.
[John:] tie up with things like velocity?
[Graham:] Well yo you still getting a ratio you're getting a ratio of distance and erm time.
[John:] Okay so in a in a fixed time
[Graham:] You go so far.
[John:] you go so far. If you're going further in a set over a a set
[Graham:] Mhm.
[John:] time then you're going faster.
[Graham:] You're going faster.
[John:] If you're not going as far you're going slower and then you take that up a level all right
[Graham:] Mhm.
[John:] a level of e abstraction if you like your gradient you've got so you you're very happy with this yeah?
[Graham:] Yeah....
[John:] So that's time going along... along there.
[Graham:] Mhm.
[John:] And that's distance
[Graham:] That's distance.
[John:] there or it's displacement. And as time goes on... you find he's gone further and further away.
[Graham:] Mhm.
[John:] And it's a nice straight line so the gradient is constant. The gradient
[Graham:] Mhm.
[John:] the D S by D T
[Graham:] Mhm.
[John:] the velocity... velocity is constant.
[Graham:] Mhm.
[John:] Then that that that sort of makes sense You know
[Graham:] Yeah.
[John:] that's practical and you can you can get your head round it.
[Graham:] Mhm.
[John:] Then they take it... sort of up a level
[Graham:] Yeah.
[John:] if you like there's time going along there. Now what we're plotting up here is D S by D T well we don't call it that it's a bit confusing call it velocity. We're plotting velocity
[Graham:] Mhm.
[John:] up there and we've got say another straight line here erm... this is a straight line which means that D V by D T... is... How is velocity changing if constant?
[Graham:] Yeah.
[John:] Okay erm now you you've... used to the equations of motion for a constant acceleration are you?
[Graham:] Erm.
[John:] V squared equals U squared plus two A S and
[Graham:] We've come across them yeah
[John:] You've come across them okay.
[speaker001:] [cough]
[John:] You can derive all those
[Graham:] Mhm.
[John:] starting from the acceleration. Acceleration's a constant
[Graham:] Yeah.
[John:] integrate it it gives you the velocity. Integrate it again it gives you the distance you integrate with respect to time.
[Graham:] Mhm.
[John:] Each time you do it. So this is why we get interested in differentiation. Erm
[Graham:] I've done stuff like this actually I remember doing exactly them sort of questions.
[John:] Right so the more you can spot the tie up and say well this isn't something I'm going to take this off. This isn't something totally artificial
[Graham:] Yeah.
[John:] It it has got a real purpose and it does tie up very well with erm velocity and
[Graham:] Here we are... there it is. Doing velocity is a certain time
[John:] Mhm.
[Graham:] Velocity at time zero acceleration at a
[John:] Mhm.
[Graham:] certain time. Acceleration at
[John:] Mhm.
[Graham:] time zero.
[John:] Right. Okay.
[Graham:] So I've covered bits and pieces like that.
[John:] So how about... you you've you've just had have a quick glance at that.
[Graham:] Mhm.
[John:] Okay.... So if I put something like this erm... okay?
[Graham:] Mhm.
[John:] Could you could you give me an expression for if that's the acceleration
[Graham:] Mhm.
[John:] Okay. The velocity would be what?
[Graham:] Your velocity would now be... D S by D T.
[John:] Right... and your distance?... would just be? S.
[Graham:] Mhm.
[John:] Right I don't know what I don't know I didn't have the function a formula
[Graham:] Yeah.
[John:] to let me work this out what the distance was as time went on. And I didn't have a formula to work out what the velocity was all I did ha know was the acceleration was a constant.
[Graham:] Yeah. [tape ends]
[John:] derive these equations by what what what trick?
[Graham:] Well the easiest way would be to integrate that.
[John:] Right okay so there we are a horrible looking thing to integrate.
[Graham:] [LAUGHTER] Yeah.
[John:] K some constant where where where do I start here? Well talk about it.
[Graham:] Right. Well it's a constant value... and I know that it's to the power of one.
[John:] Whoa whoa whoa whoa hang on hang on hang on hang on right think about what you just said it's a constant
[Graham:] Mm.
[John:] Right now I'm asking you to integrate it which is work backwards. The answer was a constant, now what did I differentiate? Think in terms of X if you like. The answer was a constant what did I differentiate?...
[Graham:] It'd need to be something like K X.
[John:] K X right okay. So right K X something like K X. Now is it only K X that could have given me that? If I differentiate K X I'll get K
[Graham:] Mhm.
[John:] What about if I had K X plus three?
[Graham:] That'd do.
[John:] Mhm so
[Graham:] That'd be the same.
[John:] Right... so is erm Y or... that's the velocity
[Graham:] Mhm.
[John:] right velocity up there and T along here. We're integrating with respect to T so it's going to be K... it's not going to be K X it's K...
[Graham:] K T.
[John:] Right okay so if we differentiate K T with respect to T we get K.
[Graham:] Mhm.
[John:] If we differentiated So there's... there's velocity equals K T... it's supposed to be
[Graham:] Yeah.
[John:] Velocity equals K T plus
[Graham:] Plus something yeah.
[John:] Velocity equals K T minus four.
[Graham:] It makes no odds because the gradient's still the same.
[John:] So plus any constant
[Graham:] Yeah.
[John:] Okay. Right well that's found the velocity now can you find the distance? What would you do to find the distance?
[Graham:] The same again.
[John:] Okay integrate again....
[Graham:] Now this is where it gets interesting. Erm...
[John:] don't forget when they're added when they're added you can do them one bit at a time.
[Graham:] Mm.
