Pratique du Shadowing: Big Picture: Derivatives - Apprendre l'anglais à l'oral avec la vidéo
Création de la leçon...
1
OK, hi.
2
This is the second in my videos about the main ideas, the big picture of calculus.
3
And this is an important one because I want to introduce and compute some derivatives.
4
And you remember the overall situation is we have pairs of functions,
5
distance and speed, Function one and function two, height of a graph, slope of the graph,
6
height of a mountain, slope of a mountain.
7
And it's the connection between those two that calculus is about.
8
And so our problem today is you could imagine we have an airplane climbing.
9
Its height is y as it covers a distance x.
10
And its flight recorder will, well, probably it has two flight recorders.
11
Let's suppose it has.
12
Or your car has two recorders.
13
One records the distance, the height, the total amount achieved up to that moment,
14
up to that time t or that point x.
15
The second recorder would tell you at every instant what the speed is.
16
So it would tell you the speed at all times.
17
Do you see the difference?
18
The speed is like what's happening at an instant.
19
The distance or the height y is the total accumulation of how far you've gone, how high you've gone.
20
And now I'm going to suppose that this speed, this second function, the recorder is lost.
21
But the information is there and how to recover it.
22
So that's the question.
23
How if I have a total record, say, of height?
24
Let me, I'll stay mostly with y of x.
25
I write these two so that you realize that letters are not what calculus is about.
26
It's ideas.
27
And here is the central idea.
28
If I know the height as I go along, I know the height.
29
It could go down.
30
How can I recover from that height what the slope is at each point?
31
So here's something rather important.
32
That's the notation that Leibniz created, and it was a good idea for the derivative.
33
And you'll see where it comes from.
34
But somehow I'm dividing height distance up by distance across.
35
And that ratio of up to across is a slope.
36
So let me develop what we're doing.
37
So one thing we can do and now will do is,
38
for the great functions of calculus, a few very special, very important functions,
39
we will actually figure out what the slope is.
40
These are given by formulas, and I'll find a formula for the slope, dy dx equals.
41
And I won't write it in yet, let me keep a little suspense,
42
But this short list of the great functions is tremendously valuable.
43
The process that we go through takes a little time, but once we do it, it's done.
44
Once we write in the answer here, we know it.
45
And the point is that other functions of science,
46
of engineering, of economics, of life,
47
come from these functions by multiplying.
48
I could multiply that times that.
49
And then I need a product rule, a rule for the derivative, the slope of a product.
50
I could divide that by that.
51
And so I need a quotient rule.
52
I could do a chain, and you'll see that's maybe the best and most valuable, e to the sine x.
53
So I'm putting e to the x together with sine x in a chain of functions, e to the sine of x.
54
Then we need a chain rule.
55
That's all coming.
56
Let me start, well, let me even start by giving away the main facts for these three examples.
57
Because there are three you want to remember and might as well start now.
58
The x to the nth, so that's something, if n is positive, x to the nth climbs up.
59
Let me draw a graph here of y equal x squared.
60
Because that's one we'll work out in detail, y equal x squared.
61
So this direction is x, this direction is y, and I want to know the slope,
62
and the point is that that slope is changing as I go up.
63
So the slope depends on x.
64
The slope will be different here, so it's getting steeper and steeper.
65
I'll figure out that slope.
66
For this example, x squared when n is 2 and it's climbing.
67
If n was minus 2, just because that's also on our list,
68
n could be negative, the function would be dropping.
69
You remember x to the minus 2, that negative exponent means divide by x squared.
70
x to the minus 2 is 1 divided by x squared, and it'll be dropping.
71
Here, so n could be positive or negative here.
72
So I tell you the derivative.
73
The derivative is easy to remember.
74
It's that number n.
75
It's another power of x, and that power is one less.
76
One down.
77
You lose one power.
78
So I'm going to go through the steps here for n equal to 2.
79
So I hope to get the answer 2 times x to the 2 minus 1 will just be 1, 2x.
80
But what does the slope mean?
81
That's what this lecture is really telling you.
