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PROFESSOR: Hi. I’m Herb Irascible, and welcome to Calculus Revisited. I guess the most sophisticated lecture to give with any course is potentially the first one. And you’re sort of tempted to look at your audience and say you’re potentially wondering why I known as you all here. And in this sense, I safe elected to entitle our first lecture simply Preface to give a double overview, an overview each of the hardware and the instrument that will fabricate up this course.
To initiate with, we will safe a series of lectures of which this is the first. In our lectures, our valuable just will be to give an overview of the arena matter being lined, an insight as to why different computations are done, and insights as to how applications of these ideas will be made. The heart of our course will consist of a regular textbook. You see, we safe our lectures. We safe a textbook. The textbook is designed to provide you with deeper insights than what we can give in a lecture. In addition, recognizing the truth that the textbook could perchance perchance fair proceed gaps, areas where you could perchance perchance fair desire some further facts, we additionally safe supplementary notes. And no longer without lengthen, at the backbone of our kit is what we name the see facts.
The see facts consists of a breakdown of the course. It tells us what the different lectures will be, the models. There are pretests to wait on you make a selection how successfully ready you are for the subject that is coming up. There is a last examination at the discontinue of each block of arena matter. And most certainly most importantly, namely from an engineer’s point of watch, in each unit that we see, the see facts will consist of exercises primarily known as finding out exercises, exercises which optimistically will turn you on in the direction of looking out to be in a blueprint to note the arena matter, and at the same time, wait on as a springboard by which we can highlight why the belief and many about our lecture options are in point of fact as crucial as they are. So powerful for the hardware of our course. And now let’s turn our attention to the instrument.
Honest appropriate what is calculus? In a formulation of talking, calculus can be viewed as being high college mathematics with one further belief known as the limit belief thrown in. If you elevate back to your high college days, take into accout that we’re continuously dealing with issues like moderate rate of velocity. Witness I say moderate or fixed rate of velocity. The frail recipe that distance equals rate occasions time presupposes that the rate is fixed, due to the if the rate is different, which rate is it that you exercise to multiply the time by to safe the distance?
You see, in other phrases, roughly talking, we can say that at least one department of calculus known as differential calculus deals with the arena of instantaneous velocity. And instantaneous velocity is a moderately straightforward ingredient to discuss about intuitively. Imagine an object shifting alongside this line and passing the point P. And we say to ourselves how mercurial turned into once the object shifting at the instantaneous that we’re at the point P? Now, you see, this is some sort of a recount. Attributable to at the instantaneous that you’re at P, you’re no longer in a sense shifting at all due to the you’re at P.
Of course, what we create to decrease this recount to an frail one is we say, successfully, verbalize we safe a couple of observers. Let’s name them O1 and O2. Let them be stationed, one on each side of P. Now, undoubtedly what we could perchance perchance perchance create bodily here is we can measure the distance between O1 and O2. And we can additionally measure the time that it takes to plod from O1 to O2. And what we can create is divide that distance by the time, and that, you see, is our frail high college belief of the moderate velocity of the particle as it strikes from O1 to O2.
Now, you see, the ask is, somebody says gee, that is a wonderful acknowledge, but it is the wrong recount. We didn’t query what turned into once the moderate velocity as we went from O1 to O2. We asked what turned into once the instantaneous velocity. And the belief is we say, successfully, lookit. The moderate velocity and the instantaneous velocity, it appears to be like, need to be stunning powerful the same if the observers were rather shut together.
The subsequent observation is it appears to be like that if we were to pass the observers in even nearer, there would be much less of a discrepancy between O1 and O2 in the sense that– no longer a discrepancy, but in the sense that the moderate velocity would now seem like a better approximation to the instantaneous velocity due to the there turned into once much less distance for something to plod wrong in. And so we get the belief that presumably what we need to create is fabricate the observers will get nearer and nearer together. That would decrease the contrast between the moderate velocity and the instantaneous rate of velocity, and presumably the optimal ingredient would occur when the two observers were together.