[John:] Now if they're multiplied or divided then you can't say, Oh well I'll just take this bit and do that and then I'll come back for the other one. You've got to take the whole lot as a unit. And you do something pretty clever with it usually but if they're just added, one bit at a time. Well... how did how did you manage to get a C you had something here and you differentiated it
[Graham:] It'd need to be like C X.
[John:] So and you were working the variable we were working with is not X but T.
[Graham:] not T so
[John:] Okay so that's C T. Now don't forget... as we did here it's not just
[Graham:] Mhm.
[John:] C T that could give us C C T plus any other form of constant
[Graham:] Erm. N.
[John:] N okay right so that's sorted that bit out now where did we get a K T from?
[Graham:] It was that
[John:] Okay but we're we're we're working up from
[Graham:] You
[John:] this way we started I started here... I wrote in here S equals some function of T.
[Graham:] Mhm.
[John:] Right? And differentiated it and it gave me K T well forget about the K it's only a number it's only
[Graham:] Mhm.
[John:] a constant. I had something here I differentiated it and it gave me T so what what was it?
[Graham:] Another number.
[John:] Er let's go back to there
[Graham:] Hang on erm
[John:] let's g let's go to the X's
[Graham:] Let me think.
[John:] Let's take these two out to the X's and just look at that that bit there. I had something we'll forget about that
[Graham:] Mhm.
[John:] cos we've done that sorted
[Graham:] Yeah.
[John:] that out. I had Y equals something don't know what we had
[Graham:] Mhm.
[John:] and I differentiated it and it gave me D Y by D X equals X. Okay so what did I have here so that when I differentiated I got X?...
[Graham:] You'd need... you need to have that to the power of something.
[John:] Yep okay and what power?...
[Graham:] Erm... one.
[John:] Try try this one I had Y equals something
[Graham:] Mhm.
[John:] and I brought it this way round the normal way that we go
[Graham:] Mm.
[John:] and I found D Y by D X and I found that it came to erm X squared.
[Graham:] Mhm.
[John:] What did I start with there so that when I differentiated it I got X squared....
[Graham:] A third X cubed.
[John:] Right okay if it was X cubed that would have given me three times too much
[Graham:] Mhm.
[John:] so it must have been a third X cubed
[Graham:] Third
[John:] Brilliant okay now here I started off with something I don't know what it was but I know that when I differentiated it I finished up with X
[Graham:] Ah now
[John:] so what did I start with?
[Graham:] Right to do that... mm...
[John:] you've just done it for there.
[Graham:] Yeah.
[John:] Instantly right away bang you did that. Now is it throwing you because this is an X and it doesn't look like
[Graham:] It doesn't have a power at all.
[John:] it is X to the one
[Graham:] power of one.
[John:] okay. So what did I start with?
[Graham:] Half X squared.
[John:] Right. Okay?
[Graham:] Mhm.
[John:] Half X squared. So if I started off with a half X squared and I differentiated it I would get two times a half times X
[Graham:] A half it cancels.
[John:] cancels out just X. Right we're not working with X
[Graham:] We're working with T
[John:] I have something and I differ forget about the K cos it's only a like a three times or a fives times
[Graham:] Mhm.
[John:] it's only a coefficient we can adjust that later. I had some function of T and I finished up with T what did I start with?
[Graham:] Ah now I see how it's tying in.
[John:] Right.
[Graham:] Yeah that's exactly the same as what you're I was trying to do before so it'd be the equivalent of that.
[John:] Right which would be?
[Graham:] Yeah I see how that's going now.
[John:] Okay.
[Graham:] Mhm.
[John:] So I finished up with T to the one, what did I start up with?
[Graham:] You started off with say half
[John:] T squared.
[Graham:] T squared yeah.
[John:] So it's K times a half T squared....
[Graham:] Mhm.
[John:] Right so we've got our equation now... S is equal to a half some constant let's call it G
[Graham:] Mhm....
[John:] T squared plus C T plus some other constant erm say C one T plus C two. You recognize that from your equations of motion? Now we got there just by starting off from D two S D T squared equals K.
[Graham:] Mm.
[John:] So it's a very inconve if someone said to to you come up er okay this you put initial values in
[Graham:] Mhm.
[John:] erm to make some of these terms disappear if you like to make
[Graham:] Yeah.
[John:] some of your constants go but your important thing that comes out is this
[Graham:] Is the
[John:] half G T half G T squared plus some constant times time normally your... the the G will be a negative half A T squared but someone had said come up with that equation, and you've said well what are you going to give me to go on well the acceleration's constant.
[Graham:] And they've given me that.
[John:] You say, Oh yeah so what that doesn't help at all.
[Graham:] Yeah.
[John:] And it it doesn't without integration you just
[Graham:] Mhm.
[John:] you just give up.
[Graham:] Yeah.
[John:] As people have mathematicians have done for centuries. They've said well I can't make head or tail of it.
[Graham:] Mhm.
[John:] So how did you feel when you first did fractions? Can you remember?
[Graham:] Oh God that's going back a bit.
[John:] Mhm but how did you feel?
[Graham:] Erm confused as hell until somebody told me about having a pie.
[John:] Right until you eventually
[Graham:] You're cutting it up into bits.
[John:] Right until you eventually got something physical to relate it to.
[Graham:] Yeah.
[John:] How did you feel about negative numbers when you first met them? can't remember.
[Graham:] I wasn't too bad with them as far as I can remember I was fairly okay.
[John:] have a look have a look see how many cars are outside. Minus fifteen.
[Graham:] [LAUGHTER] Yeah. []
[John:] Definitely yes I agree with you.
[Graham:] No I remember from what I can remember anyway when I first started using negative numbers it was on a graph.
[John:] Mm really? that's
[Graham:] Started off with a graph
[John:] If it's if it's below the table we'll say it's it's minus two centimetres you know.
