Big Picture of Calculus

Big Picture of Calculus

PROFESSOR: Hi, I’m Gilbert
Strang, and this is the very first in a series of videos
about highlights of calculus. I’m doing these just because
I hope they’ll be helpful. It seems to me so easy to be
lost in the big calculus textbooks and the many, many
problems and in the details. But do you see the
big picture? Well, I hope this will help. For me, calculus is about the relation between two functions. And one example for those two
functions, one good example, is function 1, the distance,
distance traveled, what you see on a trip meter in a car. And function 2, the one that
goes with distance, is speed, how quickly you’re going, how
fast you’re traveling. So that’s one pair
of functions. Let me give another pair. I could get more and more, but
I think if we get these two pairs, we can move forward. So in this second pair,
height is function 1, how high you’ve climbed. If it’s a graph, how far the
graph goes above the axis. Up, in other words. So that’s height, and then
the other one tells you how fast you climb. The height tells how
far you climbed. It could be a mountain. And then the slope tells you how
quickly you’re climbing at each point. Are you going nearly
straight up? Flat? Possibly down? So distance and speed, height
and slope will serve as good examples to start with. And let me give you some
letters, some algebra letters that you might use. Distance, maybe I would
call that f of t. So f for how far or for
function, and the idea is that t is the input. It’s the time when you’re
asking for the distance. The output is the distance. Or in the case of height,
maybe y of x would be the right one. x is how far you go across. That’s the input. And at each x, you have an
output y how far up? So f is telling you how far. y is telling you the
height of a graph. That’s function 1, two examples
of function 1. Now, what about slope? Well, luckily, speed and slope
start with the same letter, so I’ll often use s for the speed
or the slope for this second– oh, it even stands for
a second function. But let me tell you
also the right– the official– letters that make the connection
between function 2 and function 1. If my function is a function of
time, the distance, how far I go, then the speed is– the right letters are df dt. Everybody uses those letters. So let me say again how
to pronounce: df dt. And Leibniz came up with
that notation, and he just got it right. And what would this one be? Well, corresponding to this, it
looks the same, or dy dx. Again, I’ll just repeat how
to say that: dy dx. And that is the slope, and we
have to understand what those symbols mean. Right now, I’m just writing
them down as symbols. May I begin with the most
important and the simplest example of all? Let me take that case. OK, so the key example here,
the one to get completely straight is the case
of constant speed, constant slope. I’ll just graph that. So here I’m go to graph. Shall I make it the speed? Yeah, let’s say speed. So time is going
along that way. Speed is up this way. And I’m going to say in this
first example that the speed is the same. We’re traveling at the
constant speed of let’s say 40. So it stays at the
height of 40. Oh, properly, I should add units
like miles per hour or kilometers per hour or meters
per second or whatever. For now, I’ll just write 40. OK, now if we’re traveling at a
speed of 40 miles per hour, what’s the distance? Well, let me start with the
trip meter at zero. so this is time again,
and now this is going to be the distance. After one hour, my
distance is 40. So if I mark t equal to
1, I’ve reached 40. That’s height of 40. At t equal to 2, I’ve
reached 80. At t equal to 1/2, half an
hour, I’ve reached 20. Those points lie on a line. The graph of distance covered
when you’re just traveling at a steady rate, constant rate,
constant speed is just a straight line. And now I can make
the connection. I’ve been speaking here about
distance and speed. But now let me think of
this as the height– 40 is that height. 80 is that height– and ask about slope. What is slope? So let’s just remember what’s
the connection here. What’s the slope if that’s the
distance if I look at my trip meter and I know I’m traveling
along at that constant speed, how do I find that speed? Well, slope, it’s the distance
up, which would be 40 after one hour, divided
by the distance across, 40/1, or 80/2. Doesn’t matter, because we’re
traveling at constant speed, so the slope, which is up,
over, across is 40/1, 80/2, 20 over 1/2. I’ll put 80/2 as one
example: 40. Oh, let me do it– that’s arithmetic. Let me do it with algebra. We don’t need calculus
yet, by the way. Calculus is coming
pretty quickly. This is the step we can take. Because the speed is constant,
we can just divide the distance by the time to find– and this slope, let me
right speed also. Up, over, across, distance
over time, f/t, that gives us s. This is s. OK, what about– calculus goes both ways. We can go both ways here. We already have practically. Here I went in the direction
from 1 to 2. Now, I want to go in
