PROFESSOR: Hi. Well, this is sort of a summary

lecture for the big group about differential

calculus. And it’s got a fancy title, Six

Functions, that we know. Well, five of them

that we know. And a new one– of course,

there has to be something new– Six Rules, and Six Theorems.

So I haven’t emphasized theorems, but it seemed like

this was an occasion where we could see the main points of the

math behind the functions and the rules. OK. So, here are my first five

functions, all familiar. And, what I’m happy about is

that, if we understand those five and the rules to create

more out of them, we get practically everything,

everything we frequently use. OK. So, I wrote down function

one, power of x, and its derivative, function

two and its derivative, function three. Function four has,

a little bit, something that is important. If it’s e to the x,

then we know the derivative is e to the x. But, if it’s e to the c, x

of factor c, comes down. Important case, you could

say the chain rule. The derivative is that times

the derivative of what’s inside, which is the c. And, finally, the natural

logarithm with the great derivative of 1/x. And now, oh, I left space

to go from function one backwards, to remember the

function that came before it. So, what function has

this derivative? I’m looking here at the other

generation, the older generation. Well, the function with that

derivative is we need the power to be one higher, right? And then, the derivative of

that, we need to divide by n plus 1 so that, when we take the

derivative, the n plus 1 comes down, cancels this, and

gives us x to the n-th. The function that comes before

sine x will be– oh, there was cos x

in that direction. In this direction, we need

minus cos x because the derivative of minus cos

x is plus sine x. But, for this guy, cosine x,

that came from sine x. And, what about this one? What function has

this derivative? Well, with exponentials, we

expect to see that exponential always, e to the c, x again,

but, since this would bring down a c and here we don’t

want it, we’d better divide by that c. So then, if I take that, that’s

e to the c, x divided by c, so the c will come

down, cancel the c, just the way here. And, oh, we’ve never

figured out log x. That’ll be something novel to

do for integral calculus. But, I think, if I write down

the answer, I think it’s x times log x minus x. I believe that works. I would use the product

rule on that. x times the derivative of that

would be a 1 minus that. And the derivative of that

would be a 1, so two ones would cancel, and the product

rule would leave me with log x times the derivative of that. It works. It works. And notice the one beautiful

thing in this list, that the case here is great unless

I’m dividing by 0. If n is minus 1,

I’m in trouble. If n is minus 1, I don’t have

here something whose– if n is minus 1, I can’t

divide by 0. I don’t get x to the minus

1 out of x to the 0. That rule fails at

n equal minus 1. But look, here, is exactly

fills in that whole. Wonderful. Here is the minus 1 power, and

here is where it comes from. So that log just filled in the

one hole that was left there. OK. Otherwise, you know

these guys. But here’s a new one:

a step function. A step function, it’s 0 and it

jumps up to 1 at x equals 0. So, here’s x. The function is 0 until

it gets to that point. So it’s level, then it takes

a step up, a jump up, to 1. And let’s say it’s 1 at that

point, so it takes that jump. All right. OK. That’s a function that’s

actually quite important. And it’s sort of like a two-part

function, it’s got a part to the left and a

part to the right. And they don’t meet, it’s a

non-continuous function. Can I figure out what is it

that– so here will be the old generation, what graph do

I put there so that the derivative is 0 and then 1? Well, that’s not too hard. If I put 0’s here, the

derivative will be 0. And now, over here, I want the

derivative to be a constant 1. And we know that the derivative

of x is what I need, so this is 0 and then

x, two parts again. And the derivatives of those

parts are 0 and then 1. And I often call that a ramp

function because it looks a little like a ramp. OK. What about going this way? Ah, that’s a little more

interesting because what’s the derivative of a step function? What’s the slope of

a step function? Well, the slope here is

certainly 0, and the slope along here is certainly 0,

so, is the answer 0? Well, of course not. All the action is

at this jump. And what’s the derivative

there? Now, a careful person

would say there is no derivative there. The limit of delta f/delta x,

you don’t get a correct answer there because delta f jumps

by one, and delta x could be very small. And, as delta x goes to 0, we

have 1/0, we have infinite. Well, I say, what? Let’s go for infinite. So my derivative is