[Graham:] Yeah.
[John:] If it's below sea-level we'll just say that if it's you owe me rather I owe you
[Graham:] Mhm.
[John:] it's below zero on a It it's just a little trick there's no such thing as minus three.
[Graham:] It's just another way of putting it that's all.
[John:] It's it just means we go like we got mixed up with our zero and we should have started lower down but we're not going to change it now.
[Graham:] Mm. [LAUGHTER]
[John:] That's what minus three means it does it's not real. What about erm the square root of minus one then?
[Graham:] The square root of minus one oh God. [LAUGHTER]
[John:] Some number you multiply it by itself and it comes to minus one. Have you done complex numbers?
[Graham:] Erm probably somewhere along the road.
[John:] Right.
[Graham:] Erm
[John:] So when you first mean these things they're weird [LAUGHTER]
[Graham:] Mm.
[John:] erm now professional mathematicians people who did nothing but that the whole the community used to bring them
[Graham:] Mhm.
[John:] their food and support them they say around all day having very weird discussions about
[Graham:] Mhm.
[John:] sort of funny little numbers. They spent thousands of years before they could understand fractions
[Graham:] Mhm.
[John:] or negative numbers.
[Graham:] Yeah.
[John:] Right just fractions or negative numbers forget about integration
[Graham:] Mhm.
[John:] integrating cos squared just to get to that stage they had thousands of years they were full time professionals did nothing else but it. You are expected you know
[Graham:] Yeah.
[John:] you're given about four or five years to play with one and one makes two, Yes he's doing
[Graham:] Mhm.
[John:] very well he can count to twenty. all of a sudden in the next couple of years you've got to go through thousands of years of evolution.
[Graham:] [LAUGHTER]
[John:] That mathematicians had a very very hard time and they got a lot of things wrong they went off down the wrong track for hundreds of years before
[Graham:] Mhm.
[John:] people put them right. So it's not obvious and it's not natural unless as you said fractions cut up your pie.
[Graham:] Yeah.
[John:] Right then you know where you are I always I always have a pie with me just in case I get hungry.
[Graham:] [LAUGHTER]
[John:] Because there are so many people who have trouble with fractions
[Graham:] Mhm.
[John:] adults particularly they're alright with a quarter and a half but after that you know
[Graham:] After that it gets confusing yeah.
[John:] So... relate it to something like that and it's real now if we can relate this to something that you are happy with and you understand it will become real for you
[Graham:] Mhm.
[John:] and you'll know what you're doing. At the moment there's so much... new stuff being shoved in. If I'm
[Graham:] Mm.
[John:] explaining this to you in Chinese you know you don't understand the words
[Graham:] No no
[John:] I'm using. If I'm explaining it to you in terms of cos squared and idea what cos squared is I'll have a guess then it's not explaining.
[Graham:] Mhm.
[John:] You're you're getting you know
[Graham:] Confused terms even more confused. [LAUGHTER]
[John:] Explaining it in terms of something you don't understand.
[Graham:] Yeah.
[John:] You know if if if for example you didn't understand anything about engines
[Graham:] Mhm.
[John:] and erm someone was trying to explain simple harmonic motion They start piston going up and down inside an engine and you think well what what what's a what's a piston?
[Graham:] [LAUGHTER] [LAUGHTER] That's exactly what we done. []
[John:] What's an engine? And if you don't know that
[Graham:] Yeah.
[John:] well it's just useless trying to
[Graham:] You've had it [LAUGHTER] yeah [].
[John:] explain it like that okay?
[Graham:] Mhm.
[John:] So it's trying to get this tied down and get its limits clearly defined so you know what you're talking you know exactly
[Graham:] Yeah.
[John:] what you're talking about. Erm somebody comes in some prat comes in to your garage and he's telling
[Graham:] Mhm.
[John:] you about erm well of course erm I've got this diesel car and it's exactly the same principle as a petrol engine and I just put petrol in it. Erm
[Graham:] [LAUGHTER] I have a fit yeah. []
[John:] hang on hang on now if you're talking to a mathematician and you're sort of saying cos squared is well erm you know if I integrate that ah I'm going to get cos so if erm well I'll just change that there and I'll fiddle
[Graham:] Mm.
[John:] fiddle I'll just have a bit of a fiddle well perhaps I'll try running it on lead free petrol and see if that
[Graham:] Yeah [LAUGHTER]
[John:] works better. Erm it's similar yes I mean
[Graham:] Mm.
[John:] there is a there are more similarities between a petrol engine and a diesel engine than there are differences really.
[Graham:] Yeah there are very true.
[John:] a lot of similarities but
[Graham:] Mhm.
[John:] if you try try and treat one as the other you're gonna come very unstuck
[Graham:] It doesn't yeah
[John:] Yeah erm like say mixing up your low tension and your high tension circuit with your electrics erm so
[Graham:] [LAUGHTER]
[John:] or electricity isn't it you know. Not going to make a lot of difference surely. Erm more similarities than differences but the
[Graham:] Mhm.
[John:] differences are important enough to make
[Graham:] Yeah.
[John:] th make it
[Graham:] Make it a definite similarity. [LAUGHTER]
[John:] totally catastrophic if you start trying to treat one as the other and this is
[Graham:] Mhm.
[John:] why you know okay you're happy with cars you're happy
[Graham:] Mm.
[John:] with mechanics with electrics and you you wouldn't dream of trying to say well these two are the same.
[Graham:] Yeah,
[John:] So you've got to learn now when you can use the similarities and
[Graham:] Mhm.
[John:] when you have to watch out for the differences
[Graham:] Yeah.
