1
00:00:36,247 --> 00:00:37,334
Here we go.
2
00:00:41,244 --> 00:00:44,167
- Cool; I was gettin' worried there.
- Me, too.
3
00:01:01,032 --> 00:01:04,268
Okay. They're comin' in now.
I've got... 11.
4
00:01:14,497 --> 00:01:18,261
I'll wait until noon, because that's
when I set this one to start.
5
00:01:18,261 --> 00:01:21,251
And, um... then I'll begin.
6
00:01:40,612 --> 00:01:43,123
This time, I didn't need
password to log in.
7
00:01:43,123 --> 00:01:44,073
Good.
8
00:01:51,953 --> 00:01:57,011
All right, I've got Manon; I've got...
Richard; I've got Causey.
9
00:02:01,986 --> 00:02:03,330
Ross. Yeah.
10
00:02:06,391 --> 00:02:07,629
I have Jordan...
11
00:02:12,251 --> 00:02:13,352
Don't have Jordan.
12
00:02:16,381 --> 00:02:17,609
[student] There you go. 15.
13
00:02:21,014 --> 00:02:23,497
[student] Oh my gosh; I was
panicking for the last 30 minutes.
14
00:02:23,497 --> 00:02:26,500
- Oh, well—never panic, because—
- [laughs]
15
00:02:26,500 --> 00:02:30,741
I promise, I—even if there's
nobody here, I can record it...
16
00:02:30,741 --> 00:02:34,624
You know, just by myself.
- Oh, sweet. Okay.
17
00:02:36,032 --> 00:02:40,137
[instructor] I'm kind of waiting on Jordan,
because he was having trouble.
18
00:02:45,503 --> 00:02:47,227
Is that Luke, Luke, Luke...?
19
00:02:49,400 --> 00:02:50,163
Nope.
20
00:02:58,359 --> 00:02:59,381
There's Jordan.
21
00:03:03,682 --> 00:03:07,191
- [student] Finally.
- Heh. Do I have Audrey?
22
00:03:42,072 --> 00:03:43,648
[student] So, what happened there?
23
00:03:45,511 --> 00:03:47,264
[instructor] Your guess
is as good as mine.
24
00:04:40,972 --> 00:04:41,762
Okay.
25
00:04:42,747 --> 00:04:47,255
Um, so, it is 12:01, so I'm
gonna go ahead and start.
26
00:04:47,255 --> 00:04:50,755
And hopefully you printed
out, or, you know,
27
00:04:50,755 --> 00:04:54,071
copied the handouts that
have been on Blackboard.
28
00:04:54,071 --> 00:04:56,500
Since we lost so much time today,
29
00:04:56,500 --> 00:04:59,761
I really think we'll probably
just get through 4.1,
30
00:04:59,761 --> 00:05:03,623
and then try to save the
2.8 until next time,
31
00:05:03,623 --> 00:05:08,640
and maybe I can find a way to
combine 2.8 with 4.2. We'll see.
32
00:05:09,005 --> 00:05:10,762
But 4.1, I'm glad you're here for me,
33
00:05:10,762 --> 00:05:15,267
because this one is a section on
applications. It's called related rates.
34
00:05:15,267 --> 00:05:19,376
And it... can be a little bit challenging;
but it's also really fun.
35
00:05:19,376 --> 00:05:24,262
So let me get my screen-sharing
going for ya, here.
36
00:06:03,126 --> 00:06:04,094
Wow.
37
00:06:04,094 --> 00:06:06,586
Okay. That was a very long delay.
38
00:06:07,245 --> 00:06:11,217
I don't know if it's just that there's
so many people on Zoom, and it's slow?
39
00:06:11,217 --> 00:06:12,625
I don't know.
40
00:06:12,625 --> 00:06:13,946
Hope this works.
41
00:06:13,946 --> 00:06:15,242
So, here's where we are.
42
00:06:15,242 --> 00:06:17,761
Today, I'm gonna definitely
get through 4.1, and like I said,
43
00:06:17,761 --> 00:06:22,508
if I need to try to squeeze 2.8
into Wednesday, I'll do that.
44
00:06:22,508 --> 00:06:25,007
All right. And here's our
handout for today.
45
00:06:25,007 --> 00:06:27,259
So this one is related rates.
46
00:06:27,859 --> 00:06:31,033
Try to work on my... focus.
47
00:06:31,033 --> 00:06:33,623
And this is the famous balloon problem.
48
00:06:33,623 --> 00:06:39,354
If I had a balloon, I would blow one
up for you; but I don't have a balloon.
49
00:06:39,964 --> 00:06:42,757
And I didn't want to go
to the store to get one.
50
00:06:42,757 --> 00:06:44,937
Hashtag: coronavirus.
51
00:06:45,430 --> 00:06:49,180
So this is the famous balloon problem
which we shall just try to imagine.
52
00:06:49,630 --> 00:06:54,512
It says, "Suppose I can blow up a balloon
at a rate of 3 cubic inches per second."
53
00:06:54,512 --> 00:06:57,490
So, [puffing noises] blowing it up;
it's getting bigger.
54
00:06:58,030 --> 00:07:03,251
This is a unit of volume.
So, (dv/dt) equals 3.
55
00:07:03,251 --> 00:07:06,002
It's the derivative of volume
with respect to t,
56
00:07:06,002 --> 00:07:08,511
because it's a rate of change in volume.
57
00:07:09,559 --> 00:07:12,512
Both volume and radius are changing
as you blow that thing up;
58
00:07:12,512 --> 00:07:15,634
the radius is increasing; well,
everything is increasing.
59
00:07:15,634 --> 00:07:18,750
The radius; the diameter; the
circumference; the surface area;
60
00:07:18,750 --> 00:07:20,649
the volume; everything.
61
00:07:20,649 --> 00:07:25,353
The rate at which the radius is changing,
we're gonna call that (dr/dt).
62
00:07:25,353 --> 00:07:28,261
Remember, a derivative
is a rate of change.
63
00:07:28,751 --> 00:07:31,364
Now, at first, the radius grows quickly.