0 and 0, and, at this point, it’s infinite. It’s a spike, or sometimes

called a delta function. It’s 0, and then infinite at

one point, and then 0. And the oddball thing is that

the area under that one-point tower, spike, is supposed

to be 1. Because, do you remember– and we’ll do more areas if we

get to integral calculus– but, the area under

this function is supposed to be this one. The area under the cosine

function is sine x. The area under this function

should be this one, so the area is 0 here. Run along here. No area under it. Then, I have a one-point

spike, and the area is supposed to jump to 1

under that spike, at that single point. That spike is infinitely tall,

and it actually has a little area under it. Ah, well, your teacher may say

get that function out of here. That’s not a function. And I’m afraid that’s

a true fact that it’s not a real function. So you could say I don’t

want to see this thing, clear it out. But, actually, that’s

very useful. It’s a model for something that

happens very quickly: an instant, an impulse, so

I’ll leave it there. I’ll leave it there,

but I’ll go on. So, if you don’t like it, you

don’t have to look at it. OK. So those were the

six functions, now for the six rules. Nothing too fancy here. I don’t think I really

emphasized the most important and simplest rule that, if you

have as a combination, like you add two functions, then

the derivatives add. Or, if you multiply that

function by 2 and that function by 3 before you add,

then you multiply the derivatives by 2 and

3 before you add. It’s that fact that

allowed us– I mean, you’ve used

it all the time. If you integrated x plus x

squared, you used the sum rule to integrate– ah, sorry– took the derivative. If you want the slope of x plus

x squared, you would say oh, no problem: 1 plus 2x. 1 coming from the first

function, 2x from the x squared function. So the slope of a sum is just

the sum of the slopes. You constantly use that to build

many more functions out of the simple, anything, x

squared plus x cubed plus x 4, if you know its derivative and

you’re using this rule. Now, the product rule,

we worked through. You’ve practiced that. The quotient rule is a little

messier with this minus sign and the division by g squared. It’s a fraction. And then, a little more

complicated, was this inverse function. Do you remember that if you

start from y equals f of x– which is what we always

have been doing– and then you say all right,

switch it so that x isn’t the input anymore, it’s now

the output, and the input is the y. So you’re reversing

the function. You’re flipping the graph. We did this to get between

e to the x and log x. That was the most important

case of doing this flip between y equals e to the

x and x equals log of y. And the chain rule tells us that

the derivative of this inverse function is 1 over the

derivative of the original. Nice rule. And here’s the full-scale

chain rule. Oh, that deserves to be put

inside a box or something because this is a really

great way to create new functions as a chain. You start with x. You do g of x, and then

that’s the input to f. You will know that chain rule. And you remember that that

produces a product, the derivative of f times the

derivative of g, but there was this little trick, right? This g of x was the y. I’ll just remind you that this

g of x is the y, and you have to get y out of the answer. Use this to get an answer

in terms of x. Wherever you see y, you

have to put in g of x. So, that’s the chain rule. And then the final rule that

I want to mention is this L’hopital rule about– well, a lot of calculus is about

a ratio of f of x to g of x when it’s going to 0/0. What do you do about 0/0? Well, as we’re going to some

point, like x equals a, if this is going to 0/0, then

you’re allowed to look. The slopes will tell you how

quickly each one is going to 0, and the ratio becomes a

ratio of the two slopes. So, normally then, this answer

would be the derivative at a divided by the derivative

at a. If we’re lucky, this 0/0 thing,

when we look at the slopes, isn’t 0/0 any more. It’s good numbers,

and L’hopital gets the answer right. OK. That’s a review of L’hopital’s

rule, just really remembering that that’s an important rule

that came directly from the idea of the derivative. We’re using the important part

of the function because the constant term in that

function is 0. Good. OK, are you ready for

six theorems? That is a handful, but

let’s just tackle it. Why not? Why not? OK. So, six functions were easy. Well, we start with the big

theorem, the big theorem, the fundamental theorem

of calculus. The fundamental theorem

of calculus, OK, that ought to be important. And what does it say? It says that the two operations

of going from function one to two by taking

the derivative, the slope, the speed, is the reverse of

going the other way, from two back to one. It’s really saying that, if

I start with a function– Here, this would be one way. If I start with a function, f,