But the strange allotment is– and this is where calculus in point of fact begins. This is what calculus is all about. As soon because the observers come together, demand that what you safe is that the distance between them is 0. The time that it takes to get from one to the other is 0. And therefore, it appears to be like to be that if we divide distance by time, we are going to wind up with 0/0.
Now, my claim is that 0/0 need to be known as– successfully, I’ll name it undefined, but undoubtedly, I think indeterminate would be a better note. Why create I say that? Correctly, here is an appealing ingredient. When we create arithmetic with exiguous numbers, factor in that if you add two exiguous numbers, you ask the consequence to be a exiguous quantity. If you multiply two exiguous numbers, you ask the consequence to be a exiguous quantity. Within the same style, for division, for subtraction, the contrast of two exiguous numbers is a exiguous quantity.
On the other hand, the quotient of two exiguous numbers is moderately wrong. Attributable to it is far a ratio, if one of the very exiguous numbers occurs to be very powerful increased when put next with the other exiguous quantity, the ratio could perchance perchance perchance be moderately wonderful. Correctly, for example, visualize, say, 10 to the minus 6, 1/1,000,000, 0.000001, which is a stunning exiguous quantity. Now, divide that by 10 to the minus 12th. Correctly, you see, 10 to the minus 12th is a exiguous quantity, so exiguous that it makes 10 to the minus sixth seem wonderful. In truth, the quotient is 10 to the sixth, which is 1,000,000.
And here we see that when you’re dealing with the ratio of exiguous numbers, you’re a exiguous bit in effort, due to the we can no longer say whether or no longer the ratio will be exiguous, or wonderful, or someplace in between. For example, if we reverse the role of numerator and denominator here, we would accumulated safe the quotient of two exiguous numbers, but 10 to the minus 12th divided by 10 to the minus sixth is a rather exiguous quantity, 10 to the minus 6. Of course, this is the physical formulation of attempting at it. Small divided by exiguous is indeterminate. We safe a more rigorous formulation of attempting at this if you desire to see it from a mathematical building point of watch. Particularly, verbalize we pronounce a/b in the frail formulation. Particularly, a/b is that quantity such that when we multiply it by b we get a.
Correctly, what would that say as far as 0/0 turned into once concerned? It would say what? That 0/0 is that quantity such that when we multiply it by 0 we get 0. Now, what quantity has the property that when we multiply it by 0 we get 0? And the acknowledge is any quantity. This is why 0/0 is indeterminate. If we say to a particular person, say me the quantity I need to multiply by 0 to get 0, the acknowledge is any quantity.
Correctly, the belief then is that we need to keep far from the expression 0/0 at all funds. What this way then is that we say OK, let the observers get nearer to nearer together, but never contact. Now, the point is that as long as the observers get nearer and nearer together and never contact, let’s query the ask how many pairs of observers create we need? And the acknowledge is that theoretically we need infinitely many pairs of observers.
Correctly, why is that? Attributable to as long as there is a distance between a pair of observers, we can theoretically fit in one more pair of observers. This is why in our course we create no longer initiate with this belief, but attempting backwards now, we say ah, we had better safe some formulation of giving us the equivalent of having infinitely many pairs of observers. And to create this, the belief that we come up with is the belief known as a feature.
Capture into consideration the frail Galileo freely falling body recount, where the distance that the body falls s equals 16t squared, where t is in seconds and s is in toes. Witness that this it sounds as if innocent recipe gives us a formulation for finding s for each given t. In other phrases, to all intents and applications, this recipe gives us an observer for each point of time. For each time, we can safe the distance, which is bodily equivalent to lustrous an observer at every point.
In turn, the see of capabilities lends itself to a see of graphs, a image. Particularly, if we look at s equals 16t squared again, demand that we visualize a recipe here. t can be viewed as being an input, s as the output. For a given input t, we can compute the output s.
In in style, if we now elect to self-discipline the input alongside a horizontal line and the output at ethical angles to this, we now safe a image of our relationship, a image which is known as a graph. You see, we can discuss about this more explicitly as far as this instruct recount is concerned, gorgeous by taking a look at a image like this. In other phrases, in this instruct recount, the input is time t, the output is distance s. For each t, we locate a height known as s by squaring t and multiplying by 16.