64
00:07:31,364 --> 00:07:34,693
So imagine, you know, when you've got
the balloon, it's about this long.
65
00:07:34,693 --> 00:07:38,358
And you put in that first couple
of [puff-puff-puffff].
66
00:07:38,358 --> 00:07:43,197
And then it grows fast. Like, all of
a sudden, it's a round shape.
67
00:07:44,377 --> 00:07:48,752
But as the balloon gets larger,
the radius grows more slowly.
68
00:07:48,752 --> 00:07:52,756
So imagine you got a big balloon here,
and I put in a couple more puffs of air.
69
00:07:52,756 --> 00:07:53,486
[puff-puff]
70
00:07:53,486 --> 00:07:56,859
You hardly notice any change
in the shape of the balloon.
71
00:07:57,659 --> 00:08:01,519
So, as the balloon gets larger,
the radius grows more slowly,
72
00:08:01,519 --> 00:08:06,987
even though the rate at which the
volume is changing remains constant.
73
00:08:06,987 --> 00:08:10,929
So, it's always 3 cubic inches per second.
74
00:08:10,929 --> 00:08:14,752
It's just that that's more noticeable
when the balloon is this big;
75
00:08:14,752 --> 00:08:18,022
less noticeable when
the balloon is this big.
76
00:08:19,245 --> 00:08:25,020
Volume and radius are related
by the formula V = (4/3) π r³.
77
00:08:25,020 --> 00:08:27,768
That's just the formula
for volume of a sphere.
78
00:08:28,484 --> 00:08:33,017
And A says, "At what rate is the radius
increasing with respect to time,
79
00:08:33,017 --> 00:08:36,896
when the radius is 2 inches?"
80
00:08:39,752 --> 00:08:42,266
Okay. So here's my little handout here.
81
00:08:42,266 --> 00:08:45,630
"At what rate is the radius increasing
with respect to time
82
00:08:45,630 --> 00:08:48,023
when the radius is 2 inches?'
83
00:08:48,608 --> 00:08:49,354
Okay.
84
00:08:49,354 --> 00:08:52,033
So what we need to find, is this.
85
00:08:52,514 --> 00:08:55,515
We need (dr/dt).
86
00:08:55,515 --> 00:09:00,814
Because this is the rate
at which the radius is increasing.
87
00:09:00,814 --> 00:09:02,389
We need (dr/dt).
88
00:09:02,947 --> 00:09:04,697
Okay.
89
00:09:04,697 --> 00:09:11,899
So. I'm gonna take the formula
that I have, which is V = (4/3) π r³;
90
00:09:11,899 --> 00:09:17,319
and we're gonna differentiate
that with respect to t.
91
00:09:17,319 --> 00:09:18,676
Not r, but t.
92
00:09:18,676 --> 00:09:21,395
So, this is an implicit differentiation.
93
00:09:21,395 --> 00:09:23,612
I want it with respect to t.
94
00:09:24,254 --> 00:09:28,260
So I'm gonna find (dv/dt),
95
00:09:28,260 --> 00:09:33,025
and that will be the derivative with
respect to t of the right-hand side,
96
00:09:33,025 --> 00:09:36,240
which is (4/3) π r³.
97
00:09:36,240 --> 00:09:40,024
So what I want you to notice here
is that r is the wrong letter.
98
00:09:40,024 --> 00:09:44,252
t is the right letter;
r is the wrong letter.
99
00:09:44,252 --> 00:09:46,760
So, with this implicit differentiation,
100
00:09:46,760 --> 00:09:48,758
when I differentiate
that right-hand side,
101
00:09:48,758 --> 00:09:53,501
I've got to follow it with an r prime,
'cause it's the wrong letter.
102
00:09:53,501 --> 00:09:57,976
Now on the left-hand side, you know,
we just leave that, (dv/dt),
103
00:09:57,976 --> 00:10:00,604
and here we go on the
right-hand side differentiating.
104
00:10:00,604 --> 00:10:03,754
So we'll do 3 times (4/3) is 4.
105
00:10:03,754 --> 00:10:06,748
And that's π r².
106
00:10:06,748 --> 00:10:10,494
Now, the chain rule says, multiply
by the derivative of the inside.
107
00:10:10,494 --> 00:10:12,319
And that inside is the r.
108
00:10:12,319 --> 00:10:16,409
So since it was the wrong letter,
this is what we'd follow it with; r prime.
109
00:10:16,409 --> 00:10:18,043
Now here's the deal.
110
00:10:18,043 --> 00:10:23,272
So, instead of using r prime,
which is a perfectly fine symbol;
111
00:10:23,272 --> 00:10:24,363
but instead of using that one—
112
00:10:24,363 --> 00:10:26,384
- Professor?
- Uh-huh.
113
00:10:26,384 --> 00:10:31,774
[student] Um, how did you get the r²?
The derivative function?
114
00:10:31,774 --> 00:10:34,879
Minus one?
- [instructor] So, the 3 ti—mhmm.
115
00:10:34,879 --> 00:10:40,675
3 times (4/3) is 4,
π, and then r ⁿ - 1. So r².
116
00:10:40,675 --> 00:10:42,363
[student] Okay. Gotcha.
117
00:10:42,363 --> 00:10:44,376
[instructor] Instead of
using this r prime,
118
00:10:44,376 --> 00:10:47,510
we're gonna use this
notation for the derivative.
119
00:10:47,510 --> 00:10:51,926
The only reason is because with
(dr/dt), it really makes it noticeable
120
00:10:51,926 --> 00:10:56,497
that we're talking about a rate of change
of radius with respect to time here.
121
00:10:56,497 --> 00:11:00,248
And with r prime, that's not
as obvious what the intent is.
122
00:11:00,248 --> 00:11:04,045
So we'll follow this with (dr/dt).
123
00:11:04,513 --> 00:11:09,108
So that's implicit differentiation
involving a chain rule.
124
00:11:09,108 --> 00:11:11,750
Now, what we need is (dr/dt).
125
00:11:11,750 --> 00:11:16,022
Well, we've got an equation here;
we're just going to isolate (dr/dt)
126
00:11:16,022 --> 00:11:17,753
in this equation.