I take the derivative to get function two, the speed,

the slope. Then, if I go backwards– which is this integrating that

integration symbol that’s the core in integral calculus– if I take the derivative

and then take the integral, I’m back to f. And what you actually get

in this number is f at– it depends. It’s like a delta f, really. It’s the f at the end minus

the f at the start. Maybe you’ll remember that. When we talked about it, there

was one lecture on big picture of the integral, and there may

be more coming, but that was the one where we had

this kind of thing. And, in the other direction, if

I start with function two, do its integral to get function

one, take the derivative of that, then I’m

back to function two. Actually, you’re going to say

I knew that: function one to two, back to one. Or start with two, go to one,

then back to two, that’s the fundamental theorem. That those two operations,

of taking the derivative, that limit– You remember what’s tricky about

all that is that this d,f, d,x, involves a limit

as delta x goes to 0. And this integral will also

involve a limit as delta x goes to 0. So that’s the point at which it

became calculus instead of just algebra. Well, important. I should say, let’s assume here,

that these functions are all continuous functions. And I’m going to assume that

these theorems will apply to continuous functions. And do you remember

what that meant? Basically, it meant that that

jump function is not continuous. And that delta function is– well, that’s not even

a function. The ramp function is continuous

but, of course, the derivative isn’t. OK. All right. So, we’ve got functions that

we can draw without raising our pen, without lifting

the chalk. And here’s the fact about

them, that if I have a continuous function on an

interval– so, here is some point, a, and here is some

point, b, and my function goes like that. Oh, it doesn’t do that. It goes like that. Then this thing says that this

maximum is actually reached, and this minimum is

actually reached. And any value in-between,

anywhere between this height and this height, there are

points where the function equals that. The continuous function hits its

maximum, hits its minimum, hits every point in-between. Where, if it wasn’t continuous,

you see it could go up, and then, suddenly,

never reach that point, suddenly drop to there. There’s a function not

continuous, of course, because it fell down there. And it never reached m because

it was this close, as close as it could be. But it never got there because,

at the last minute, it jumped down. OK. So, that’s sort of a good

theoretical bit about continuous functions. OK. So, that’s new. That was not mentioned before. But you can see it by just

drawing a picture where it hits the max, hits the min, hits

all values in-between. And then, you see the point,

y, continuous was needed because, if you let it jump,

the result doesn’t work. OK. Here’s another thing. This is now called the

mean value theorem. That’s a neat theorem. OK. Oh. Now, here, our function is going

to have a derivative over some region. That function probably

had a derivative. OK. OK. So, that function, or this

function, f of x, here’s the idea of the mean

value theorem. This is like delta f/delta

x for the whole interval from a to b. Delta x is b minus a,

the whole jump. Delta f is f at the end

minus f at this end. So that delta f/delta x is like

your average speed over the whole trip. Like you went on the

MassPike, right? And you entered at 1:00 o’clock

and came out at 4:00 o’clock, so you were on the

pike for three hours. And your trip meter

shows 200 miles. So your average speed, average

speed, was 200 divided by 3, that number of miles per hour. Yeah, about 66 miles– well,

probably illegal. OK. A little over 66 miles an

hour: 200/3, so you’re slightly over the speed limit. Well, the mean value theorem