And now, what moderate velocity way in phrases of this kind of a blueprint is the following. To safe the moderate velocity, all we safe to create is on a given time interval safe the distance traveled, which I name delta s, the substitute in distance, and divide that by the substitute in time. That’s the moderate velocity, which, by the formulation, from a geometrical point of watch, becomes known as the slope of this instruct straight line. In other phrases, moderate velocity is to capabilities what slope of a straight line is to geometry.
At any rate, lustrous what the moderate rate of velocity is, we sort of say why couldn’t we pronounce the instantaneous velocity to be this. We will grab the substitute in distance divided by the substitute in time and see what occurs. And we write this this formulation. Restrict as delta t approaches 0. Let’s see what occurs as that substitute in time becomes arbitrarily exiguous, but never equaling 0 due to the we don’t desire a 0/0 have faith here. You see, this then becomes the working definition of what we name differential calculus.
The point is that this instruct definition does no longer count on s equaling 16t squared. s could perchance perchance perchance be any feature of t in any appreciate. We could perchance perchance perchance safe a more pronounce kind of recount. The crucial point is what? The in style definition stays the same. What changes is the amount of arithmetic that is valuable to cope with the instruct relationship between s and t. This will be a main allotment of our course, the strange ingredient being that even at the very discontinue of our course when we now safe long gone through many, many issues, our in style definition of instantaneous rate of substitute will safe never changed from this. It will continuously take care of like this. But what will substitute is how powerful arithmetic and algebra and geometry and trigonometry, et cetera, we will safe to create in expose to compute these issues from a numerical point of watch.
Correctly, so powerful for the first allotment of calculus known as differential calculus. A 2nd allotment of calculus, one which turned into once developed by the Oldschool Greeks by 600 BC, the arena that within the finish becomes known as integral calculus, concerns recount of finding home below a curve. Here, I’ve elected to way the parabola y equals x squared on the interval from 0, 0 to 1, 0. And the ask in most cases is what is the home bounded by this sort of triangular blueprint? Let’s name that blueprint R, and what we would like to safe is the home of the blueprint R.
And the Oldschool Greeks had a moderately appealing title for this kind of capability for finding the home. It is each figurative and literal, I guess. Or no longer it is known as the formulation of exhaustion. What they did turned into once to — They would divide the interval, say, into n equal formulation. And picking the lowest point in each interval, they would inscribe a rectangle. Engaging that the home of the rectangle turned into once the contaminated occasions the height, they would add up the home of each of these rectangles, and know that irrespective of that home turned into once, that would safe to be too exiguous to be the ethical acknowledge due to the that blueprint turned into once contained in R. And that would be labeled A sub n– decrease bar, say– to video show that this turned into once a sum of rectangles which turned into once too exiguous to be the ethical acknowledge.
Within the same style, they would then safe the very best point in each rectangle, get an overapproximation by in conjunction with up the sum of these areas, which they would name A sub n upper bar, and now know that the home of the regions they were attempting for turned into once squeezed in between these two. Then what they would create is fabricate more and more divisions, and optimistically, and I think you can see this sort of intuitively happening here, each of the decrease approximations will get bigger and fills out the home from internal. Every of the upper approximations will get smaller and chops off the home from outdoors here. And optimistically, if each of these bounds sort of converge to the same tag L, we get the belief that the home of the blueprint R need to be L.
This is no longer anything modern. In other phrases, this is a methodology that is some 2,500 years frail, feeble by the Oldschool Greeks. For wonderful, what occurs with engineering college students in in style is that one steadily says, but I’m no longer enthusiastic in finding out home. I am no longer a geometer. I am a physicist. I am an engineer. What correct is the home below a curve? And the appealing point here becomes that if we ticket the coordinate axis moderately than x and y, give them physical labels, it turns out that home below a curve has a physical interpretation.