127
00:11:17,753 --> 00:11:20,078
So if I isolate this equation...
128
00:11:20,078 --> 00:11:24,353
Let's see. How can I do this?
I can... ummm... hmmm.
129
00:11:24,353 --> 00:11:29,931
I can first substitute in
place of (dv/dt) the 3,
130
00:11:29,931 --> 00:11:35,604
because I was told that
(dv/dt) equals 3.
131
00:11:35,604 --> 00:11:41,754
So now, 3 equals 4π r² (dr/dt).
132
00:11:41,935 --> 00:11:48,520
And I say we divide everything by the
4πr², and then we've got (dr/dt) all alone!
133
00:11:49,373 --> 00:11:56,402
So we'll say (dr/dt) = (3/4 π r²).
134
00:11:56,402 --> 00:11:57,752
Hmmmm.
135
00:11:57,752 --> 00:12:00,678
What was the r in this problem?
136
00:12:00,678 --> 00:12:02,259
What was the r.
137
00:12:03,029 --> 00:12:04,081
See right here?
138
00:12:04,081 --> 00:12:08,001
- [student] Two?
- When the radius is 2 inches.
139
00:12:08,001 --> 00:12:11,787
So now we'll just substitute in
the 2; we'll have a number.
140
00:12:11,787 --> 00:12:15,954
3 over 4 times π times r².
141
00:12:15,954 --> 00:12:18,386
So, 2 squared is another 4.
142
00:12:18,386 --> 00:12:21,766
This point, I'm gonna go to
my calculator; see what that is.
143
00:12:25,369 --> 00:12:26,554
And we'll clear.
144
00:12:26,554 --> 00:12:29,524
Okay. So I want 3 divided by...
145
00:12:30,514 --> 00:12:32,630
And I'm gonna put that
denominator in parentheses,
146
00:12:32,630 --> 00:12:36,250
so the calculator understands
I'm dividing by all of this stuff.
147
00:12:36,760 --> 00:12:40,113
So that would be a 16π down there.
148
00:12:40,113 --> 00:12:48,769
And I get... (dr/dt) is
approximately 0.0597.
149
00:12:48,769 --> 00:12:52,900
I just chose to round that
to four decimal places.
150
00:12:55,230 --> 00:12:56,892
I need a unit, though.
151
00:12:56,892 --> 00:13:00,988
So this was a rate of change of
the radius with respect to time.
152
00:13:01,528 --> 00:13:04,743
So the unit for radius was...
153
00:13:04,743 --> 00:13:07,188
-Inches per second.
[Prof] -Inches.
154
00:13:07,188 --> 00:13:10,150
And for time, it was seconds.
155
00:13:10,150 --> 00:13:12,496
So, I'll circle this by itself.
156
00:13:12,496 --> 00:13:18,182
So, (dr/dt) equals
0.0597 inches per second.
157
00:13:18,182 --> 00:13:21,613
The rate of change of the radius
with respect to time.
158
00:13:22,550 --> 00:13:24,244
Okay, so that was the first example.
159
00:13:24,244 --> 00:13:26,508
And that's the way all of
these are gonna work.
160
00:13:26,508 --> 00:13:28,260
You're gonna have a formula.
161
00:13:29,000 --> 00:13:30,759
Sometimes you're given the formula,
162
00:13:30,759 --> 00:13:34,646
and sometimes you have to figure
the formula out; that's comin' up.
163
00:13:35,276 --> 00:13:38,108
And then once you get that formula,
you're gonna be doing an, um—
164
00:13:38,108 --> 00:13:42,641
implicit differentiation with
respect to time on all of these.
165
00:13:42,641 --> 00:13:47,179
With respect to time. And then,
solving for an unknown.
166
00:13:47,736 --> 00:13:51,310
Okay, so let's look at part B.
Still about the same problem.
167
00:13:51,310 --> 00:13:53,281
It s-says—"It s-ssays."
168
00:13:53,281 --> 00:13:57,746
It says, "Suppose I increase my
effort when r equals 2 inches,
169
00:13:57,746 --> 00:14:01,500
and begin to blow air into
the balloon at a faster rate.
170
00:14:01,500 --> 00:14:03,956
A rate of 4 cubic inches per second.
171
00:14:03,956 --> 00:14:06,665
Well, how fast is the
radius changing now?"
172
00:14:07,250 --> 00:14:13,765
Okay. So in this case, the (dv/dt)
is now equal to 4.
173
00:14:14,765 --> 00:14:20,270
So, I had (dv/dt) from above,
so I can just write that down again.
174
00:14:20,270 --> 00:14:22,173
Instead of reinventing the wheel.
175
00:14:22,173 --> 00:14:28,336
And it was 4 times π
times r² times (dr/dt).
176
00:14:28,336 --> 00:14:33,130
And now I'm going to substitute
the 4 in place of the (dv/dt).
177
00:14:33,130 --> 00:14:35,744
So we've got that faster rate
of change of volume.
178
00:14:36,120 --> 00:14:39,744
So 4 equals 4 times π.
179
00:14:39,744 --> 00:14:45,606
That radius is still 2, as it says;
when r equals 2 inches.
180
00:14:45,606 --> 00:14:51,852
So that's a 2² (dr/dt), and we just
need to solve for (dr/dt) again.
181
00:14:53,252 --> 00:14:59,515
So, to do that, I'll divide both
sides by the 4π times 2²?
182
00:15:00,729 --> 00:15:02,259
2² is 4.
183
00:15:02,259 --> 00:15:04,430
4 times 4 is 16.
184
00:15:04,430 --> 00:15:07,713
So, there's that 16π again.
185
00:15:07,713 --> 00:15:09,752
So I'll bring out my calculator.
186
00:15:10,570 --> 00:15:13,113
And... turn it on.
187
00:15:13,113 --> 00:15:15,371
Ooh, I got a bad glare on that; sorry.
188
00:15:15,763 --> 00:15:18,754
Okay, so this one'll be 4 divided by.