[John:] this sort of stuff because that's where most of the traps are what happens to everyone who tries fractions? What's a half plus a third? that's no problem add the top two add the bottom it's going to be two fifths it's obvious isn't it. Well okay it might be obvious but it doesn't give you the right answer. [LAUGHTER] So []
[Graham:] Yeah.
[John:] I mean it's obvious that you can put petrol in a diesel engine and diesel in a petrol engine
[Graham:] Mm.
[John:] It might be obvious it might
[Graham:] [LAUGHTER] Won't do you any good like but []
[John:] it might be obvious but it's not a good idea
[Graham:] [LAUGHTER]
[John:] yeah. So it's find out what works and what doesn't so that's integration at work if you like why it's useful you've just done simple harmonic motion.
[Graham:] Mhm.
[John:] Right now this time I'm not telling you that the acceleration... for some we won't bother writing it like that right we'll write it this way. D two S D T squared
[Graham:] Mhm.
[John:] right is equal to minus K X. Minus K is it minus K X?
[Graham:] Mhm.
[John:] So acceleration is always working the opposite way to whichever way you're measuring X and trying to pull it back. That's that's it
[Graham:] Yeah mhm. I'm with you.
[John:] Okay that's your equation for simple harmonic motion.
[Graham:] Mm.
[John:] Now have you seen it like that?
[Graham:] Never [LAUGHTER].
[John:] Oh.
[Graham:] I have never seen it like that.
[John:] Well that's that's erm have you heard a disht a definition of simple harmonic motion?... Give me a definition if you can remember one.
[Graham:] I couldn't
[John:] if you've got a book handy or notes handy it'd be worth looking it up.
[Graham:] Erm see if I can remember what on earth I did with the damn thing....
[John:] Don't worry don't worry if you can't.
[Graham:] I think it's in here.
[John:] I'm I'm I'm interested in you you you doing more of the work than me if we can.
[Graham:] I'm with you.... I'm trying to remember where on earth it was. I know for certain it's in here somewhere. [LAUGHTER]
[John:] Mhm.
[Graham:] Erm...
[John:] Don't don't
[Graham:] Now we're close here.
[John:] Oh good.... There's another nother thing about any course but particularly this one where there's sort of lots of subjects and a bit of overlap it is difficult to get a good system for organizing your notes but it is essential.... Okay well
[Graham:] Er... basically this one drags. [LAUGHTER]
[John:] What about this bit right at the start?
[Graham:] Mhm....
[John:] Just read that bit. It's it moves
[Graham:] Acceleration is always
[John:] it moves with S H M when
[Graham:] its acceleration is directed towards a fixed point in its path
[John:] Yeah.
[Graham:] and is proportional to its distance from this point.
[John:] Okay that's it.
[Graham:] Mhm.
[John:] That's the definition of S H M.
[Graham:] Mhm.
[John:] That's another way of writing it. Acceleration right
[Graham:] Yeah.
[John:] is proportional to equals K
[Graham:] Mhm.
[John:] the distance from a point
[Graham:] Mhm.
[John:] and always works the opposite way.
[Graham:] Mm.
[John:] Okay so just from that definition of simple harmonic motion we've got acceleration equal to minus K X.
[Graham:] Yeah....
[John:] D two S D T squared equals minus K X.... Now it's your turn. What are we going to do with that? Oh I'm not interested in its acceleration I want to find out its speed. Its velocity at any point in time. So what do we do?
[Graham:] Okay.... Need to take that down to D...
[John:] Right.
[Graham:] D S by D T.
[John:] Yeah.
[Graham:] Right then.
[John:] Now I differentiated something forget about the K you can
[Graham:] Mhm.
[John:] put the K down right away so we'll keep that there. Now we're only bothered about the X. I differentiated something and I finished up with X what did I start from? What would you differentiate that would give you X?...
[Graham:] Erm... X squared.
[John:] Okay but that will give us too much
[Graham:] So half X squared.
[John:] Right so K times a half times X squared. Right... that's the velocity at any point.
[Graham:] Now S.
[John:] Okay now the other thing is
[Graham:] C.
[John:] Right good.... There's a whole family of equations that could have been differentiated to give that we don't know what C is. But we always fiddle the C by choosing suitable starting conditions suitable initial conditions okay.
[Graham:] Right then... What I'll do is I'll split it up again. Make life easier.... the same as the other before.
[John:] Mm.
[Graham:] Leaves me with this.
[John:] Okay and your half K you can leave your half K out. You keep that sort of to one side....
[Graham:] Wouldn't that need to be to the power of X? Er not to the power of X sorry erm times X?
[John:] Er don't quite follow well I don't I don't see why you want it to be?
[Graham:] I'm trying to see where
[John:] Okay let's have a look at this. We've got let's take let's take the minus K out completely.
[Graham:] Mhm.
[John:] We can the the constant is no problem we can always if we've got ten times it five times it minus point six times it
[Graham:] Mhm.
[John:] no problem we just multiply that one or divide
[Graham:] Mhm.
[John:] okay. So the problem we've got here is that we've X squared D Y by D X equals X squared so somewhere up here we had Y equals what? And we differentiated it and we got D Y by D X is X squared. So what did we start with?
[Graham:] No. X?
[John:] We we er okay
[Graham:] Mm.
[John:] now there's another big thing that comes into it integration involves
[Graham:] Mhm.
[John:] flipping backwards and forwards
[Graham:] Mhm.
[John:] between differentiating and integrating and erm integrating and differentiating and it's ver one of the most common faults is to forget which way you're going.
[Graham:] Mm.
[John:] Instead of going there and back sort of go go back and back again.
[Graham:] Yeah.
[John:] Or sort of there and further on. So what are we doing? right this is why it's a good ides to draw a line down the side and think well I'll just play about over here till I
[Graham:] Mhm.