82
I'll tell you the answer for if it's sine x going in beautifully,
83
the derivative of sine x is cos x, the cosine.
84
The derivative of the sine curve is the cosine curve.
85
You couldn't hope for more than that.
86
And then we'll also at the same time find the derivative of the cosine curve, which is minus the sine curve.
87
It turns out a minus sine comes in because the cosine curve drops at the start.
88
OK.
89
And would you like to know this one?
90
e to the x, which I will introduce in a lecture coming very soon, because it's the function of calculus.
91
And the reason it's so terrific is the connection between the function,
92
whatever it is, whatever this number e is to whatever the x power means,
93
the slope is the same as the function, e to the x.
94
That's amazing.
95
The slope, as the function changes, the slope changes, and they stay equal.
96
OK, that's just, really my help is just to say, if you know those three, you're really off to a good start,
97
plus the rules that you need.
98
All right, now I'll tackle this particular one and say, what does slope mean?
99
OK.
100
So I'm given the recorder that I have.
101
This is function one here.
102
This is function one, the one I know.
103
And I know it at every point.
104
If I only had, like, the trip meter after an hour or two hours or three hours,
105
well, calculus can't do the impossible.
106
It can't know if I only knew the distance reached after each hour,
107
I couldn't tell what it did over that hour, how often you had to brake, how often you accelerated.
108
I could only find like an average speed over that hour.
109
That would be easy.
110
So averages don't need calculus.
111
It's instant stuff.
112
What happens at a moment?
113
What is that speedometer reading at the moment, say, x equal 1?
114
What is the slope?
115
Yeah.
116
Let me put in x equal 1 on this graph and x equal 2, say.
117
And now x squared is going to be at height 1.
118
If x is 1, then x squared is also 1.
119
If x is 2, x squared would be 4.
120
What's the average?
121
Let me just come back one second to that average business.
122
The average slope there would be in a distance across of 1, I went up by how much?
123
3. I went up from 1 to 4.
124
I have to do a subtraction.
125
Differences, derivatives, that's the connection here.
126
So it's 4 minus 1, that is 3.
127
So I would say the average is 3 over 1. But that's not calculus.
128
Calculus is looking at the instant thing.
129
And let me begin at this instant, at that point.
130
What does the slope look like to you at that point?
131
At x equals 0, here's x equals 0, and here's y equals 0.
132
We're very much 0.
133
You see it's climbing, but at that moment,
134
it's like it just started from a red light, whatever.
135
The speed is 0 at that point.
136
And I want to say the slope is 0.
137
That's flat right there.
138
That's flat.
139
If I continued the curve, if I continued the x squared curve for x negative, it would be the same as for x positive.
140
Whoa, it doesn't look very the same.
141
Let me improve that.
142
It would start up the same way.
143
It would be completely symmetric.
144
Everybody sees that at that zero position, the curve has hit bottom.
145
Actually, this will be a major, major application of calculus.
146
You identify the bottom of a curve by the fact that the slope is zero.
147
It's not going up, it's not going down.
148
It's zero at that point.
149
But now what do I mean by slope at a point?
150
OK.
151
Here comes the new idea.
152
If I go way over to one, that's too much.
153
I just want to stay near this point.
154
I'll go over a little bit.
155
And I call that a little bit delta x.
156
So that letter delta signals to our minds small.
157
Some small, actually probably smaller than I drew out there.
158
And then, so what's the average?
159
I'd like to just find the average speed or average slope.
160
If I go over by delta x, and then how high do I go up?
161
Well, the curve is y equal x squared.
162
So how high is this?
163
So the average is up first divided by across.
164
Across is our usual delta x.
165
How far did it go up?
166
Well, if our curve is x squared and I'm at the point delta x, then it's delta x squared.
167
That's the average over the first piece,
168
over the first piece of the curve.
169
Out from here, out to delta x.
170
OK.
171
Now, again, that's still only an average because delta x might have been short.
172
I want to push it to 0.
173
That's where calculus comes in.