189
00:15:18,754 --> 00:15:21,600
And then in parentheses, (16π).
190
00:15:22,260 --> 00:15:25,561
That would also reduce to ¼π.
191
00:15:25,561 --> 00:15:26,494
Dun't matter.
192
00:15:26,494 --> 00:15:35,876
At any rate, this (dr/dt) is
approximately 0.0796.
193
00:15:35,876 --> 00:15:39,379
And again, that would be
inches per second.
194
00:15:39,379 --> 00:15:42,258
So you notice, when you
compare the two,
195
00:15:42,258 --> 00:15:47,331
that your radius here is changing
at a faster rate than it did here.
196
00:15:47,331 --> 00:15:50,850
Obviously; because you
were increasing your effort,
197
00:15:50,850 --> 00:15:54,272
and the rate of change in
your volume was higher.
198
00:15:55,762 --> 00:15:57,979
Okay. So, um.
199
00:15:58,813 --> 00:16:00,752
Hold on one second, y'all.
200
00:16:02,360 --> 00:16:06,910
I've got to mute you for a second
because my dog needs to go out.
201
00:16:06,910 --> 00:16:07,438
[exaggerated whisper]
I'm so sorry.
202
00:16:12,490 --> 00:16:16,724
[Student] Hey, I just came back.
My house just had a rolling blackout.
203
00:16:17,943 --> 00:16:19,750
[Instructor] A rolling blackout?
204
00:16:20,680 --> 00:16:23,547
[Student] Yeah, I've been gone
for, like, five minutes. [Chuckles]
205
00:16:23,547 --> 00:16:26,501
[Instructor] Whoaaa.
So, where do you live?
206
00:16:27,971 --> 00:16:32,514
[Student] Uh, right next to the
Crow Bar. On South Congress.
207
00:16:33,340 --> 00:16:35,386
[Instructor] Oh, maaaan...
208
00:16:35,386 --> 00:16:38,249
Okay. That's all we need,
are blackouts.
209
00:16:38,249 --> 00:16:39,029
[Student] Yep.
210
00:16:39,829 --> 00:16:40,749
[Instructor] How fun.
211
00:16:42,255 --> 00:16:43,290
Sorry.
212
00:16:43,290 --> 00:16:47,248
So, now, for part C it says, "At what
rate is the volume increasing
213
00:16:47,248 --> 00:16:52,336
with respect to the radius, when
the radius is 1 inch or 3 inches?"
214
00:16:52,336 --> 00:16:56,416
"At what rate is the volume increasing."
215
00:16:56,416 --> 00:17:01,503
Okay. So, I'm gonna write a little note
here to be careful with this one.
216
00:17:01,503 --> 00:17:03,433
You've got to read it carefully.
217
00:17:03,433 --> 00:17:07,531
"At what rate is the volume."
So, now, underline that.
218
00:17:07,531 --> 00:17:11,260
—"increasing with respect
to the radius."
219
00:17:11,260 --> 00:17:16,743
So, what we want here, is (dv/dr).
220
00:17:17,503 --> 00:17:21,398
This is the rate of change of the
volume with respect to the radius.
221
00:17:21,398 --> 00:17:24,693
We need (dv/dr).
222
00:17:24,693 --> 00:17:28,384
So, we're gonna start with the
formula we were given again.
223
00:17:28,384 --> 00:17:33,630
And that was V = (4/3) π r³.
224
00:17:33,630 --> 00:17:36,246
That was our formula for
volume of that sphere.
225
00:17:36,246 --> 00:17:37,963
We need (dv/dr).
226
00:17:37,963 --> 00:17:44,743
So, if you'll notice. When we get
(dv/dr), uh... r is the right letter.
227
00:17:44,743 --> 00:17:48,176
So this one's not gonna require
an implicit differentiation.
228
00:17:48,176 --> 00:17:50,187
This one's pretty straightforward.
229
00:17:50,187 --> 00:17:58,246
(dv/dr). 3 times (4/3) is 4. Times π.
Times r², and there you have it.
230
00:17:58,246 --> 00:18:00,383
That's the rate of change of volume.
231
00:18:00,383 --> 00:18:04,774
Don't need to do—
following it by a (dr/dt),
232
00:18:04,774 --> 00:18:07,793
because r was the correct
letter in the first place.
233
00:18:08,570 --> 00:18:14,091
So, now we just need to evaluate this when
r is one inch, and when r is 3 inches.
234
00:18:14,736 --> 00:18:17,128
So, let's see if I can get
a little more room here.
235
00:18:20,710 --> 00:18:21,509
There we go.
236
00:18:21,509 --> 00:18:29,181
So, at r = 1, (dv/dr) is equal to 4π.
237
00:18:30,379 --> 00:18:34,987
Okay? If I'm looking for my units here,
this is a rate of change of volume
238
00:18:34,987 --> 00:18:37,253
with respect to the radius.
239
00:18:37,253 --> 00:18:41,033
The units of volume were inches cubed.
240
00:18:41,743 --> 00:18:45,765
The units for the radius were inches.
241
00:18:45,765 --> 00:18:49,503
Now. I don't want you to reduce
that to inches squared. [Chuckles]
242
00:18:49,503 --> 00:18:50,900
Don't do that.
243
00:18:50,900 --> 00:18:53,275
So, this is a rate of change of volume.
244
00:18:53,275 --> 00:18:57,131
So what it says is that
when the radius is 1 inch,
245
00:18:57,131 --> 00:19:01,225
that your volume is changing
at a rate of 4π cubic inches
246
00:19:01,225 --> 00:19:04,631
for every 1-inch change in radius.
247
00:19:04,631 --> 00:19:07,508
So, leave this be; it means something.
248
00:19:07,508 --> 00:19:10,515
It's describing how the
volume is changing
249
00:19:10,515 --> 00:19:13,250
with respect to how
the radius is changing.
250
00:19:13,250 --> 00:19:14,770
Does that make sense to y'all?
251
00:19:18,995 --> 00:19:20,142
[Student] Yeah.
252
00:19:21,755 --> 00:19:23,622
All right. Let's try r = 3.