catches you because you could say well, but when did

I pass the limit? When was I going more than 65? And the mean value theorem says

there was a time, there was a moment when your speed,

when the speedometer, itself, was exactly. This instant speed equaled

the average speed. Shall I say that again? If you travel with a smooth

changes of speed, no jumps in speed, then, if I look at the

average speed over a delta t, there is some point inside that

one where the average speed agrees with the

instant speed. Or you could say, if

you prefer slope– Suppose the average slope, the

up over a cross, is 10, So in the time at cross, you

eventually got up 10. Then there will be some

point when your climbing rate was 10. There’d be some point when that

instant slope is also 10. OK. That’s the mean value theorem. This is called the mean value. Mean value is another

word for average. So the mean value equals the

instant value at some point. But we don’t know, that point

could be anywhere. OK. Now, I’m ready for the last two

theorems. And the first one is called Taylor Series,

the Taylor’s theorem. And we have touched on that. And what is Taylor

Series about? Taylor Series is when you know

what’s going on at some point x equal a, and you want to know

what the function is at some point x near a. So x is near a. And, to a very low

approximation, f of x is pretty close to f of a. This is the constant term. That’s where the trip started. So this is like a trip meter

for a very short trip. The first thing would be to know

what was the trip meter reading at the start. But then the correction term, so

this is the calculus term, it’s the speed at the start

times the time of the trip. If you only keep this, the

trip meter isn’t moving. When you add on this, you’re

like following a tangent line. If I try to describe it, you’re

pretending the speed didn’t change. Here, you’re pretending the

trip meter didn’t change. Nothing happened. Here is the next term. But now, of course, this speed

normally changes too. So calculus says there is

a term from the second derivative, there’s

a bending term. This, we would be correct to

stop right there on a straight line: constant speed. But now, if the speed is

increasing, your trip meter graph is bending upwards, you’d

better have a correction from the second derivative. That’s the slope of the slope,

the rate of change of the rate of change. It’s the acceleration. So, if I had constant

acceleration, like I drop this chalk, it accelerates. So, from where I drop it, that

gives me its original height. Its original speed might be

0, if I hold onto it. But then, this term would

account for the second derivative, the acceleration. And that would give me the right

answer, the right answer to the next term, but now I’ve

drawn the famous three dots. So three dots is the way to

say there are more terms because the acceleration

might not be constant. What’s the next term? If you know the next

term, then you and Taylor are square. The next term will be

1/3 factorial, 1/6. It’ll be a third derivative of

f at the known point times this x minus a cubed. You see that these terms

are getting, typically, for a nice function– and we saw this for

e to the x. We saw the Taylor Series

for e to the x. Can I remind you of the Taylor

Series for e to the x around the point 0 because e to the x

is the greatest function I’ve spoken about, at all? So, if this was e to the x, it

would start out at e to the 0, which is 1. Its slope is 1, so this

is 1 times x. Its second derivative

is, again, 1. And a is 0 here, so

this would be 1/2, 1/2 factorial x squared. And then that next three-dot

term would be 1/3 factorial x cube. And you remember what

it looks like. So the Taylor Series just

looks messy because I’m writing any old f. I’m allowing it to be the start

point, to be a, and not necessarily 0. But, typically, it’s 0. And the e to the x series

is the best example. But I want to show you

one more example. That’ll be my last theorem. I just mention it here because

it’s just like the mean value theorem. If I do stop, suppose I stop

here and I don’t include the x cube term, the third derivative

term, then I’ve made an error. And, of course, that error

depends on what the third derivative is, the one I

skipped, the x minus a cube, the thing I skipped, and

the 1/3 factorial. And this third derivative

is, at some point, between a and x. That’s a lot to put in, but the

mean value theorem said you could take the derivative at

some point in-between, some point along the MassPike. And this is just the same thing,

but I’m keeping more terms. I’m quitting at any

point, and then I would take the next derivative at somewhere

along the MassPike. What should you learn

out of that? I think the idea is

Taylor Series. And, of course, we have

two possibilities. Either we cut the series off and

we make some error, but we get a pretty good answer, or we

let the series go forever. And then comes the question. Then we have an infinite number