174
Taking the limit of shorter and shorter and shorter pieces in order to zoom in on that instant,
175
that moment, that spot where we're looking at the slope and where we're expecting the answer 0 in this case.
176
And you see that the average in this happens to be especially simple.
177
Delta x squared over delta x is just delta x.
178
So the average slope is extremely small.
179
And I'll just complete that thought as so the instant slope at 0,
180
at x equals 0.
181
I let this delta x get smaller and smaller.
182
I get the answer 0, which is just what I expected.
183
And you could say, well, not too exciting.
184
But it was an easy one to do.
185
It was the first time that we actually went through the steps of computing a, this is like a delta y.
186
This is the delta x.
187
Instead of 3 over 1, starting here I had delta x squared over delta x.
188
That was easy to see, was delta x.
189
And if I move in closer, that average slope is smaller and smaller, and the slope at that instant is zero.
190
No problem.
191
The travel, the climbing, began from rest, but it picked up speed.
192
The slope here is certainly not zero.
193
We'll find that slope.
194
We need now to find the slope at every point.
195
OK.
196
So we've got, that's a good start.
197
Now I'm ready to find the slope at any point.
198
Instead of just x equals 0,
199
which we've now done, I better draw a new graph of the same picture climbing up.
200
Now I'm putting in a little climb.
201
At some point x here, I'm up at a height x squared, and I'd like to know at that,
202
I'm at that point on the current climb, I'd like to know the slope there at that point.
203
How am I going to do it?
204
I will follow this as the central dogma of calculus,
205
of differential calculus, function one to function two.
206
Take a little delta x.
207
Go as far as x plus delta x.
208
That will take you to a higher point on the curve.
209
That's now the point x plus delta x squared, because our curve is still y equal x squared.
210
We're not this nice, simple parabola.
211
OK.
212
So now I look at distance across and distance up.
213
So delta y is the change up.
214
Delta x is the across.
215
And I have to put, what is delta y?
216
I have to write in what is delta y.
217
It's this distance up.
218
It's x plus delta x squared.
219
That's this height minus this height.
220
So it's only, I'm not counting this bit, of course.
221
It's that that I want.
222
That's the delta y, is this piece.
223
So it's up to this, subtract x squared.
224
That's delta y.
225
Dun-dun.
226
That's important.
227
Now I divide by delta x.
228
This is all algebra now.
229
Calculus is going to come in a moment, but not yet.
230
For algebra, what do I do?
231
I multiply this out.
232
I see that thing squared.
233
I remember that x is squared.
234
And then I have this times this twice, 2x delta x's.
235
And then I have delta x squared.
236
And then I'm subtracting x squared.
237
So that's delta y written out in full glory.
238
I wrote it out because now I can simplify by canceling the x squared.
239
I'm not surprised.
240
Now, in this case, I can actually do the division.
241
Delta x is just there, so it leaves me with a 2x.
242
Delta x over delta x is 1.
243
And then here's a delta x squared over a delta x, so that leaves me with 1 delta x.
244
People who get, as you get the hang of calculus, you see the important things,
245
the important things is like this first order, delta x to the first power.
246
Delta x squared, that, when divided by delta x, gives us this, which is going to disappear.
247
That's the point.
248
This was the average over a short but still not instant range distance.
249
Now what happens?
250
Now dy dx.
251
So if this is short,
252
short over short, this is darn short over darn short.
253
That's, that d is the, is, well, it's too short to see.
254
So I don't actually now try to separate a distance dy.
255
This isn't a true division.
256
Because dy, because it's effectively 0 over 0.
257
And you might say, well, 0 over 0, what's the meaning?
258
Well, the meaning of 0 over 0 in this situation is I take the limit of this one,
259
which does have a meaning.
260
Because those are true numbers.
261
They're little numbers, but they're numbers.
262
And this was this.
263
So now here's the big step, leaving algebra behind, going to calculus.
264
In order to get what's happening at a point, I let delta x go to zero, and what is that?
265
So delta y over delta x is this.
266
What is dy dx?