253
00:19:23,622 --> 00:19:29,245
So, (dv/dr) in this case
would be 4π times 3².
254
00:19:29,245 --> 00:19:34,776
3² is 9. 9 times 4.
This'll be 36π.
255
00:19:35,646 --> 00:19:37,385
And I'm gonna leave it like that.
256
00:19:38,145 --> 00:19:42,750
And my units, again,
are inches cubed, per inch.
257
00:19:43,747 --> 00:19:46,772
Sounds better if I say "cubic
inches per inch," I think.
258
00:19:47,514 --> 00:19:48,200
Okay.
259
00:19:48,754 --> 00:19:50,620
So, there's example one.
260
00:19:50,620 --> 00:19:51,250
And that was—
261
00:19:51,250 --> 00:19:56,510
[Student] Is there a place that we can
get our... or find our graded tests?
262
00:19:56,510 --> 00:19:58,240
[Student] -Like, you have—okay.
[Instructor] -Yes. Yeah.
263
00:19:58,240 --> 00:20:02,271
[Instructor] So, um, when you
go to your gradebook,
264
00:20:02,271 --> 00:20:06,688
and go down to, like, the row
that the test is on...
265
00:20:08,278 --> 00:20:13,755
There should be a place where
you can see my feedback,
266
00:20:13,755 --> 00:20:16,397
and that's where I uploaded
your graded test.
267
00:20:17,658 --> 00:20:19,906
Can anybody else jump in
here; if you found it,
268
00:20:19,906 --> 00:20:22,051
can you explain that
better than I just did?
269
00:20:25,023 --> 00:20:28,878
[Student #2] Just next to the grade,
there's like, a little cloud thing in blue,
270
00:20:28,878 --> 00:20:30,701
which has the comment.
271
00:20:30,701 --> 00:20:34,268
[Student #1] -The little speech bubble.
-Yeah. And you can find there.
272
00:20:36,260 --> 00:20:38,604
[Instructor] -Great. Thank you.
[Student #1] -Yes, uh, thanks.
273
00:20:38,604 --> 00:20:39,544
[Instructor] Sure.
274
00:20:41,510 --> 00:20:44,757
All right; so now, related rates procedure.
275
00:20:44,757 --> 00:20:47,822
So we went through that first
example pretty slowly.
276
00:20:47,822 --> 00:20:49,082
And so now I'm gonna show you;
277
00:20:49,082 --> 00:20:52,913
this is just the general way
we're gonna handle all of these.
278
00:20:52,913 --> 00:20:56,495
So the first thing is, we're gonna
draw a picture if we can.
279
00:20:57,095 --> 00:20:59,754
Uh, I didn't really need to draw
a picture of the balloon problem.
280
00:20:59,754 --> 00:21:02,800
I could have drawn a sphere,
I guess, if I wanted, but.
281
00:21:02,800 --> 00:21:04,415
For some of these, you need
a diagram.
282
00:21:04,415 --> 00:21:07,164
You're gonna need a picture,
and you'll need to label things.
283
00:21:07,754 --> 00:21:11,053
And that's the second point,
is "label and assign variables."
284
00:21:11,053 --> 00:21:12,113
Okay.
285
00:21:12,113 --> 00:21:16,882
The third thing is, write down what
you know, and what you need to know.
286
00:21:17,382 --> 00:21:20,905
So whatever the question's asking,
that's what you need to know.
287
00:21:20,905 --> 00:21:24,754
And then what you know
is usually gonna be a formula
288
00:21:24,754 --> 00:21:27,905
associated with the shape
that you're drawing.
289
00:21:29,245 --> 00:21:33,890
Then you wanna find an equation or
a formula that relates the variables.
290
00:21:34,507 --> 00:21:39,758
So, oftentimes, this is gonna be
a formula for volume, or for area.
291
00:21:39,758 --> 00:21:42,066
It could be the Pythagorean theorem.
292
00:21:42,066 --> 00:21:45,187
Just depends on the picture
that we end up drawing.
293
00:21:46,024 --> 00:21:48,261
And then we're gonna use
implicit differentiation
294
00:21:48,261 --> 00:21:51,880
to differentiate with respect to time.
295
00:21:51,880 --> 00:21:55,023
And then the last thing is just
substitute in your known values,
296
00:21:55,023 --> 00:21:57,249
and then solve for the unknown values.
297
00:21:57,249 --> 00:22:01,598
So, we're gonna follow that pattern
on all of the rest of the problems.
298
00:22:01,598 --> 00:22:04,527
'Kay, next up is the famous
sliding-ladder problem.
299
00:22:05,017 --> 00:22:07,195
And I wish we were in a classroom,
because in a classroom,
300
00:22:07,195 --> 00:22:11,267
I bring in my meter stick,
and pretend it's a ladder,
301
00:22:11,267 --> 00:22:16,503
and then I prop it up against the wall,
and I pull it out slowly from the bottom,
302
00:22:16,503 --> 00:22:19,001
and watch it slam down on the floor.
303
00:22:19,001 --> 00:22:22,766
So, you know, the ladder
is sliding down the wall.
304
00:22:22,766 --> 00:22:26,912
And when you see it in class, I just
think this makes a little more sense, but.
305
00:22:26,912 --> 00:22:27,819
Darn it!
306
00:22:28,617 --> 00:22:30,900
So this is the famous sliding-ladder problem.
307
00:22:30,900 --> 00:22:33,620
Says, "A 10-foot ladder
rests against a wall."
308
00:22:33,620 --> 00:22:36,940
So I'm just imagining a ladder
propped up against a wall.
309
00:22:37,634 --> 00:22:40,633
If the bottom of the ladder
slides away from the wall
310
00:22:40,633 --> 00:22:43,371
at a rate of 1 foot per second—
311
00:22:43,371 --> 00:22:48,100
so that's steady, constant pulling the
bottom of that ladder away from that wall—
312
00:22:48,100 --> 00:22:51,757
—"how fast is the top of the
ladder sliding down the wall,
313
00:22:51,757 --> 00:22:55,501
when the bottom of the ladder
is 6 feet from the wall?"