the direction– suppose I know the speed. How do I recover the distance? If I know my speed is 40 and I
know I started at zero, what’s my distance? Distance or height,
either one. So these are like both. Now, I’m just going
the other way. Well, you see how. How do I find f? It’s s times t, right? Your algebra automatically says
if you see a t there, you can put it there. So it’s s times t. It’s a straight line. s times t, s times
x, y equal sx. Let me put another– the same idea with
my y, x letters. It’s that line. In other words, if that one
is constant, this one is a straight line. OK, straightforward, but
very, very fundamental. In fact, can I call your
attention to something a little more? Suppose I measured between
time 2 and time 1. So I’m looking between time 2
and time 1, and I look how far I went in that time. But what I’m trying is– I’m going to put in another
little symbol because it’s going to be really
worth knowing. It’s really the change in f
divided by the change in t. I use that letter delta
to indicate a difference between– the difference between time 2
and time 1 was 1, and the difference between height
2 and height 1 was 40. You see, I’m looking at
this little piece. And, of course, the
slope is still 40. It’s still the slope
of that line. Yeah, so that really what I’m
measuring in speed there, I don’t always have to be starting
at t equals 0, and I don’t always have to be starting
at f equals 0. Oh, let me draw that. Suppose I started
at f equals 40. My trip meter happened
to start at 40. After an hour, I’d
be up to 80. After another hour,
I’d be up to 120. Do you see that this starting
the trip meter, who cares where the trip meter started? It’s the change in the trip
meter that tells how long the trip was. Clear. OK, so that’s that example. We come back to it because it’s
the basic one where the speed is constant. And even if now I have to move
to a changing speed, you have to let me bring calculus
into these lectures. OK, I’m going to draw another
picture, and you tell me about the– yeah, let me draw function 1,
another example of function 1. So again I have time. I have distance. I’m going to start at zero, but
I’m not going to keep the speed constant. I’m going to start out at
a good speed, but I’m going to slow down. Do you see me slowing
down there? I don’t mean slowing down
with the chalk. I mean slowing down
with slope. The slope started out steep. By here, by that point,
the slope was zero. What was the car doing here? The car is certainly moving
forward because the distance is increasing. Here it’s increasing faster. Here it’s increasing barely. In other words, we’re putting
on the brakes. The car is slowing down. We’re coming to a red light. In fact, there is the red light
right at that time. Now, just stay with it to think
what would the speed look like for this problem? If that’s a picture of
the function, just let’s get some idea. I’m not going to have
a formula yet. I’m not putting in
all the details. Well, actually, I don’t plan to
put in all the details of calculus of every possible
step we might take. It’s the important ones I’m
hoping to show you and I’m hoping for you to see that
they are important. OK, what is important? Roughly, what does the
graph looks like? Well, the speed– the slope– started out somewhere
up there. Yeah, it started out at a good
speed and slowed down. And by this point, ha! Let’s mark that time
here on that graph. Do you see what is the
speed at that moment? The speed at that
moment is zero. The car has stopped. The speed is decreasing. Let me make it decrease,
decrease, decrease, decrease, and at that moment, the speed
is zero right there. That’s that point. See, two different pictures,
two different functions. but same information. So calculus has the job of given
one of those functions, find the other one. Given this function,
find that one. This way is called– from function one to function
two, that’s called differential calculus. Big, impressive word anyway. That’s function one to two,
finding the speed. Going the other direction is
called integral calculus. The step is called integration
when you take the speed over that period of time, and you
recover the distance. So it’s differential calculus
in one direction, integral calculus in the other. Now, here’s a question. Let me continue that curve
a little longer. I got it to the red light. Now imagine that the distance
starts going down from that point. What’s happening? The distance is decreasing. The car is going backwards. It’s going in reverse. The speed, what’s the speed? Negative. The speed, because distance is
going from higher to lower, that counts for negative
speed. The speed curve would
be going down here. Do you see that that’s a not
brilliantly drawn picture, but you’re seeing the– that’s the farthest it went. Then the car started backwards,
and the speed curve reflected that by going
below zero. You see, two different curves,
but same information. I’m remembering an old movie. I don’t know if you saw an
old B movie called Ferris Bueller’s Day Off. Did you see that? So the kid had borrowed