267
So in the limit, ah, it's not hard.
268
Here's the 2x.
269
It's there.
270
Here's the delta x in the limit.
271
It disappears.
272
So the conclusion is that the derivative is 2x.
273
OK.
274
I should really, so that's function 2.
275
That's function 2 here.
276
That's the slope function.
277
That's the speed function.
278
Maybe I should draw it.
279
Can I draw it above, and then I'll put the board back up?
280
So here's a picture of function 2, the derivative, or the slope, which I was calling s.
281
So that's the s function against x x is still the thing that's varying, or it could be t, or it could be whatever letter we've got.
282
And we want to, and the answer was 2x for this function.
283
So if I graph it, it starts at 0, and it climbs steadily with slope 2.
284
So that's a graph of s of x.
285
And for example, yeah, so take a couple of points on that graph.
286
At x equals 0, the slope is 0, right?
287
And we did that first.
288
And we actually got it right.
289
The slope is 0 at the start, at the bottom of the curve.
290
At some other point on the curve, what's the slope here?
291
Huh, yeah, tell me the slope there.
292
At that point on the curve, an average slope was 3 over 1, but that was the slope of this,
293
like, you know, sometimes called a chord.
294
That's over a big jump of 1.
295
Then I did it over a small jump of delta x, and then I let delta x go to 0, so it was an instant, infinitesimal jump.
296
So the actual slope, the way to visualize it is that it's more like that.
297
That's the line that's really giving the slope at that point.
298
Do you see my best picture?
299
It's not Rembrandt, but it's got it.
300
And what is the slope at that point?
301
Well, that's what our calculation was.
302
It found the slope at that point and at the particular point x equal 1.
303
The height was 2.
304
The slope is 2.
305
The actual tangent line is only there.
306
You see it?
307
It's up, up.
308
Oh, wait a minute.
309
Yeah, well, the slope is 2.
310
I don't know.
311
Yeah, maybe, yeah, it goes up to 3.
312
It's not Rembrandt.
313
But the math is OK.
314
So what have we done?
315
We've taken the first small step, and literally I could say small step almost a play on words because that's the point,
316
this step is so small, to getting these great functions.
317
Before I close this lecture, can I draw this pair, function one and function two.
318
And just see that the movement of the curves is what we would expect.
319
So let me just for one more good example, great example actually, is let me draw.
320
Here goes x.
321
I'm going to draw.
322
In fact, maybe I already drew in the first lecture that bit out to 90 degrees.
323
Only if we want a nice formula, we better call that pi over 2 radians.
324
And here is a graph of sine x.
325
This is y.
326
This is function 1 sine x.
327
And what's function 2?
328
What can we see about function 2?
329
Again, x.
330
We see a slope.
331
This is not the same as x squared.
332
This starts with a definite slope.
333
And it turns out this will be one of the most important limits we'll find.
334
We'll discover that the first little delta x, which goes up by sine of delta x,
335
has a slope that gets closer and closer to 1.
336
Good.
337
Luckily, cosine does start at 1.
338
So we're OK so far.
339
Now the slope is dropping.
340
And what's the slope at the top of the sine curve?
341
It's a maximum, but we identify that by the fact that the slope is 0,
342
because we know the thing is going to go down here and go somewhere else.
343
The slope there is zero.
344
The tangent line is horizontal.
345
And that is that point.
346
It passes through zero.
347
The slope is dropping.
348
So this is the slope curve.
349
And the great thing is that it's the cosine of x.
350
And what I'm doing now is not proving this fact.
351
I'm not doing my delta x's.
352
That's the job I have to do once, and it won't be today, but I only have to do it once.
353
And it will tell me, but today I'm just saying it makes sense.
354
The slope is dropping.
355
The slope, in that first part I'm going up, so the slope is positive, but the slope is dropping.
356
OK, and then at this point it hits zero, and that's this point, and then the slope turns negative.
357
I'm falling.
358
So the slope goes negative, and actually it follows the cosine.
359
So I go along here to that point,
360
and then I can continue on to this point where it bottoms out again and then starts up.