314
00:22:56,095 --> 00:22:58,247
So the first thing that we talk about is,
315
00:22:58,247 --> 00:23:00,999
when that ladder is
propped up against the wall,
316
00:23:00,999 --> 00:23:05,525
and you're pulling the bottom
of the ladder, pulling it out slowly,
317
00:23:05,525 --> 00:23:09,635
the top of that ladder
is also falling down.
318
00:23:09,635 --> 00:23:13,205
But would it fall at the exact same rate
319
00:23:13,205 --> 00:23:16,615
at which you're pulling the
bottom of the ladder away?
320
00:23:18,127 --> 00:23:21,020
I mean, it's all one ladder.
321
00:23:21,617 --> 00:23:25,265
So, this is my ladder.
And this is my wall.
322
00:23:25,265 --> 00:23:27,257
And I'm pulling the bottom away.
323
00:23:27,257 --> 00:23:30,261
It seems like whatever rate
I'm pulling it away,
324
00:23:30,261 --> 00:23:34,499
that the top should slide
down at that same rate.
325
00:23:35,509 --> 00:23:38,621
But if you think about it...
I mean, really think about it.
326
00:23:38,621 --> 00:23:42,909
If there really were a ladder there, and
you had a string tied around the bottom,
327
00:23:42,909 --> 00:23:44,615
and you're pulling it out,
328
00:23:44,615 --> 00:23:47,746
it's gonna slide down
the wall slowly at first,
329
00:23:47,746 --> 00:23:51,103
but what happens when
it gets close to the floor?
330
00:23:53,874 --> 00:23:54,864
[Student] Speeds up.
331
00:23:54,864 --> 00:23:58,019
[Instructor] Yeah man, that thing
is gonna smack the floor so hard,
332
00:23:58,019 --> 00:24:01,634
it's gonna damage the floor!
Unless it's on a carpet.
333
00:24:02,194 --> 00:24:03,609
So, what really happens is,
334
00:24:03,609 --> 00:24:06,900
even though we're pulling the
bottom out at a constant rate,
335
00:24:06,900 --> 00:24:10,510
the rate at which the top is
sliding down is increasing.
336
00:24:11,503 --> 00:24:13,399
Craziest thing about physics.
337
00:24:14,369 --> 00:24:15,921
So let's try to draw this.
338
00:24:15,921 --> 00:24:19,260
Says, "Draw a picture if you can."
That's the first bullet.
339
00:24:19,260 --> 00:24:24,061
So when I draw my picture,
I'm just gonna draw a wall.
340
00:24:24,431 --> 00:24:28,305
And then, here's my ladder
propped against the wall.
341
00:24:28,305 --> 00:24:30,028
This is the floor.
342
00:24:31,131 --> 00:24:32,160
Okay.
343
00:24:32,160 --> 00:24:36,751
So, that ladder. One thing
I know about it is that it's 10 feet.
344
00:24:36,751 --> 00:24:38,750
So I can label that "10."
345
00:24:38,750 --> 00:24:41,250
And of course, you notice,
I just drew a triangle.
346
00:24:41,250 --> 00:24:44,277
So, the hypotenuse of that
triangle is definitely 10.
347
00:24:45,055 --> 00:24:49,754
Now, I can think of this as
being in a coordinate system.
348
00:24:49,754 --> 00:24:54,230
And when I think of it that way,
then the base of this triangle is x,
349
00:24:54,230 --> 00:24:58,023
and so what x really represents
is the distance from the wall.
350
00:24:58,747 --> 00:25:01,740
And then y can be... Oh.
351
00:25:01,740 --> 00:25:04,512
The height of the wall
where the ladder meets.
352
00:25:05,018 --> 00:25:07,932
So, x is the distance from the base,
353
00:25:07,932 --> 00:25:12,741
and then y is the height of the
ladder propped against the wall.
354
00:25:12,741 --> 00:25:15,500
So, I've labeled it with what I know.
355
00:25:15,500 --> 00:25:18,762
And I've assigned variables
to what I don't know.
356
00:25:19,632 --> 00:25:22,007
Um. There is something else I knew.
357
00:25:22,007 --> 00:25:25,999
It says, "If the bottom of the ladder
slides away from the wall
358
00:25:25,999 --> 00:25:28,641
at a rate of 1 foot per second."
359
00:25:28,641 --> 00:25:30,493
So, here's the bottom of the ladder.
360
00:25:30,493 --> 00:25:36,253
It's being pulled away from the wall
at a rate of 1 foot per second.
361
00:25:36,253 --> 00:25:42,620
That's one of our rates. And x is
the thing that's changing there.
362
00:25:42,620 --> 00:25:49,568
So the distance from the base
is changing. That's actually (dx/dt).
363
00:25:49,568 --> 00:25:55,005
So I'm gonna write down what
I know; that (dx/dt) equals 1.
364
00:25:55,518 --> 00:25:57,073
So I really know two things about this:
365
00:25:57,073 --> 00:26:01,046
I know the length of the ladder.
And I know (dx/dt) is 1.
366
00:26:02,846 --> 00:26:05,021
So I wrote down what I know
and what I need to know—
367
00:26:05,021 --> 00:26:08,931
Oh no, I didn't. What do I need
to know? What's it asking for?
368
00:26:09,405 --> 00:26:13,386
How fast is the top of the
ladder sliding down the wall.
369
00:26:13,386 --> 00:26:15,392
Well, as the top of that
ladder slides down—
370
00:26:15,392 --> 00:26:16,884
[Student] (dr/dt).
371
00:26:16,884 --> 00:26:19,705
[Instructor] True. It's y that's changing.
372
00:26:19,705 --> 00:26:25,266
So what I want to know, what I
need to know, is (dy/dt).
373
00:26:26,343 --> 00:26:28,384
Which is why it's called "related rates."
374
00:26:28,384 --> 00:26:31,747
You're gonna have multiple
rates in the same problem.
375
00:26:31,747 --> 00:26:37,126
(dx/dt) is given to us as 1.
We wanna find (dy/dt).