[John:] know what I want and then I'll come back to what I was doing. So here trying to find out what happens who did who differentiated what sort of thing to get this X squared.
[Graham:] Mhm.
[John:] Well what happened? Someone had something some function of X here they differentiated it and it gave them X squared no what did they differentiate?
[Graham:] A third X cubed.
[John:] Right so they had a third X cubed or X cubed over three.
[Graham:] X cubed over three yeah.
[John:] A third X cubed might be a a nicer way to write cos we can keep all our constants together So it was the minus K times the half times
[Graham:] A third X cubed.
[John:] a third and then the X cubed.
[Graham:] Mhm.
[John:] And these are just sort of piling up together well they they can all be if we choose it right we can make it usually make it one.
[Graham:] Make it one figure yeah
[John:] With a bit of with a bit of fiddling. Erm okay it doesn't look like anything like what you've got here
[Graham:] Mhm.
[John:] because you're doing it in a very different coordinate system which is
[Graham:] Mhm.
[John:] a a neater way to do it because we want this sine omega T. Erm we want the tie up we want this two pi what is how does two pi come into it?
[Graham:] Two pi?
[John:] Yeah.
[Graham:] That's erm
[John:] How do you get how do you get two pi into it?
[Graham:] changing degrees to radians is all
[John:] Right.
[Graham:] Yeah basically
[John:] Okay right. Not its
[Graham:] Revs
[John:] or even even yeah revs no
[Graham:] in rads basically.
[John:] Yeah yeah revs to rads. Erm degrees don't come into it. They're a a very artificial unit very handy.
[Graham:] Mm.
[John:] They're very artificial whereas a radian is a more natural unit.
[Graham:] Yeah.
[John:] But it keeps bringing this two pi in because two pi is the ratio of all the way round to straight across.
[Graham:] Yeah I know.
[John:] It keeps coming in. Okay erm does that help to show what integration is about what you're doing?
[Graham:] It does because I've only ever done that by differentiating from S
[John:] Right.
[Graham:] and I've got to there and then to there and
[John:] Mm.
[Graham:] and that's it basically that's the
[John:] Okay.
[Graham:] way I've ever known it so it is
[John:] So if you think of those three stages then
[Graham:] Mhm.
[John:] I mean there is erm we've s we've stopped there let's go working from this way S is that that's that that's that.
[Graham:] Mhm.
[John:] Acceleration is that.
[Graham:] Mhm.
[John:] Okay er acceleration equals K X what's the rate of change of acceleration? If we differentiate acceleration with respect to time.
[Graham:] to time yeah.
[John:] What do we get when you differentiate that?...
[Graham:] Ooh erm...
[John:] Could you differentiate three X?
[Graham:] Yeah.
[John:] What would you get?
[Graham:] Erm three X to the power of one.
[John:] Uh uh okay go on.
[Graham:] Now... three.
[John:] Right
[Graham:] [LAUGHTER]
[John:] so could you differentiate minus K X the coefficient is not three this time it's minus K.
[Graham:] X it's minus K. So so it's just the minus K.
[John:] Minus K. So what's how is the acceleration changing with respect to time, how's the acceleration changing as time goes on?
[Graham:] It's the same as the constant.
[John:] It's a a constant and it's negative. Oh what do you mean what do you mean it's a negative? Well you're measuring it in the opposite direction to the way you're measuring X.
[Graham:] Mhm.
[John:] So if we're going over that way
[Graham:] Mhm.
[John:] the acceleration is always negative which means s
[Graham:] It's being measured as a deceleration
[John:] the opposite way and it's K it's sort of the rate at which the exc
[Graham:] It's a constant deceleration.
[John:] The rate of change of acceleration is a constant. In other words the acceleration itself is not changing.
[Graham:] Mhm.
[John:] So you keep you could keep on going until you got a constant and then. Now I don't know if you think this has been useful
[Graham:] Yeah yeah.
[John:] but until it's it's the same with any subject you know until you know what you're doing until you can tie it down to something physical and until you can understand I mean you always understand by similarities
[Graham:] Mhm.
[John:] and then you you oversimplify and then you refine it and you concentrate on the differences. I mean it is good to see the similarities between petrol and diesel first.
[Graham:] Mhm.
[John:] They're both engines you stick em in a
[Graham:] Yeah.
[John:] car it'll do either one of them I don't care whether you've got a diesel engine or a petrol engine as long as it'll get me from A to B.
[Graham:] Mhm.
[John:] Okay yeah and then you start concentrating on the differences. you have got one of these don't put the wrong type of stuff in. Erm that's the way everyone's brain works no matter how they think it does.
[Graham:] Mhm.
[John:] A very young kid goes out for a walk... with his parents he's only ever seen cats and dogs before they go
[Graham:] Mhm.
[John:] out and they see the big this big cow in a field he says, Ooh
[Graham:] Mhm.
[John:] look at that big dog.
[Graham:] Yeah.
[John:] And the parent says, Yes that's right yeah. I mean that's all he knows cat
[Graham:] Mhm.
[John:] and dog. Well it's he's being very intelligent cos it's much too big to be a cat.
[Graham:] Yeah.
[John:] And the only big things you get are dogs.
[Graham:] Mm.
[John:] Well it's a funny looking dog with horns on.
[Graham:] Mm.
[John:] Oh yeah a funny looking dog with horns on like that and others and everything standing in the middle of the field you call those funny sort of dogs, cows.
[Graham:] Mm.
[John:] We have a separate category for them.
[Graham:] Mhm.
[John:] But that is the way you learn, by oversimplifying over
[Graham:] Yeah.