his father’s– not borrowed, but lifted his
father’s good car and drove it a lot like so and put
on a lot of mileage. The trip meter was way up, and
he knew his father was going to notice this. So he had the idea to put the
car up on a lift, put it in reverse, and go for a
while, and the trip meter would go backwards. I don’t know if trip meters
do go backwards. It’s kind of tough to watch them
while going in reverse. But if whoever made the car
understood calculus, as you do, the speedometer– now that I think of it,
speedometers don’t have a below zero. They should have. And trip
meters should go backwards. I mean, that movie was just made
for a calculus person. Maybe I’m remembering more. I think it didn’t work
or something. And the kid got mad and kicked
the car, and it fell off the lift, went through
the glass window. Anyway, calculus would
have saved him if only the car had been– or the meters in the car had
been made correctly. All right, that’s one pair. That’s our first real pair in
which the speed changes. OK. I thought in this first video,
later, even today, I’ll get to a case where we have formulas. That’s what calculus
moves into. When f of t is given by some
formula, well, here it’s given by a formula: s times t. A simple formula. And then, knowing that, we
know that the speed is s. Later, we got more functions. But let me take an example, just
because these pairs of functions are everywhere. What could I take? Maybe height of a person. Height of a person. OK, so this is now another
example, just to get practice in the relation between the
height of a person and the rate of change of the height. So this is the height. Maybe I’ll call it y. Let me write height
of a person. And what is this going to be? What is function two? Well, slope doesn’t
seem quite right. The point about function two is
it tells how fast function one changes. It’s the rate of change
of the height. It’s the rate of change. So let me call it s, and it’ll
be the rate of change. Good if I use those words. Yeah, so I want to think
just how we grow, a typical person growing. In fact, as I wrote this on
the board, I thought of another pair. Can I just say it in words, this
other pair, and then I’ll come back to this one? Here’s another pair. This could be money in a bank. Wealth sounds better. Let’s call it wealth. That’s zippier. And then what is this one? If this is your wealth,
your total assets, what’s your worth? This would be the rate
of change, how quickly you’re saving. s could be for saving. Or if you’re down here, s
is for spending, right? If s is positive, that means
you’re wealth is increasing, you’re saving. Negative s means you’re
spending, and your wealth goes whatever, maybe– I hope– up. Height is mostly up, right? So let me come back to
height of a person. Now, where– oh, and this is time in years. This is t in years, and
this, too, of course. Actually, I realize
you started at t equals zero: birth. You do start at a certain– actually, what do I know? You don’t say tall. You say long. But then as soon as you can
stand up, it’s tall, so let’s say tall. Shall we guessed 20 inches? If that’s way off, I apologize
to everybody. Let me just say 20, 20 inches. OK, at year zero. OK, and then presumably
you grow. OK, so you grow a little. What are we headed for? About 60, 70 inches
or something. Anyway, you grow. Let’s say that’s 10 years old
and here is 20 years old. OK, so you grow. Maybe you grow faster
than that. Let’s say you’re a healthy
person here. OK, up you grow. And then at about maybe age 12
or 13, there’s a growth spurt. And maybe the point is, how do
we see that growth spurt on the two graphs? Differently, but it’s the
same growth spurt. OK, so here your height
suddenly jumps up. Boy, yeah, you catch
up with everybody. And then at about 12 or 13 well,
then unfortunately, it doesn’t do that forever, and
it kind of levels off here. It levels off, and actually
you don’t grow a whole lot more. In fact, I think when you get
to about– oh, I don’t know. Whatever. We won’t discuss this point. I say when you get too old,
you probably lose some. Let’s not emphasize that. OK, so here is the– now, what’s happening
over here? Well, it’s the slope
of that graph. So the slope might be– this is time zero, but you’re
growing right away. The s graph, the rate
of growth graph, doesn’t start at zero. It starts how fast you’re
growing, whatever you’re growing, whatever
that slope is. It’s fantastic that when we
draw graphs of things, the word “slope” is suddenly
the right word. OK, so you’re growing, maybe
at a pretty good rate here. And let me mark out
10 and 20 years. And OK, you’re doing well,
you’re coming along here, and then the growth spurt. OK, so then suddenly, your
rate of growth takes off. But it doesn’t stay
that way, right? Your rate of growth levels
off, in fact, levels way off, levels– you’ll come down to here, and
you probably don’t grow a lot. Do you see the two? This was the growth curve. This was the fast growth. But then it stopped. Up here, it slowed down. Here, it dropped. And oh, if we allow for this