of terms, and then the question is does that series add

up to a finite thing like e to the x? Or does it add up to a delta

function or something impossible? So that leads to the question

of learning about infinite series. In calculus, Taylor

Series is where infinite series come from. And, if we want to go all the

way with them, then we have to begin to think about what does

it mean for that infinite series to add up to a

number, or maybe it just goes off to infinity. Does it converge, or

does it diverge? Ah, that would be another

lecture or two. Let me complete today with one

more theorem, a famous one, the binomial theorem. So, what’s the binomial

theorem about? The binomial theorem is about

powers of 1 plus x. 1 plus x is a typical binomial:

two things, 1 and x. And we have various powers. Well, if the powers are the

first power, the second power, the third power, we can write

out, we can square 1 plus x, and we can get 1 plus x cubed. And, out of it, we get this. And there would be 1 plus

x to the 0-th power. And do you see that there’s a

whole lot of ones in the neat pattern there? And then there’s a

2, and a 3, 3. And if you’d like to know this

one, it would be 1, 4, 6, 4, 1 would be the next

row of Pascal. Pascal really had a sense of

beauty or art in this triangle of numbers. And that’s the triangle you

get, Pascal’s triangle, if you’re taking– A whole number, a power is 1

plus x to the third power, fourth power, fifth power,

sixth power, but what if you’re taking to some other

power, any power, p? So now I’m interested in this

guy to a power of p that, maybe, is not two, three,

four, five. It could be 1/2, 1 plus x square

root to the 1/2 power, or 1 plus x to the

minus 1 power. All other powers are possible

and, for those, the Taylor’s theorem. And here’s my function. And I could apply Taylor’s

theorem to find the– and I’ll do it at x equals 0,

that’s the place Taylor liked the best. So the

constant term– think of this Taylor expansion

that we just did– at x equals 0, this

thing is 1. So, the big theory starts

out with a 1 for the constant term. Then what I do for the next

term of the Taylor Series? I take the derivative and

I put x equals 0. And what do I get then? I get p times x. So this is the constant

term: f of 0. This is the derivative:

times x minus a divided by 1 factorial. Well, you didn’t see all those

things because one factorial I didn’t write. And then the next term would be

the next derivative, of p minus 1 will come down, so

you’ll have p, p minus 1. You’re supposed to divide

by 2 factorial. That multiplies x squared. Well, my point is just that

this binomial formula is Taylor’s formula. The binomial theorem, with

these, this is called a binomial coefficient. Gamblers know all about

that, you know? If you’ve got p things and you

want to take two, how many ways to do it? You know, how many ways to get

two aces out of a deck, all these things are hidden in those

numbers, which gamblers learn or lose. OK. So, I’ll make one last point

about the binomial theorem. Those were Taylor Series. This is a Taylor series. What’s the difference? The difference is these

series stop. This is a series: 1

plus x squared. That’s the Taylor Series,

but the third derivative is 0, right? The third derivative of that

function, because that function’s only going up to x

squared, the third derivative is 0, so the rest of Taylor

Series has died. It’s not there. So that’s all there is. The derivative of any of those

powers, one, two, three, four, five powers, after I take enough

derivatives, gone. But, if I take a power like

minus 1, or 1/2, or pi, or anything, then I can take

derivatives forever without hitting 0. In other words, this series

goes on, and on, and on. Those three dots– let me move that eraser so you

see those three dots– that signals an infinite series

and the question of does it add up to a finite

number, what’s going on with infinite series? But, for the moment, my point

is just this is what calculus can do. If you not only take that slope,

but the slope of the slope, and the third derivative,

and all higher derivatives, that’s what Taylor

Series tells you. OK. So that’s the, in some way, high

point of the highlights of calculus, and I sure hope

they’re helpful to you. Thank you. FEMALE VOICE: This has been

a production of MIT OpenCourseWare and

Gilbert Strang. Funding for this video was

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