ฝึกพูดภาษาอังกฤษด้วยเทคนิค Shadowing จากวิดีโอ: Big Picture: Derivatives

OK, hi.

This is the second in my videos about the main ideas, the big picture of calculus.

And this is an important one because I want to introduce and compute some derivatives.

And you remember the overall situation is we have pairs of functions,

distance and speed, Function one and function two, height of a graph, slope of the graph,

height of a mountain, slope of a mountain.

And it's the connection between those two that calculus is about.

And so our problem today is you could imagine we have an airplane climbing.

Its height is y as it covers a distance x.

And its flight recorder will, well, probably it has two flight recorders.

Let's suppose it has.

Or your car has two recorders.

One records the distance, the height, the total amount achieved up to that moment,

up to that time t or that point x.

The second recorder would tell you at every instant what the speed is.

So it would tell you the speed at all times.

Do you see the difference?

The speed is like what's happening at an instant.

The distance or the height y is the total accumulation of how far you've gone, how high you've gone.

And now I'm going to suppose that this speed, this second function, the recorder is lost.

But the information is there and how to recover it.

So that's the question.

How if I have a total record, say, of height?

Let me, I'll stay mostly with y of x.

I write these two so that you realize that letters are not what calculus is about.

It's ideas.

And here is the central idea.

If I know the height as I go along, I know the height.

It could go down.

How can I recover from that height what the slope is at each point?

So here's something rather important.

That's the notation that Leibniz created, and it was a good idea for the derivative.

And you'll see where it comes from.

But somehow I'm dividing height distance up by distance across.

And that ratio of up to across is a slope.

So let me develop what we're doing.

So one thing we can do and now will do is,

for the great functions of calculus, a few very special, very important functions,

we will actually figure out what the slope is.

These are given by formulas, and I'll find a formula for the slope, dy dx equals.

And I won't write it in yet, let me keep a little suspense,

But this short list of the great functions is tremendously valuable.

The process that we go through takes a little time, but once we do it, it's done.

Once we write in the answer here, we know it.

And the point is that other functions of science,

of engineering, of economics, of life,

come from these functions by multiplying.

I could multiply that times that.

And then I need a product rule, a rule for the derivative, the slope of a product.

I could divide that by that.

And so I need a quotient rule.

I could do a chain, and you'll see that's maybe the best and most valuable, e to the sine x.

So I'm putting e to the x together with sine x in a chain of functions, e to the sine of x.

Then we need a chain rule.

That's all coming.

Let me start, well, let me even start by giving away the main facts for these three examples.

Because there are three you want to remember and might as well start now.

The x to the nth, so that's something, if n is positive, x to the nth climbs up.

Let me draw a graph here of y equal x squared.

Because that's one we'll work out in detail, y equal x squared.

So this direction is x, this direction is y, and I want to know the slope,

and the point is that that slope is changing as I go up.

So the slope depends on x.

The slope will be different here, so it's getting steeper and steeper.

I'll figure out that slope.

For this example, x squared when n is 2 and it's climbing.

If n was minus 2, just because that's also on our list,

n could be negative, the function would be dropping.

You remember x to the minus 2, that negative exponent means divide by x squared.

x to the minus 2 is 1 divided by x squared, and it'll be dropping.

Here, so n could be positive or negative here.

So I tell you the derivative.

The derivative is easy to remember.

It's that number n.

It's another power of x, and that power is one less.

One down.

You lose one power.

So I'm going to go through the steps here for n equal to 2.

So I hope to get the answer 2 times x to the 2 minus 1 will just be 1, 2x.

But what does the slope mean?

That's what this lecture is really telling you.

I'll tell you the answer for if it's sine x going in beautifully,

the derivative of sine x is cos x, the cosine.

The derivative of the sine curve is the cosine curve.

You couldn't hope for more than that.

And then we'll also at the same time find the derivative of the cosine curve, which is minus the sine curve.

It turns out a minus sine comes in because the cosine curve drops at the start.

OK.

And would you like to know this one?

e to the x, which I will introduce in a lecture coming very soon, because it's the function of calculus.

And the reason it's so terrific is the connection between the function,

whatever it is, whatever this number e is to whatever the x power means,

the slope is the same as the function, e to the x.

That's amazing.

The slope, as the function changes, the slope changes, and they stay equal.

OK, that's just, really my help is just to say, if you know those three, you're really off to a good start,

plus the rules that you need.

All right, now I'll tackle this particular one and say, what does slope mean?

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.

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คำศัพท์และข้อสังเกตด้านการพูดสำหรับบทเรียนนี้

วิดีโอนี้มี 381 ประโยค และ 3465 คำ สำหรับฝึกพูดตาม ช่วงที่มีเสียงพูดยาว 29:39 ผู้พูดพูดด้วยความเร็วสม่ำเสมอ ประมาณ 117 คำต่อนาที เหมาะกับการฝึกพูดตาม มีเพียง 79% ของคำที่อยู่ใน 3,000 คำที่ใช้บ่อยที่สุดในภาษาอังกฤษ คำศัพท์จึงค่อนข้างยาก