person who lived too long, height actually drops. OK, there is an example
in which I don’t– also I’m sure people have
devised approximate formulas for average growth rates,
but you see, I’m not– it’s the idea of the relation
between function one and function two that
I’m emphasizing. Now, my last example, let me
take one more example, one more example for this
first lecture. So let me take a case in
which the speed is– so here will be my two– let’s use speed. Let’s use this as distance. This is distance again,
and graph two, as always, will be speed. And I’m going to take
a case in which it’s given by a formula. I’m going to let the speed
be increasing steadily. OK, so my speed graph this time
is going to go up at a constant rate. So this is the speed s. This is the time t. So this would be s equals– s is proportional to t. That’s where you get
a straight line. s is let’s say a times t. That a, a physicist, if we were
physicists, would say acceleration. You’re accelerating. You’re keeping your foot on the
gas, steadily speeding up, and so then s is proportional
to t. Now, think about the distance. What’s happening
with distance? If this is accelerating, you’re going faster and faster. You’re covering more and more
speed, more and more distance, more and more quickly. If this is slope, the
slope is increasing. Look, the graph– let’s start the trip
meter at zero. So you started with
a speed of zero. You were not really increasing
distance until you got slightly beyond zero,
and then it slightly started to increase. But then it increases faster
and faster, right? It never gets infinitely fast,
but it keeps going upwards. And the calculus question
would be can we give a formula– an equation– for the distance? Because in this case, I guess
I started with function two, and therefore, it’s function
one that I want to look at. It’s always pairs
of functions. OK, now, let’s think where this
would actually happen. If we were leaning over the
Tower of Pisa or whatever, like Galileo, and drop
something, or even just drop something anywhere, that
would be– we drop it. At the beginning, it has no
speed, but of course, instantly it picks up speed. The a would have something to
do with the gravitational constant for the Earth,
whatever, and then maybe– yeah. And what would be
the distance? OK, now can I just mention a
small miracle of calculus? A small miracle. I’m going this direction now
from speed to distance so I’m doing integral calculus. And we’ll get to that later. In the first lectures,
we’re almost always going from one to two. But here is a neat fact about
going from two to one, that if this is the time t, then, of
course, this height here will be a times t. And the amazing fact is that
this graph tells you the area under this one. Graph one– function one– tells you the area under
the graph two. And in this example with a nice
constant acceleration, steady increase in speed,
we know this. This is a triangle. It has a base of t. It has a height of at. And the area of a triangle, of
course, the area, and my point is that the area is
function one– amazing; that’s just
terrific– will be– the area of this
triangle here is 1/2 of the base times the height. That’s the area, and calculus
will tell us that’s function one. So this function is 1/2
of a times t squared. So there is a function one,
and here is df dt. If I go back to the first
letters that I mentioned, if this is my function f, then
this is my function that– and notice what kind
of a curve that is. Do you recognize that
with a square? That tells me it’s a parabola,
a famous and important curve. And, of course, it’s important
because it has such a neat formula. OK, so we have found
the function one. We’ve recovered the information
in that lost black box, the distance box, the trip
meter, from what we did find in black box two, the
speed, the record of speed. And notice, I’m using the
speed all the way from here to here. The speed kind of tells me how
the distance is piled up. The distance is kind of a
running total, where the speed at that moment is an
instant thing. Oh, we have to do that
in future lectures. The difference between a
running total of total distance covered and a speed
that’s telling me at a moment, at an instant how distance
is changing. The slope at this very point
t, that slope is this height, is at. OK, so there you have
the first– well, I’ll say the second. The first pair of calculus
was this one. f equals st, and it’s
derivative was s. Our second pair is f is this,
and will you allow me to write df dt? If f is 1/2 of at squared,
then df dt is at. You’ll see this rule again. The power two dropped
to a power one. But the two multiplied the thing
so it canceled the 1/2 and just left the a. OK, that’s a start on the
highlights of calculus. Thanks. NARRATOR: This has been
a production of MIT OpenCourseWare and
Gilbert Strang. Funding for this video was
provided by the Lord Foundation. To help OCW continue to provide
free and open access to MIT courses, please
make a donation at