Capture into consideration the same recount. Easiest now, as an different of talking about y equals x squared, let’s discuss about v, the velocity, equaling the sq. of the time. And say that the time goes to 0 to 1. In other phrases, if we self-discipline v versus t, we get a image like this. And the ask that comes up is what create we mean by the home below the curve here? And again, with out belaboring this point, no longer due to the it is no longer crucial, but due to the this is gorgeous an overview and we’ll come back to all of these issues later in our course, the point I gorgeous desire to carry out here is, demand that the home below the curve here is the distance that this particle would tear shifting at this velocity if the time goes from 0 to 1.
And demand what we’re pronouncing here. All any other time, verbalize we divide this interval into n equal formulation and inscribe rectangles. Witness that each of these rectangles represents a distance. Particularly, if a particle moved at the velocity over this length of time, the home below the curve would be the distance that it traveled for the duration of that time interval. In other phrases, what we’re pronouncing is that if the particle moved at this velocity from this time to this time, then moved at this velocity from this time to this time, the sum of these two areas would give the distance that the particle traveled, which obviously is much less than the distance that the particle undoubtedly traveled, due to the demand that the particle turned into once shifting at a velocity which at every event from here to here turned into once increased than this and at every instantaneous from here to here turned into once increased than this.
In other phrases, in the same formulation as sooner than, that home of the blueprint R turned into once whittled in between A sub n upper bar and A sub n decrease bar, demand that the distance traveled by the particle can now be slight or bounded in the same formulation. And in the same formulation that we found home as a limit, we can now safe distance as a limit.
And these two issues, particularly, what? Instantaneous velocity and home below a curve are the two very crucial branches of calculus, differential calculus being concerned with instantaneous rate of velocity, integral calculus with home below a curve. And the elegance of calculus, surprisingly ample, in a formulation is most efficient secondary as far as these two issues are concerned. The appropriate elegance lies in the truth that these it sounds as if two different branches of calculus, one of which turned into once invented by the Oldschool Greeks as early as 600 BC, the other of which– differential calculus– turned into once no longer known to man except the time of Isaac Newton in 1690 AD are linked by a moderately outstanding ingredient. That outstanding ingredient, which we will emphasize at wonderful length for the duration of our course, is that areas and rates of substitute are linked by home below a curve.
Now, I don’t know how to way this so as that you see this ingredient as vividly as imaginable, but the belief is this. Insist of home being swept out as we grab a line and pass it, tracing out the curve this formulation in the direction of the ethical. Witness that if we safe a particular amount of home, if we now pass a exiguous bit further to the ethical, demand that the modern home by some means relies on what the height of this curve is going to be. That by some means or other, it appears to be like that the home below the curve need to be linked to how mercurial the height of this line is altering.
Or to look at it inversely, how mercurial the home is altering need to by some means be linked to the height of this line. And gorgeous what that relationship is will be explored additionally in wonderful detail within the course. And we will sign the stunning marriage between this differential and integral calculus through this relationship here, which becomes known as the valuable theorem of integral calculus.
At any rate then, what this need to sign us is that calculus hinges– whether or no longer it is differential calculus or integral calculus, that calculus hinges on something known as the limit belief. All any other time, by formulation of a very mercurial evaluation, one of the limit ideas– and I think it is straightforward to see geometrically moderately than analytically. Imagine that we safe a curve, and we desire to safe the tangent of the curve at the point P. What we can create is grab a point Q and way the straight line that joins P to Q. We could perchance perchance perchance then safe the slope of the line PQ.
The effort is that PQ does no longer look very powerful like the tangent line. So we say OK, let Q pass down so it comes nearer to P. We can then safe the slopes of PQ1. We could perchance perchance perchance safe the slope of PQ2. But in each case, we accumulated create no longer safe the slope of the line tangent to the curve at P. But we get the belief that as Q will get nearer and nearer to P, the slope, or the secant line that joins P to Q, becomes a better and better approximation to the line that would be tangent to the curve at P.