361
So where is that on this curve?
362
Well, I better draw a little further out.
363
This bottom here would be the, this is our pi over 2.
364
This is our pi, 180 degrees, everybody would say.
365
So what's happening on that curve?
366
The function is dropping, and actually it's dropping its fastest.
367
It's dropping its fastest at that point, which is the slope is minus 1.
368
And then the slope is still negative, but it's not so negative, and it comes back up to 0 at 3 pi over 2.
369
So this is the point 3 pi over 2.
370
And this has come back to 0 at that point.
371
And then it finishes the whole thing at 2 pi.
372
This finishes up here back at 1 again.
373
It's climbing.
374
All right, climbing, dropping, faster,
375
slower, maximum, minimum, those are the words that make derivatives important and useful to learn.
376
And we've done in detail the first of our great list of functions.
377
Okay, thanks.
378
This has been a production of MIT OpenCourseWare and Gilbert Strang.
379
Funding for this video was provided by the Lord Foundation.
380
To help OCW continue to provide free and open access to MIT courses,
381
please make a donation at ocw.mit.edu slash donate.
📺 Même chaîne
C12
11. Introduction to Machine Learning
Business EnglishUniversity StudentsWorking Adults
C1Lec 1 | MIT 18.06 Linear Algebra, Spring 2005
TOEFL PrepCambridge Exam PrepUniversity Students
C28
Independence, Basis, and Dimension
University Students
C1Lec 1: Introduction to Principles of Microeconomics and Supply & Demand
TOEFL PrepBusiness EnglishUniversity Students
✨ Vidéo recommandée
C12
138 phrases
Derivatives: Crash Course Physics #2
Cambridge Exam PrepUniversity StudentsWorking Adults
C13
221 phrases
Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C11
181 phrases
The Limit (do not use L'Hospital rule)
Cambridge Exam PrepUniversity StudentsWorking Adults
C12
257 phrases
Video
Cambridge Exam PrepHigh School StudentsUniversity Students
C11
425 phrases
NEW- Micro Unit 1 Summary- Basic Economic Concepts
Cambridge Exam PrepUniversity Students
C12
55 phrases
Convergence and Divergence: The Return of Sequences and Series
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C12
114 phrases
Newton, Leibniz, and Usain Bolt | Derivatives introduction | AP Calculus AB | Khan Academy
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
B214
170 phrases
Why a Circle Has 360 Degrees? | The History and Logic Behind 360° Explained with Animation
Cambridge Exam PrepHigh School StudentsUniversity Students
C1252 phrases
Mes arguments philosophiques préférés
IELTS PrepTOEFL PrepCambridge Exam Prep
B12
9 phrases
Simple Machines – Inclined Planes
TOEFL PrepCambridge Exam PrepHigh School Students
C11
301 phrases
Lecture 01. Introduction to the Course - Philosophy of the Natural Sciences (Univ of New Orleans)
TOEFL PrepCambridge Exam PrepUniversity Students
Vocabulaire et conseils d’expression pour cette leçon
Cette vidéo contient 381 phrases et 3465 mots à répéter en shadowing. La partie parlée dure 29:39. Le locuteur parle à un rythme régulier d’environ 117 mots par minute, confortable pour le shadowing. Seuls 79 % des mots font partie des 3 000 mots les plus courants en anglais, le vocabulaire est donc exigeant.
Vocabulaire clé de cette vidéo
15 mots moins courants de la vidéo, avec leur prononciation et leur sens :
- slope /sloʊp/ (nom) — pente. A surface that either inclines or declines in a regular manner from the perspective of an observer.
- delta /ˈdɛltə/ (nom) — delta. The fourth letter of the modern Greek alphabet Δ, δ.
- curve /kɚv/ (nom) — courbe. A bend, such as in a road.
- calculus /ˈkæl.kjə.ləs/ (nom) — calcul. Any formal system in which symbolic expressions are manipulated according to fixed rules.
- derivative /dɪˈɹɪvətɪv/ (nom) — dérivé. Something derived.