Only registered users can comment.

  1. Thank you MIT for uploading Professor Strang's Calculus Student's study guide. Just saw it few hours ago. I had commented about it and where to find it.

  2. God bless this great man : How I wish I had him as my Math teacher when I was young. He would have made it more interesting. When I was a kid I used to wonder why on earth would this be of so much importance in my life and brushed it off ( also my math teachers were cunts who never drilled on the subject well ) This man will straight go to the Lord's world . He is mind blowing

  3. Wow! This guy just made calculus look like a toy – no limits, no infinitesimals, no epsilons needed. Mind Blown!

  4. Taking calculus this year as a grade 12. I've always been good at math and I'm fairly good when it comes to calculus but somehow an MIT professor managed to make these concepts difficult for me to understand????

  5. Gilbert is amazing. I remember one course i took where my professor was so bad. He could write two boards full of math and end up with 2=1 haha. I had to stop going to his lectures. It just so happened that all the topics we read were in a course gilbert taught that was available on youtube. I passed the test even though english is a foreign language to me.

  6. at 34:30 he states that speed = acceleration multiplied by time; however acceleration = speed squared and the area under the line cannot be one half times speed squared times time squared. The area is one half times speed times time. Something doesn't add up here …

  7. My idea regarding calculus enhanced dramatic. I am self learner who is fond of maths and physics and I have been learning a lot from professor Strang.

  8. ha ha deciduous trees function of do you season. temperature inclination of the planet .strange arrange ha ha change niel bohr Wolfgang Pauli 'tippe top' elementary up down top bottom strange charm. philosophical tree k-now grow go. spin doctor child play

  9. This Tutor can me exactly, what is this again? I know you are myself, Please Valen smith is my Wife in Black skin.

  10. I majored in Math, in the early 70's. I "thought" I had good professor's…now I know, perhaps they could have done a better job. EVERY Calculus professor, should begin his first Calc1 class with these examples. Actually, I'm embarrassed to say this, but I really didn't understand the "application" of Calculus, until AFTER I earned my BS/Math degree, and studied Civil Engineering. YES, I WAS EMBARRASSED. Physics taught me the "application" of Calculus!

  11. Why, America??? Why do you spew out these dumbed down, tabloid newspaper style deliveries??? Why not have a normal person speaking normally???

  12. Thank you sir for such great videos. Now I am finally able apply calculus in mechanics. Love from India

  13. Thank you very much proff you are a definition of maths…God bless you Banda castern from Zambia.

  14. I like how you explain that calculus is the relationship between functions and you show how the function can be represented by other letters other than f and that the variable function can be represented by any letter inside the parentheses. Made a lot of sense. Everything you discuss in this video was helpful and enlightening. Thank you so much.

  15. The last two minutes were a little confusing for a beginner like me. Otherwise great explanation. Taking notes 40 years after high school. 🙂

  16. I've decided to donate MIT OCW, once everything starts to payoff(as i'm still a student), thank you MIT..these lectures solved my biggest mystery!❤

  17. Laughed at the 20 inches for height/length comment- he's right on the money!

  18. Last night I was crying because I can't understand calculus,I watched many videos of teachers explains it,but They're explains the mathmatical side only,now is the first time I feel that I'm begaining to understand claculus:")💖
    And so excited to do that!
    Thank you professor,thank you from all of my heart💜

  19. I failed calculus the first time I took it because I got a bad teacher. When I got a good one it was so easy to learn. It just seems so easy once you really understand it. Every college student should learn it. If you put in the time anyone can learn it.