คำศัพท์สำคัญในวิดีโอนี้

คำที่พบไม่บ่อย 15 คำจากวิดีโอ พร้อมคำอ่านและความหมาย:

delta /ˈdɛltə/ (คำนาม) — เดลตา. The fourth letter of the modern Greek alphabet Δ, δ.
calculus /ˈkæl.kjə.ləs/ (คำนาม) — แคลคูลัส. Any formal system in which symbolic expressions are manipulated according to fixed rules.
climb /ˈklaɪ̯m/ (คำกริยา) — ปีน, ขึ้น. To traverse a slope or contour, either upwards (ascending) or downwards (descending).
graph /ɡɹæf/ (คำนาม) — กราฟ. 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/ — ลบ. Made less or reduced by (followed by an expression of number or quantity).
lecture /ˈlɛk.t͡ʃə/ (คำนาม) — การบรรยาย, เล็กเชอร์. A spoken lesson or exposition, usually delivered to a group.
formula /ˈfɔː.mjʊ.lə/ (คำนาม) — สูตร. Any mathematical rule expressed symbolically.
algebra /ˈæl.d͡ʒɪ.bɹə/ (คำนาม) — พีชคณิต. Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations.
introduce /ˌɪntɹəˈdus/ (คำกริยา) — แนะนำ. To cause (someone) to be acquainted (with someone else).
leaf /lif/ (คำนาม) — ใบ, ใบไม้. The usually green and flat organ that represents the most prominent feature of most vegetative plants.
disappear /dɪsəˈpɪɹ/ (คำกริยา) — หาย, สูญ. To vanish.
steep /stiːp/ (คำคุณศัพท์) — ชัน. Of a near-vertical gradient; of a slope, surface, curve, etc. that proceeds upward or downward at an angle approaching vertical.
economics /ˌiːkəˈnɒmɪks/ (คำนาม) — เศรษฐศาสตร์. The study of resource and wealth allocation, consumption, and distribution, of capital and investment, and of management of the factors of production.
discover /dɪˈskʌvə/ (คำกริยา) — ค้นพบ. To find or learn something for the first time.
ratio /ˈɹeɪ.ʃoʊ/ (คำนาม) — สัดส่วน, อัตราส่วน. A number representing a comparison between two named things.

กริยาวลีที่คุณจะได้ยิน

come back (คำกริยา) — กลับมา. To return to a place.
pick up (คำกริยา) — ยก, หยิบ. To lift; to grasp and raise.

การออกเสียงที่ควรระวัง

ผู้พูดใช้รูปย่อและรูปลดเสียง 65 ครั้ง เช่น I'm, I'll, we're ให้พูดแบบสั้นตามที่ได้ยิน

เสียง “sh” และ “zh”: conclusion /kənˈkluːʒən/, ratio /ˈɹeɪ.ʃoʊ/, donation /doʊˈneɪʃən/, differential [dɪfəˈɹənʃəɫ], calculation /ˌkælkjuˈleɪʃn̩/
คำยาว — ลงเสียงหนักให้ถูกพยางค์: derivative /dɪˈɹɪvətɪv/, economics /ˌiːkəˈnɒmɪks/, mathematics /mæθ(.ə)ˈmæt.ɪks/, beautifully /ˈbju.tɪ.fə.li/, horizontal /ˌhoɹəˈzɑn.təl/

วิธีฝึกกับวิดีโอนี้

ฟังวิดีโอให้จบหนึ่งรอบโดยยังไม่ต้องพูด แล้วจดคำที่ยังไม่รู้จัก
พูดตามทีละประโยคด้วยความเร็วปกติ ทำซ้ำแต่ละประโยคจนจังหวะของคุณตรงกับผู้พูด
อัดเสียงตัวเองแล้วเทียบกับต้นฉบับ โดยสังเกตคำอย่าง delta, calculus, climb เป็นพิเศษ

ไวยากรณ์ในวิดีโอนี้

โครงสร้างที่ผู้พูดใช้บ่อยที่สุด พร้อมคำพูดจริงจากวิดีโอ:

โครงสร้าง	ในวิดีโอ
Present perfect have/has + กริยาช่อง 3 — เหตุการณ์ในอดีตที่ยังเกี่ยวข้องกับปัจจุบัน	you've gone · has hit · we've now done
Passive voice be + กริยาช่อง 3 — เน้นสิ่งที่เกิดขึ้น ไม่ใช่ผู้กระทำ	is lost · are given · is squared
Relative clauses who / which + อนุประโยค — ข้อมูลเพิ่มเติมเกี่ยวกับคนหรือสิ่งของ	curve, which is · People who get · this, which is

เทคนิค Shadowing คืออะไร?

Shadowing เป็นเทคนิคการเรียนรู้ภาษาที่ได้รับการรับรองทางวิทยาศาสตร์ พัฒนาขึ้นสำหรับการฝึกนักแปลมืออาชีพ วิธีการนี้เรียบง่ายแต่ทรงพลัง: คุณฟังเสียงภาษาอังกฤษจากเจ้าของภาษาและพูดตามทันที — เหมือนเงาที่ตามผู้พูดด้วยช่วงเวลาห่าง 1-2 วินาที การวิจัยแสดงว่าเทคนิคนี้ปรับปรุงความแม่นยำในการออกเสียง ทำนองเสียง จังหวะ การเชื่อมเสียง การฟังเข้าใจ และความคล่องแคล่วในการพูดได้อย่างมีนัยสำคัญ

เทคนิค shadowing: อ่านคู่มือฉบับเต็มทีละขั้นตอน →

ฝึกพูดภาษาอังกฤษด้วยเทคนิค Shadowing จากวิดีโอ: Big Picture: Derivatives

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คำศัพท์และข้อสังเกตด้านการพูดสำหรับบทเรียนนี้

คำศัพท์สำคัญในวิดีโอนี้

กริยาวลีที่คุณจะได้ยิน

การออกเสียงที่ควรระวัง

วิธีฝึกกับวิดีโอนี้

ไวยากรณ์ในวิดีโอนี้

เทคนิค Shadowing คืออะไร?