376
00:26:37,953 --> 00:26:41,506
So now, the next thing says,
"Find an equation or a formula
377
00:26:41,506 --> 00:26:43,959
that relates all the variables."
378
00:26:43,959 --> 00:26:46,506
So, back to our draw-ring.
379
00:26:46,506 --> 00:26:49,636
We have a 10, an x, and a y.
380
00:26:49,636 --> 00:26:53,505
What's a formula you know
that relates these three numbers?
381
00:26:54,260 --> 00:26:55,751
[Student] Pythagoras theorem.
382
00:26:56,671 --> 00:26:58,752
[Instructor] Thank you, Pythagoras.
383
00:26:58,752 --> 00:27:01,439
Pythagoras makes our lives easy.
384
00:27:01,439 --> 00:27:04,133
"Py-tha-gor...as."
385
00:27:04,133 --> 00:27:06,624
So thank you, Pythagoras.
And here's your theorem.
386
00:27:06,624 --> 00:27:12,355
It says that x² plus y²
is equal to 10².
387
00:27:13,465 --> 00:27:14,355
Okay.
388
00:27:14,355 --> 00:27:15,925
So that's our equation.
389
00:27:15,925 --> 00:27:18,874
And that's the thing
we need to differentiate.
390
00:27:18,874 --> 00:27:20,621
So once we have that equation,
391
00:27:20,621 --> 00:27:26,029
we use implicit differentiation to
differentiate with respect to time.
392
00:27:26,969 --> 00:27:29,259
So, that means on the left-hand side,
393
00:27:29,259 --> 00:27:35,508
we want the derivative with
respect to t, of x² plus y².
394
00:27:35,508 --> 00:27:40,753
On the right-hand side, the derivative
with respect to t of 100.
395
00:27:42,240 --> 00:27:44,758
So now on the left,
we're gonna split it up.
396
00:27:44,758 --> 00:27:47,250
We want the derivative of that sum.
397
00:27:47,250 --> 00:27:53,820
So I'm gonna write it as: the
derivative with respect to t of x²
398
00:27:53,820 --> 00:28:00,359
plus the derivative with
respect to t of y², equals.
399
00:28:00,359 --> 00:28:04,171
And then, what is the derivative
with respect to t of 100?
400
00:28:04,851 --> 00:28:05,927
What is that?
401
00:28:06,736 --> 00:28:07,755
[Student #1] -Zero.
[Student #2] -Zero.
402
00:28:07,755 --> 00:28:10,998
It's the constant, so we're gonna
get a zero on the right-hand side.
403
00:28:10,998 --> 00:28:15,015
Now, on the ladder problems, when
you know the length of the ladder,
404
00:28:15,015 --> 00:28:18,319
you'll have the constant on that
side of the Pythagorean theorem,
405
00:28:18,319 --> 00:28:20,762
and that derivative is
always going to be zero.
406
00:28:21,500 --> 00:28:23,507
Now, on the left,
we need to differentiate.
407
00:28:23,507 --> 00:28:27,948
So getting the derivative
with respect to t of x²?
408
00:28:28,748 --> 00:28:30,298
x is the wrong letter.
409
00:28:30,938 --> 00:28:38,229
So, we'll do our 2x, all right, but then
we've got to follow it by... (dx/dt).
410
00:28:38,229 --> 00:28:39,625
And that's the chain rule.
411
00:28:39,625 --> 00:28:44,500
Since x is the wrong letter, 2 times
x is the derivative of the outside.
412
00:28:44,500 --> 00:28:46,647
This is the derivative of the inside.
413
00:28:47,317 --> 00:28:49,759
Now for the y² term?
Same thing.
414
00:28:49,759 --> 00:28:51,376
y is the wrong letter.
415
00:28:51,376 --> 00:28:58,294
So we'll do 2y, followed by
(dy/dt) is equal to 0.
416
00:28:58,294 --> 00:28:59,148
Great.
417
00:28:59,148 --> 00:29:03,392
So now that we've differentiated, we're
gonna sub in the things that we know.
418
00:29:04,031 --> 00:29:06,530
So, what do we know here?
419
00:29:07,119 --> 00:29:11,244
Um, let's see. It says,
"10-foot ladder"...
420
00:29:11,244 --> 00:29:16,014
A rate of 1 foot per second;
that was (dx/dt).
421
00:29:16,014 --> 00:29:19,518
And we also are stopping this;
we're looking at this
422
00:29:19,518 --> 00:29:23,388
when the bottom of the ladder
is 6 feet from the wall?
423
00:29:24,128 --> 00:29:28,433
Okay. So it's 6 feet from
the wall right now. That's x.
424
00:29:29,243 --> 00:29:37,118
So I'll say 2 times 6 times (dx/dt),
which is... what, now?
425
00:29:38,041 --> 00:29:40,045
[Students] -One.
[Instructor] -One. Yeah.
426
00:29:40,045 --> 00:29:46,251
And then plus 2 times yyyy.
427
00:29:47,482 --> 00:29:49,980
[Student] You can find that
using Pythagoras' theorem.
428
00:29:49,980 --> 00:29:51,746
[Instructor] Exactly right.
429
00:29:51,746 --> 00:29:55,747
So, 2 times y, and then
times (dy/dt) equals 0.
430
00:29:55,747 --> 00:29:59,754
We don't know what y is,
but we can find it.
431
00:30:00,273 --> 00:30:04,951
So I'm gonna go back over here,
and rewrite my Pythagorean theorem,
432
00:30:04,951 --> 00:30:08,988
which is x² + y² = 10².
433
00:30:08,988 --> 00:30:12,060
I know what x is; x is 6.
So this is—
434
00:30:12,060 --> 00:30:13,973
[Student] -Professor?
[Instructor] -Yes.
435
00:30:13,973 --> 00:30:19,388
[Student] So the 2 times 6 times 1.
Is the 1 a derivative of the x?
436
00:30:20,178 --> 00:30:25,196
[Ins.] Yes. That was the rate of change
of x with respect to time. It was a 1.
437
00:30:25,196 --> 00:30:26,089
[Student] Okay.