  20. why is he winking? I dont know what he's trying to express, do I know something I should? Is he dropping double sense jokes in the class? I DONT GET IT!!!

  21. Where were u when I needed you sir makes me coz when I was I did love maths but teachers couldn’t convey what calculus ReAly is bit sir you av managed to dimistify this subject for me some what I wish I can study maths again

  22. Wow great teacher you are Prof. Strang….I have really learnt a lot from this video…Thumbs up MIT for making this video public…Am gonna watch all of them….

  23. I got C score on Calculus about 17 years ago. And just understood the fundamental of Calculus by watching this. Lol. Its never too late to fix the failure. 💪

    Thank you so much.

  24. If comments collectively 'is'( not 'are' since using 'collectively') a function, namely 'c', of viewers reflections(thoughts/opinions) over time 't', thus d{sub}c/d{sub}t = a*t(comments area{a) traversed over time{t}), then youtube's overall 'comments' section is the 'integral'(function) of that collective over time, thus graphically a 'parabola' alphanumerically represented as 1/2a*t∧2, where '∧' means to the power of. If youtube operates to man's distinction, or independent of man, unto infinity, so will the integral(function) be represented by a 'parabola' stretching to that infinity.

  25. I think I didn’t understand calculus years ago due to my teachers who weren’t that good
    They didn’t understand it themselves
    There is a say: if you understand something, you will be able to explain it to someone else . and if they don’t understand it, it means you didn’t grasp it yourself

  26. What I've discovered through experimentation: This is true for all disciplines. There is a huge difference between being a specialist and knowing some Calculus. For example, I'm pretty good at life science, but I'm not a specialist. I am, however, a specialist at mathematics, and I'm a half way decent physicist too. Calculus is not just a big word. It's hard to master. The same is true for Chemistry, Biology, etc. Mathematics is my specialty area, and I'd say I do fairly well in physics too.

  27. Calculus could also be called "the perceived condensation of Time", the Spacetime coordination program of the ordinary everyday Universe.

  28. In a car what’s important is the magnitude, or the absolute value, of the milagre |x| and not the direction or the sign; which in turn is the measure of how much a car has been used (be it in the forward or the reverse direction). In other words, if the wheels are turning, you’re putting mileage on the car, regardless of the direction. When it comes to a car, in fact, you can’t look at the algebraic sum of the mileage (in the forward or reverse directions) but the overall magnitude of the mileage that’s been put on the car. If you use calculus, in this particular example, you’re either fooling yourself, or else cheating a potential buyer, with regards to the total number of miles (the absolute value) that the car has being utilized.

  29. Brilliant; you activated my neurons – well, lack of neurons-, on this subject since I'm naive about it. But now I don't see it as difficult as before, gives me the desire to continue. Thank you.

  30. Consider myself so very fortunate to have free uni. lectures such as this on the internet. Thank you Prof. Strang.

  31. No matter how many “big picture” videos i watch. I still don’t understand lol. Maybe when i start my job i will

  32. I visited courses on an actually very well reputational university, but they never teach you the basics … they always go right into the guts and details, and you wonder why you are even here, watching all those details. When watching videos like these i sometimes wonder if it is not intentional dumb keeping. Well, let's praise the internet, and foremost, let's praise MIT opencoursware and Gilbert Strang for these amazing things we are able to witness right now! I am truly blessed to live in these times, where information is so freely available.

  33. From mannerisms alone, one can tell of the level of dedication to Mathematics this professor has, I think it's amazingly beautiful!!!!

  34. the realization of – "at birth you do start at a certain height" i loved this lmao (also "when you stand up it's tall" i love his humor 😂😂)

Leave a Reply

Your email address will not be published. Required fields are marked *