- climb /ˈklaɪ̯m/ (verbe) — grimper, monter. To traverse a slope or contour, either upwards (ascending) or downwards (descending).
- instant /ˈɪnstənt/ (nom) — instant. A very short period of time; a moment.
- graph /ɡɹæf/ (nom) — graphique. A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or…
- minus /ˈmaɪ.nəs/ — moins. Made less or reduced by (followed by an expression of number or quantity).
- divide /dɪˈvaɪd/ (verbe) — diviser, fendre. To split or separate (something) into two or more parts.
- recorder /ɹiˈkɔɹ.dəɹ/ (nom) — enregistreur. An apparatus for recording; a device which records.
- lecture /ˈlɛk.t͡ʃə/ (nom) — conférence, cours magistral. A spoken lesson or exposition, usually delivered to a group.
- formula /ˈfɔː.mjʊ.lə/ (nom) — formule. Any mathematical rule expressed symbolically.
- algebra /ˈæl.d͡ʒɪ.bɹə/ (nom) — algèbre. Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations.
- introduce /ˌɪntɹəˈdus/ (verbe) — présenter. To cause (someone) to be acquainted (with someone else).
Les verbes à particule que vous entendrez
- go up (verbe) — monter, s'élever. To move upwards.
- go down (verbe) — descendre. To descend; to move from a higher place to a lower one.
- look at (verbe) — regarder. To observe or watch.
- figure out (verbe) — cerner. To come to understand; to discover or find a solution; to deduce.
- go through (verbe) — traverser. To travel from one end of something to the other.
- turn out (verbe) — se finir, se terminer. To end up; to result.
- back up (verbe) — reculer. To move backwards, especially for a vehicle to do so.
- come back (verbe) — revenir. To return to a place.
Prononciation à surveiller
Le locuteur utilise 65 contractions et formes réduites, comme I'm, I'll, we're. Prononcez-les sous leur forme courte, telles que vous les entendez.
- Les sons « sh » et « zh »: conclusion /kənˈkluːʒən/, ratio /ˈɹeɪ.ʃoʊ/, donation /doʊˈneɪʃən/, differential [dɪfəˈɹənʃəɫ], calculation /ˌkælkjuˈleɪʃn̩/
- Mots longs — placez bien l’accent: derivative /dɪˈɹɪvətɪv/, economics /ˌiːkəˈnɒmɪks/, mathematics /mæθ(.ə)ˈmæt.ɪks/, beautifully /ˈbju.tɪ.fə.li/, horizontal /ˌhoɹəˈzɑn.təl/
Comment s’entraîner avec cette vidéo
- Écoutez la vidéo en entier une fois sans parler et notez les mots que vous ne connaissez pas.
- Répétez phrase par phrase à vitesse normale, en reprenant chacune jusqu’à ce que votre rythme corresponde à celui du locuteur.
- Enregistrez-vous et comparez avec l’original, en faisant attention à des mots comme slope, delta, curve.
La grammaire de cette vidéo
Les structures que le locuteur utilise le plus, avec les mots exacts de la vidéo :
| Structure | Dans la vidéo |
|---|---|
| Present perfect have/has + participe passé — une action passée qui compte encore maintenant | you've gone · has hit · we've now done |
| Voix passive be + participe passé — l’accent est mis sur ce qui arrive, pas sur qui le fait | is lost · are given · is squared |
| Propositions relatives who / which + proposition — une précision sur une personne ou une chose | curve, which is · People who get · this, which is |
Qu'est-ce que la technique du Shadowing ?
Le Shadowing est une technique d'apprentissage des langues fondée sur la science, développée à l'origine pour la formation des interprètes professionnels. Le principe est simple mais puissant : vous écoutez de l'anglais natif et le répétez immédiatement à voix haute — comme une ombre suivant le locuteur avec un décalage de 1 à 2 secondes. Les recherches montrent une amélioration significative de la précision de la prononciation, de l'intonation, du rythme, des liaisons, de la compréhension orale et de la fluidité.