[John:] overgeneralizing. Now... if you
[Graham:] Get the basics the rest will follow through. [LAUGHTER]
[John:] can understand that you can do the rest on your own
[Graham:] Mhm.
[John:] you can do it because as you read through your notes as someone's talking to you in the lecture you will be saying, Yeah of course of course.
[Graham:] Mm.
[John:] Rather than, What the hell is he talking about? [LAUGHTER]
[Graham:] That's the [LAUGHTER] yeah []
[John:] And you'll be and you'd be thinking, Ah I can see where he's going now yeah I can see what's coming up next he's going to now he's he's integrated once
[Graham:] Mhm.
[John:] he's going to go integrate again he's going to find out what you know if you're working one way he's going to find out what the distance is.
[Graham:] Mhm.
[John:] Erm or he's going to find out what the acceleration is. Tying it up to something real. Now... X's and Y's
[Graham:] Yeah.
[John:] thetas omega T doesn't matter it's all the same thing
[Graham:] Mhm.
[John:] The maths that you do ties up in some way with usually ties up with reality there's some reason for it. They
[Graham:] Yeah.
[John:] don't always tell you at the time they just say
[Graham:] [LAUGHTER]
[John:] learn that.
[Graham:] Yeah.
[John:] Learn that. Erm why would I need to use the square root of minus one? Well it's brilliant at solving problems in electrical theory.
[Graham:] Mm.
[John:] Erm... what is it? Oh don't know [LAUGHTER] any more than we know what minus one is, how
[Graham:] Mhm.
[John:] find the square root of minus one we don't even understand minus one.
[Graham:] Yeah.
[John:] So we just say it's it does something.
[Graham:] Mhm.
[John:] It's an operator that say
[Graham:] It's you use it
[John:] rotates a vector so that it's pointing in completely the opposite
[Graham:] Mhm.
[John:] well in minus one points a vector in completely
[Graham:] Yeah.
[John:] the opposite way.
[Graham:] Mhm.
[John:] Now the square root of minus one rotates it to ninety degrees
[Graham:] Ninety degrees.
[John:] and again rotates it through another so square root of minus one times the square root of minus one is the same effect as multiplying by minus one.
[Graham:] Mhm.
[John:] It's a trick but it helps it took all the
[Graham:] I never though of that one. [LAUGHTER]
[John:] mathematicians a long time but it does help. Erm you don't have to exactly understand the tools that you're using and how they work but
[Graham:] Mhm.
[John:] you do have to understand their limitations.
[Graham:] How to use them.
[John:] Right you
[Graham:] Mhm.
[John:] don't you might not know what happens when you press the accelerator on
[Graham:] Mm.
[John:] a car
[Graham:] Yeah.
[John:] you might not know what happens when you waggle this stick about and press your left foot up and down.
[Graham:] Mhm.
[John:] But if don't do it properly [LAUGHTER]
[Graham:] Yeah.
[John:] you know you finish up with a wrecked
[Graham:] [LAUGHTER]
[John:] a wrecked gearbox or or or what's more likely to happen on this sort of thing
[Graham:] Mhm.
[John:] You're too scared about changing gear so you go all the way from here to Glasgow in first.
[Graham:] Yeah.
[John:] Right when you could be going up and down the gears and using whatever is appropriate
[Graham:] Mhm.
[John:] for what you're doing or you might even find that it's been left in reverse and you go all the way from here to Glasgow in reverse.
[Graham:] Mm.
[John:] Or in fact you go to London because it happens to be in reverse and it's not really the way you want to go in go but it's carrying you that way instead of you
[Graham:] Mhm.
[John:] in control. So it's getting in control of this you can only control it if you understand it.
[Graham:] Mhm.
[John:] Right when you know what the terms are what people are talking about what they mean when they say petrol engine or a diesel engine.
[Graham:] Yeah.
[John:] [whispering] They're all the same aren't they engines [] you know.
[Graham:] [LAUGHTER] I've heard that one many times. []
[John:] Yeah?
[Graham:] [LAUGHTER]
[John:] Yeah so when people are saying, Well simple harmonic motion or erm you know throwing a stone up and down that sort of thing roughly isn't it sort of one of them goes up and down a lot like that one of them just goes up once and down same equation. It's not.
[Graham:] as it happens. [LAUGHTER]
[John:] It would be nice if it would be nice if
[Graham:] It'd be easier if it was.
[John:] As you say if there was just one equation that covered all eventualities. So definition of simple harmonic motion you find anything that's moving so that the acceleration is always measured the opposite way to
[Graham:] Mhm.
[John:] the way you're measuring the distance and is proportional to that distance
[Graham:] Mhm.
[John:] you've got simple harmonic motion. Erm you'll find this on exam questions show that the resulting motion is simple harmonic motion.
[Graham:] Mhm.
[John:] Erm for example if we ha if I gave you that equation.
[Graham:] Yeah.
[John:] Right and said erm a particle moves so that its displacement is given by this function of T
[Graham:] Mhm.
[John:] they would they those X squareds there'd be would be T squared watch that by the way because I did say that's D S by D T
[Graham:] Yeah.
[John:] and these tend to come in so I put them in there. Right so everywhere there's an X there it should be T's
[Graham:] Yeah.
[John:] Erm
[Graham:] It's cos I'm used to er using D Y by D X now.
[John:] So if I gave you that and said show that that is simple harmonic motion.
[Graham:] Mhm.
[John:] Differentiate it.
[Graham:] Try and work it back to that.
[John:] No well well well
[Graham:] Well work it forward to the
[John:] for work it forward this is the easy way when they
[Graham:] Yeah.
[John:] say they give you something and it'll be hideous looking you know Er
[Graham:] Mhm.