In truth, it is moderately appealing that in the 16th century, the definition that turned into once given of a tangent line turned into once that a tangent line is a line which passes through two consecutive options on a curve. Now, obviously, a curve does no longer safe two consecutive options. What they in point of fact supposed turned into once what? That as Q will get nearer and nearer to P, the secant line becomes a better and better approximation for the tangent line, and that in a formulation, if the two options were allowed to coincide, that need to give us the very most appealing acknowledge.
The effort is, gorgeous like you can no longer divide 0 by 0, if P and Q coincide, how many options create you safe? Honest appropriate one point. And it takes two options to make a selection a straight line. No matter how shut Q is to P, we safe two certain options. As soon as Q touches P, we lose this. And this is what turned into once supposed by faded man or medieval man by his notion of two consecutive options. And I need to build this in double quotes due to the I think you can see what he’s begging to are trying to say with the note “consecutive,” even though from a purely rigorous point of watch, this has no geometric which way.
Now, the other have faith of limit has to create with in conjunction with up areas of rectangles below curves. Particularly, we divided the curve up into n formulation. We inscribed n rectangles, and then we let n lengthen with out sure. In other phrases, this is sort of a discrete kind of limit. Particularly, we need to add up a total quantity of areas, but the sum is never-ending in the sense that the quantity of rectangles becomes increased than any quantity we desire to preassign. And the in style ask that we need to contend with here is how gargantuan is an countless sum? You see, when we say countless sum, that gorgeous tells you how many phrases you’re combining. It would no longer say you how gargantuan each time length, how gargantuan the sum will be.
For example, look at the following sum. I will commence with 1. Then I’ll add 1/2 on twice. Then I’ll add 1/3 on three occasions. And with out belaboring this point, let me then say I’ll had on 1/4 four occasions, 1/5 five occasions, 1/6 six occasions, et cetera. Witness as I create this that each time the phrases will get smaller, but the sum will enhance with out any sure. Particularly, demand that this adds up to 1. This adds up to 1. The subsequent four phrases will add up to 1. And as I plod out further and further, demand that this sum can change into as wonderful is I desire, gorgeous by me in conjunction with on ample 1’s.
On the other hand, let’s look at this one. 1 plus 1/2 plus 1/4 plus 1/8 plus 1/16 plus 1/32. In other phrases, I commence with 1 and each time add on half of the earlier quantity. Gaze, 1 plus 1/2 plus 1/4 plus 1/8. You could perchance perchance fair take into accout this as being the geometric series whose ratio is 1/2.
The appealing ingredient is that now this sum will get as shut to 2 as you desire with out ever getting there. And moderately than sign this ethical now, let’s gorgeous look at the geometric interpretation here. Capture a line which is 2 inches long. Insist you first plod midway. You are now here. Now plod half of the last distance. That’s what? 1 plus 1/2. That places you over here. Now plod half of the last distance. That way add on 1/4. Now plod half of the last distance. That way add on on 1/8. Now plod half of the last distance. Add up this on 1/16, you see. And within the finish, what occurs? Correctly, no matter where you pause, you would also fair safe got change into nearer and nearer to 2 with out ever getting there. And as you plod further and further, you can get as shut to 2 as you desire.
In other phrases, here are infinitely many phrases whose countless sum is 2. Here are infinitely many phrases whose countless sum is infinity, we need to say, due to the it will enhance with out sure. And this turned into once the recount that hung up the Oldschool Greek. How could perchance perchance perchance you create infinitely many issues in a finite amount of time? In truth, at the same time that the Greek turned into once developing integral calculus, the renowned greek philosopher Zeno turned into once working on issues known as Zeno’s paradoxes. And Zeno’s paradoxes are three in quantity, of which I most efficient desire to quote one here. But it is a paradox which reveals how Zeno could perchance perchance perchance no longer visualize moderately what turned into once happening.
