Introduction to Calculus (1 of 2: Seeing the big picture)

the thing I need to say before we actually begin dig into this is I know many of you have met this topic already okay I just want you to forget everything you have ever heard about this just like slope okay um the biggest danger especially with a class like this is to approach this this topic is just like a set of rules basically set of rules memorize the rules know how to apply them to you know hundreds thousands maybe situations and then that's it you know calculus nothing could be further from the truth okay um the design and the use of calculus is such a creative process that to just reduce it down to okay like you will learn rules we will learn rules but if learning rules is what calculus is to you then you turn to what calculus is okay so now pick up your hand and let's think about what this is calculus is actually an abbreviation the full name that goes is infinitesimal calculus now the reason why like this is what we're actually going to be infinitesimally calculus but as you can see it doesn't exactly roll off the tongue okay uh because of such a long awkward phrase calculus is just what it came to know but that's a bit weird calculus is a very general word it's a the same word where we get the word calculator from and calculate in fact um before the 1600s calculus if she said calculus now basically being the same as maths it's it's is calculating stuff right the reason why calculus means maths is because calculus it's actually a latin word it means um is latin for our small pebble and you're like what does it see what does that have to do with anything and um there goes my school I think small pebbles small pebbles are what we used to calculate with in the early days in ancient Rome right they get a whole bunch of um a whole bunch of small pebbles right and some some smart guy and some some country I don't know its name put all these pebbles on like rows and move them around and Colton abacus okay tall pebbles are useful for counting and for calculations right um by the way it's also where we get the same word chalk talk calc you get it in fact if any of you ever become calcium that's exactly right if any of you become dentists or go to the dentist they might say oh you have calculus on your teeth but by the way not a good thing because you're not good to have small pebbles on your teeth like it's a it's a it's you know all of the gross stuff which is not meant to be Darrin it's damaging your teeth okay so calculus right it's really this word for calculation and the fact that it took over like what we're about to do this little subtopic it took over from like can you imagine like okay mathematics is a really really big word right and if just one little topic in mathematics became called mathematics right what does that say about all of the other things right it says like this is really really important we have many branches of mathematics right calculus is just one of them each of them is sort of keyed into one single big idea okay you think about this with me geometry for instance we're familiar geometry when you think geometry you think shapes right you think shapes things that take up space right shapes plane and space right that's what geometry is about dimensions are like empathy our drive this is a bit trickier right but all those X's and Y's and pronin was a nun bones what they're really about is quantities that are related to each other things that change together in proportion to each other in inverse proportion to each other all those kinds of things they're related the algebra will help you understand probability that's a bit of an easier one it's the mathematics of what change chance uncertainty right when you're like I'm not sure something is going to happen but I can know like the basics of like this is more likely to happen less likely to happen am I going to win the lottery or not we're gonna get need tired no nope no 5 etc each branch of mathematics has its key idea calculus calculus is the mathematics of all things that change all things that change and you know what there's lots of things in the world any of the universe that change so you can kind of imagine why cockers kind of took over in fact number one the preliminary nature see courses that you're doing mathematics and mathematics extension one are called the calculus courses they're good 7080 percent plus calculus and it's like subtopics okay in fact the equivalent and what you guys are learning in America is called AP calculus the AP stands for advertisement right it means it means where you know you guys we don't mess with you guys you go to Souter that yeah that's okay the course is called calculus in some ways like so is ours it's just it's just not given it yes right there's there's precalculus and then there's calculus right it's like this is the things prepare you for this okay our cactus you need to know you need to know it was are developed not jointly but independently by two guys in two countries at exactly the same time there's a huge argument between these two as to who invented it are by the way before was called calculus called the calculus the beans like a you know capital C cutters are the two guys are one you may have heard of a guy named Isaac Newton guy in England right you might you might have heard of him the other guy the other guy you call a doctor cuddle Gottfried live knits okay German guy Newton in England Leibniz in Germany and both of them jointly so not only simultaneously develop this this big idea okay and I'll tell you later on that's what you could remind me why it's unusual that no one's heard of magnets and everyone's heard of new okay so here's the problem and this is the famous part that on though he's trying to work out so um Newton well actually yes Newton was sitting under a tree my or so the story goes right and IV saw the Apple fall from the apple tree it fell to the ground is not what happened it's a it's a story ah the apple fell to the ground where where it actually you know hit is up to debate or whether water actual app or not it's on the debate but the poem is that he watched the Apple fall and he watched her fall the ground he's like okay I'm grabbing grab is a thing it makes the Apple fall but then as he watch the Apple fall he thought the moon is way up there right and it's also being held in place by the same thing that makes the Apple fall right gravity but it doesn't it doesn't hit the earth why does it do that why does gravity do do this to this object but that object just stays way up there and he's like how do I have I woke this up so what he did was um try as we've been doing to take a problem which you don't know how to phrase and turn it into home which you can phrase because then you can solve it you ran into a problem gravity does something unusual it does something like this um gravity like most forces in the universe is are inversely proportional to the force that it exerts on something is even worse ly proportional based on the UM I was going to any business it's really the square of the distance you are away from something okay so the further away you get something from something so if I have over here distance as your distance increases the force of gravity and actually everything really if you think of the electrostatic force it's our Coulomb's law I think it gets it gets smaller and smaller as well it drops off like this okay drops off quite fast okay now here is his problem right he was trying to work out okay how do these things relate to each other right how does how does gravity change over distance right this is the problem he was trying to solve and this is tricky because this thing this thing is changing at different rates everywhere that you look right for example if you just compare it with this straight line you can draw one of these for me okay first straight line if you want to know how this straight line is changing okay that's not hard to work out that's not hard to work out you just see for example at a particular point in distance time whatever value you compare it to another point okay and if you've got two points on there you can just say look I'll just compare how much it changes in one quantity versus how much it changes in the other you just get them as a ratio right we know that measure ratio by the name start of the G radians right are they armed on a Cartesian plane you'd say look this is vertical so you call that rise and this is horizontal we call that run okay and the gradient which is M still who knows why okay the gradient is just the ratio between those two things okay but there's a problem here there's a problem here no matter which two points you pick you're always going to get a different value so the thing about this the reason why this works is because I can calculate that or I can calculate between these two points or I can calculate between these two points and you're always going to get the same value I always get the same gradient it's going to be this constant right we know how to read that off but here we have some trouble here we have some trouble so here is what you can did and it's ingenious think about think about a circle okay yeah thumb battle all right now what he really wanted was how much how much am i changing I mean any given point right now by grading you guys already know in terms of like geometry like what does that look like it's the steepness of the graph right if you can know how steep something is at a given point that it's green so he wanted to know like I know obviously it's steep over here and it's not so steep over here but how much okay so what he was really after the quantity he was after was the gradient of the tangent okay this is a really critical a deep breath agree imitator the problem with working out the gradient of a tangent is that gradient is rise over run you have to have some run you have to have two points like to compare between y2 minus y1 over x2 minus x1 you need a y2 and x2 but a tangent by definition does it have two points it's just got one right in fact that's why it's the word tangent is even though just like all of this right we have another one now you wish languages tongue which comes from the same root as tangent is there no it is don't use very often admit of it and here tangerine tangible tangible what does tangible me it means you can touch it you can touch it right so a tangent is something which just touches one point okay so this presents problems for us and mutant trying to work on like this try to work out what's the gradient at that point because there are no two points to compare by definition so here's what they both you

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Keywords: math, maths, mathematics

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Length: 12min 10sec (730 seconds)

Published: Fri Jun 26 2015