[John:] it won't be nice and simple like this it's when they give you the simple one and say integrate it that it's hard. When they give you the horrible one which is the answer
[Graham:] Mhm.
[John:] and say now what was the question erm erm
[Graham:] Then you can just chop them off as they become constants.
[John:] Yeah. So you look at that and you say okay differentiate it once to get the V differentiate it again to get acceleration and it'll come to minus K X. So it comes to minus K X and you say as the acceleration is proportional to X but in the opposite direction
[Graham:] Mhm.
[John:] erm we've got simple harmonic motion because that's the definition of it. Now you might be give all sorts of polar coordinates and you'll be given working with sines and coses which is where we came in.
[Graham:] Yeah.
[John:] Right?
[Graham:] Mhm.
[John:] Erm now if I'd given you the same thing and you'd
[Graham:] Mhm.
[John:] had sine squared and cos squared.
[Graham:] Mhm.
[John:] Are you with you wouldn't have been so happy with it would you?
[Graham:] No not at all.
[John:] But it would have been exactly the same problem simple harmonic motion
[Graham:] Mhm.
[John:] looking at it from a different point of view and measuring different things. So going back to where we came in. You're not rushing off anywhere are you?
[Graham:] No no.
[John:] It wasn't in there I'll scribble that out.
[Graham:] [LAUGHTER]
[John:] Looking at cos erm I tend to use a lot of paper particularly if it's yours it's usually my own but never mind.
[Graham:] [LAUGHTER]
[John:] So yes you can what I want you to do with these is look through them
[Graham:] Mhm.
[John:] some of them you might just ignore and say Oh I'm not bothered about that others
[Graham:] Yeah.
[John:] you might think Ah there was a good point now put it down in your own words don't copy
[Graham:] Mhm.
[John:] what I wrote or the example that I worked out or even the way you did it yourself here
[Graham:] Mhm.
[John:] try to paraphrase it as though you were explaining it to someone who doesn't understand quite as much as you.
[Graham:] Mhm.
[John:] Okay someone else in your own class say who's got a fair idea about what's going on but
[Graham:] Mm.
[John:] doesn't quite understand it as well as you do and you're trying to explain it to him.
[Graham:] Mhm.
[John:] Because you'll understand it fairly well at the moment but when you come to read your notes
[Graham:] Yeah it's a different thing.
[John:] What on earth was I talking about here? [LAUGHTER] I obviously understood this very well and I had all sorts of
[Graham:] Mhm.
[John:] funny abbreviations and lots of and so obviously.
[Graham:] Yeah. [LAUGHTER]
[John:] seem obvious a month later when you when you've
[Graham:] Mhm.
[John:] skipped half the stages or you know how do you feel when he's working on the board and he skips several lines?
[Graham:] Yeah.
[John:] and you're thinking how did he get from there and by the time you've worked out how he got from there to there you've
[Graham:] Mhm.
[John:] missed where he's going to.
[Graham:] Yeah that happens
[John:] Erm
[Graham:] quite often.
[John:] Yeah so the big thing about this is is getting up to date getting yourself understanding the terms when he says cos squared when he says simple har when he says S H M you think yeah.
[Graham:] Simple harmonic motion straight away.
[John:] I know what that means it means if you get ten times further away you've got ten times the acceleration but it's pulling back in towards
[Graham:] Mhm.
[John:] the centre. It's pulling it against the direction we're measuring S
[Graham:] Mhm.
[John:] or X or anything else we might have to call it. So if we... If I gave you this to integrate... erm... what do you think of that?... Apart from how kind he is to me giving me something so easy.
[Graham:] Two words pop into me head and they're oh hell. [LAUGHTER]
[John:] [LAUGHTER]
[Graham:] Erm right okay... well the first thing I'd do is handle that one.
[John:] What does that look like?
[Graham:] That would be sine three X squared.
[John:] Right but don't forget to put your brackets on it's not sort of sine of nine X squared it's
[Graham:] Yeah.
[John:] it's... that
[Graham:] Mm all squared.
[John:] Okay.
[Graham:] Yeah.
[John:] so there's all sorts of things this could be and that's the one it is.
[Graham:] Mhm.
[John:] It's find the sine of three X and then square your answer.
[Graham:] Square the lot.
[John:] Right okay so it's that times...
[Graham:] Times cos three X
[John:] cos of three X.... Okay.
[Graham:] Right then.
[John:] Now where did that come from? What was I differentiating to finish up with that weird looking thing?... Have you spotted anything?...
[Graham:] It's a function of a function again.
[John:] Right that's what it is. it's raised to some power
[Graham:] Mhm.
[John:] right sine three X differentiated
[Graham:] Cos three X.
[John:] Right. It might not actually be exactly cos three X erm
[Graham:] It might be plus erm
[John:] It's more likely to be times. Right so now we've got something to start on okay.
[Graham:] Okay.
[John:] We've got a we've got this this is what integrating is about.
[Graham:] Mhm.
[John:] Look at it what on earth have we got here what a horrible mess. Now what
[Graham:] Mhm.
[John:] could we have possibly started out with not to give us exactly this cos it's too it's too much to find
[Graham:] Mhm.
[John:] exactly this. But to give us something along these lines. So we might have started out with something like Y equals sine three X squared.
[Graham:] Mhm.
[John:] Right okay so if you'd like to differentiate that see what it gives.... Now you're doing all this in your head?
[Graham:] Mm.
[John:] Mm. Mm. Okay well soon I will be asking you to do them in your head but
[Graham:] Yeah.
[John:] to start with erm same as any any job you do... stripping an engine whatever you like when you first do it you go through very slowly one bit at a time to make sure you're doing everything right and then
[Graham:] Mhm.