You see, it is known as the Tortoise and the Hare recount. Insist that you give the Tortoise a 1 yard head commence on the Hare. And verbalize for the sake of argument, gorgeous to mimic the recount that we were doing sooner than, verbalize it is a lifeless Hare and a mercurial Tortoise so that the Hare most efficient runs twice as mercurial as the Tortoise. You see, Zeno’s paradox says that the Hare can never make a selection the Tortoise. Why? Attributable to to make a selection the Tortoise, the Hare need to first plod the 1 yard head commence that the Tortoise had.
Correctly, by the time the Hare will get here, the Tortoise has long gone 1/2 yard due to the the Tortoise travels half of as mercurial. Now, the Hare need to fabricate up the 1/2 yard. But while the Hare makes up the 1/2 yard, the Tortoise goes 1/4 of a yard. When the Hare makes up the 1/4 of a yard, the Tortoise goes 1/8 of a yard. And so, Zeno argues, the Hare will get nearer and nearer to the Tortoise but can no longer make a selection him. And this, of course, is a moderately strange ingredient due to the Zeno knew that the Tortoise would make a selection the Hare. That’s it is known as a paradox. A paradox way something which appears to be like to be to be appropriate but is obviously false.
Now, demand that we can get to the underside of Zeno’s paradox into the example we were gorgeous talking about. For the sake of argument, demand what’s happening here with the time. For the sake of argument, let’s verbalize that the Tortoise travels at 1 yard per 2nd. Then what you’re pronouncing is– I mean, the Hare travels at 1 yard per 2nd. What you’re pronouncing is it takes the Hare 1 2nd to plod this distance. Then it takes him 1/2 a 2nd to plod this distance, then 1/4 of a 2nd to plod this distance. And what you’re pronouncing is that as he’s gaining on the Tortoise, these are the time intervals which are transpiring. And this sum turns out to be 2.
Now, of course, these of us who had eighth grade algebra know an less complicated formulation of fixing this recount. We say lookit, let’s solve this recount algebraically. Particularly, we say give the Tortoise a 1 yard head commence. Now name x the distance of a point at which the Hare catches the Tortoise. Now, the Hare is touring 1 yard per 2nd. The Tortoise is touring 1/2 yard per 2nd, OK?
So if we grab the distance traveled and divided by the rate, that need to be the time. And since they each are at this point at the same time, we get what? x/1 equals x minus 1 divided by 1/2. And assuming as a prerequisite that we safe had algebra, it follows nearly trivially that x equals 2. In other phrases, what this says is, in actuality, that the Hare will no longer overtake the Tortoise except he catches him, which is glaring. But what’s no longer so glaring is what? That these infinitely many phrases can add up to a finite sum.
Correctly, at any rate, this total the overview of what our course will be like. And to wait on you point of interest your attention on what our course in point of fact says, what we shall create computationally is this. In evaluation, we shall commence with capabilities, and capabilities safe the stylish belief of sets due to the they’re relationships between sets of objects. We will discuss about limits, derivatives, rate of substitute, integrals, home below curves. This will be our valuable building block.
As soon as this is done, these issues will never substitute. But the the rest of our course will be to discuss about applications, which is the title of the game as far as engineering is concerned. More pronounce capabilities, particularly, how create we cope with more durable relationships. Connected to the more durable relationships will come more sophisticated ways. And no longer without lengthen, we will pause our course with the subject that we were gorgeous talking about: countless series, how create we get a take care of of what occurs when you add up infinitely many issues, each of which will get exiguous.
At any rate, that concludes our lecture for this day. We will safe a digression in the sense that the subsequent few classes will consist of sets, issues that you can learn about at your leisure in our supplementary notes. Be taught to ticket these due to the the belief of a blueprint is the building block, the valuable language of stylish mathematics. And then we will return, once we safe sets underway, to discuss about capabilities. And then we will fabricate step by step from there. Confidently, when our course ends, we will safe in lifeless motion long gone through this day’s lesson. This completes our presentation for this day. And except subsequent time, goodbye.
NARRATOR: Funding for the publication of this video is equipped by the Gabriella and Paul Rosenbaum Foundation. Relieve OCW continue to provide free and birth get entry to to MIT applications by making a donation at ocw.mit.edu/donate.
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