[John:] you build up a pattern then you build up a rhythm and eventually you know [whistling] do it
[Graham:] Yeah.
[John:] as you're talking to someone. So let's do that's good so you've got a three in it three cos cubed X. That's not quite D Y by D X yet is it?
[Graham:] Mm no.
[John:] So let's take it through erm the way we would do that okay? Y equals that
[Graham:] Mhm.
[John:] can't do that
[Graham:] Mhm.
[John:] That's it
[Graham:] Yeah.
[John:] I cannot I cannot do I cannot differentiate that I'll tell you what I could do I'd have no problem at all if you gave me something Y equals U squared no problem
[Graham:] Mm.
[John:] so I'm going to fix it so that's the that's what I do cos I'm just going to say let Y equal U
[Graham:] Mhm.
[John:] So okay what's... or hang on
[Graham:] U equals
[John:] Right.
[Graham:] sine
[John:] That's it.
[Graham:] three X.
[John:] Sine three X. So what can you find from that?... Yeah that's correct....
[Graham:] Am I on the right line there?
[John:] You definitely are on along the right lines.
[Graham:] Now I can never remember whether you need to put the minus in or not.
[John:] Well... Erm I haven't got a a definite way a a sort of little trick for remembering that erm but the big thing to remember is s before you just put it in no matter how sure you are stop and think now am I differentiating or integrating?
[Graham:] Mhm.
[John:] Because one way it's one way round the other way it's the other.
[Graham:] It's the other way around.
[John:] Which is the the one that you're sort of that you came across first of all the trig functions?
[Graham:] Erm... sine.
[John:] Sine and then cos.
[Graham:] Mhm.
[John:] Differentiate sine you get cos.
[Graham:] Right.
[John:] Okay they're the straightforward ones if you do it the other way round if you differentiate cos you'll get minus sine.
[Graham:] Minus sine.
[John:] And if you're integrating it will be the other way round. So there's quite a lot of things to think of and the
[Graham:] Yeah.
[John:] best way is to just thick of one of them
[Graham:] Mhm.
[John:] and get that solid
[Graham:] Yeah.
[John:] and use that as a reference so you can work out the others from it instead of trying to remember the lot. Cos you've got
[Graham:] Yeah
[John:] so many things to remember in all subjects not just in maths
[Graham:] Mhm.
[John:] that you need some sort of anchor points that you think
[Graham:] Yeah.
[John:] Right I know that's okay so I can build on that.
[Graham:] Mhm.
[John:] So you've worked out D U by D X... right and what do you get from here? You can differentiate that with respect to
[Graham:] D Y by D U.
[John:] Okay.
[Graham:] Right...
[John:] Hang on you're differentiating.... You're differentiating.
[Graham:] Mm.
[John:] Not integrating.... You're not
[Graham:] Oh.
[John:] right and as I said this is this is so
[Graham:] Yeah.
[John:] easy to do.
[Graham:] Mm.
[John:] Because of that it's a very common peril.... So Y equals X squared you differentiate that two X okay?
[Graham:] Mm yeah. [LAUGHTER]
[John:] And it's always the dead easy ones like Y equals X where people think it's too easy too
[Graham:] Yeah.
[John:] differentiate and get one so they integrate and get a half X squared.
[Graham:] You integrate it to get the difficult one.
[John:] Yeah. Right now what about this one here let's have a little look at this. We've got U equals sine three X.
[Graham:] Mhm....
[John:] Anyone ever shown you how to in how to differentiate sine of three X can't do that. I could do
[Graham:] Erm
[John:] sine X but not sine of three X.
[Graham:] Oh hang on a minute now I'm
[John:] Mm?...
[Graham:] There's something [LAUGHTER] in the back of me mind [] which is saying there should be a figure there.
[John:] Right okay. Well I can't do that I'm afraid
[Graham:] Mm.
[John:] Tell you what I could do if I had something like U equals... sine... T or if
[Graham:] Mhm.
[John:] you don't like T I can d what what other letter would you like Z okay?
[Graham:] Mm.
[John:] U equals sine Z I could
[Graham:] Yeah.
[John:] do that. That's be D U by D Z.
[Graham:] Mhm.
[John:] Would be now differentiate differentiate sine you get a cos okay.
[Graham:] Mhm.
[John:] Integrating integrating a sine you get a minus cos so cos Z okay
[Graham:] Mhm.
[John:] where Z equals three X and D Z
[Graham:] Mhm.
[John:] by D X is what?
[Graham:] Erm
[John:] If Z equals three X so if I differentiate that with respect to X what do I get?... Differentiate three X with respect to X.
[Graham:] You get erm... I'm getting messed up now.
[John:] Yeah you're getting you're you're getting saturated at the moment you've
[Graham:] Mhm.
[John:] had a lot of stuff thrown at you in one go. Erm Y equals five X differen D Y by D X.
[Graham:] [sigh] Right
[John:] Draw a picture Y equals five X differentiate think of some coordinates.
[Graham:] Erm
[John:] Y equals five X if X equals two what would Y be?
[Graham:] Y would be... ten....
[John:] Y equals five X the gradient
[speaker001:] equals five.
[Graham:] Mhm.
[John:] Okay it's the easy ones that go.
[Graham:] Yeah.
[John:] And the big thing that happens is you you suddenly in the middle of differentiating start integrating or in the middle of an integrating start differentiating or you start... doing taking you've had enough of this so you start doing little short cuts like erm differentiating sine three X just as if it was sine X
[Graham:] Mhm.
[John:] which it's not. to just three.
[Graham:] Yeah.
[John:] So this one... come to so the D Y the D U by D X that we're looking for here
[Graham:] Mhm. Mhm.
[John:] by a function of a function of a function. Mm good [tape ends]