438
00:30:27,037 --> 00:30:31,746
So, down here, 6² + y² = 100.
439
00:30:31,746 --> 00:30:36,393
That's y² = 100-36.
440
00:30:36,393 --> 00:30:39,644
y² is equal to... uh...
441
00:30:40,912 --> 00:30:42,621
Make that 64.
442
00:30:43,751 --> 00:30:46,511
And y must be 8.
443
00:30:46,511 --> 00:30:48,076
Negative-8 wouldn't make sense,
444
00:30:48,076 --> 00:30:51,511
so we're going with the
positive square root of 8.
445
00:30:51,511 --> 00:30:54,248
So now, I can plug in that unknown.
446
00:30:54,248 --> 00:30:56,594
And this is somethin' that
commonly happens.
447
00:30:56,594 --> 00:31:00,748
So, once you've got your equation,
you do your implicit differentiation;
448
00:31:00,748 --> 00:31:03,012
you fill in the stuff you know.
449
00:31:03,012 --> 00:31:08,160
A lot of times, there's another unknown
variable that you've gotta go find,
450
00:31:08,160 --> 00:31:11,254
but you will be given the
information to find it,
451
00:31:11,254 --> 00:31:13,998
and it's usually from your formula.
452
00:31:13,998 --> 00:31:17,258
So you'll plug something in
to find something else;
453
00:31:17,258 --> 00:31:22,250
then you can sub it all in, and finally,
just be left with that one unknown,
454
00:31:22,250 --> 00:31:25,491
which is (dy/dt),
and that's what we want.
455
00:31:25,911 --> 00:31:28,158
Well, 2 times 6 is 12.
456
00:31:28,158 --> 00:31:34,758
12 plus 2 times 8 is 16;
times (dy/dt) equals 0.
457
00:31:34,758 --> 00:31:42,957
Let's try to isolate (dy/dt),
so I have 16 (dy/dt) = -12;
458
00:31:42,957 --> 00:31:46,886
and the last step is just
dividing both sides by 16.
459
00:31:46,886 --> 00:31:54,113
(dy/dt) is equal to -12 over 16,
and that is negative...
460
00:31:54,783 --> 00:31:57,246
-It's like, three-fourths?
[Student] - Yep.
461
00:31:58,056 --> 00:32:03,330
So negative 0.75.
And then our units for this,
462
00:32:03,330 --> 00:32:07,265
since this is a change in y
with respect to t,
463
00:32:07,265 --> 00:32:10,391
is gonna be feet per second.
464
00:32:11,068 --> 00:32:14,612
The units of y were feet;
the units of time were seconds,
465
00:32:14,612 --> 00:32:19,505
so in (dy/dt), units are feet per second.
Man, I've almost run out of—
466
00:32:19,505 --> 00:32:22,495
[Student] Would you have a
preference on fraction or decimal?
467
00:32:22,495 --> 00:32:24,745
[Instructor] Oh no, I don't. Nah.
468
00:32:25,915 --> 00:32:28,964
To me, on these kind of problems,
though, the decimals...
469
00:32:30,070 --> 00:32:32,968
I guess I like 'em better because
I can imagine that better.
470
00:32:33,743 --> 00:32:38,022
Like, I have an idea of -0.75 feet
per second, but -3/4—
471
00:32:38,022 --> 00:32:39,756
Well, I guess it wouldn't matter.
472
00:32:40,246 --> 00:32:41,376
I don't care.
473
00:32:42,768 --> 00:32:44,355
Whatever makes you happy.
474
00:32:49,303 --> 00:32:51,855
Okay. Now, that was the famous
ladder problem, and—
475
00:32:51,855 --> 00:32:55,406
[chuckles] because, in every calculus
book since the history of calculus,
476
00:32:55,406 --> 00:32:58,161
there has been a ladder problem.
477
00:32:58,161 --> 00:33:01,096
And you will have more ladder
problems in your homework.
478
00:33:01,096 --> 00:33:04,831
And you will most likely have
a ladder problem on your next test.
479
00:33:04,831 --> 00:33:05,975
[Whispers] It's famous.
480
00:33:06,940 --> 00:33:10,021
Okay, the last question says,
"How fast is the top moving down
481
00:33:10,021 --> 00:33:12,919
when the ladder is
9 feet from the wall."
482
00:33:12,919 --> 00:33:15,090
How about 9.9 feet.
483
00:33:15,090 --> 00:33:18,630
How about 9.99999999999 feet?
484
00:33:19,503 --> 00:33:21,998
So in other words:
the ladder's only 10 feet.
485
00:33:21,998 --> 00:33:25,123
So, when you're pulling it out.
When it's 9 feet—
486
00:33:25,123 --> 00:33:29,185
I mean, most of the ladder is down.
It only has another foot to fall;
487
00:33:29,185 --> 00:33:33,084
so we're looking at the speed
at which it's falling at that point.
488
00:33:33,706 --> 00:33:34,473
Okay.
489
00:33:35,509 --> 00:33:39,996
So, let's go back to when x equals 9.
490
00:33:39,996 --> 00:33:43,266
Because we need to figure out
what y is at that point.
491
00:33:43,266 --> 00:33:45,749
'Cause, you know, if I'm
drawing a picture of it...
492
00:33:46,639 --> 00:33:49,742
It now looks like that, right?
493
00:33:49,742 --> 00:33:52,757
So it's almost all the way on the ground.
494
00:33:52,757 --> 00:33:56,130
So when x is 9, let's
figure out what y is.
495
00:33:56,130 --> 00:34:01,263
So using our Pythagorean
theorem, x² + y² = 10².
496
00:34:02,003 --> 00:34:04,779
That is, 9² is 81.
497
00:34:05,371 --> 00:34:07,747
Plus y² equals 100.
498
00:34:07,747 --> 00:34:13,250
So then y² is 100 minus 81,
which would be 19,
499
00:34:13,250 --> 00:34:17,011
and y will be the square root of 19.
500
00:34:17,633 --> 00:34:23,260
So then, we'll go back to
our "(dy/dt) equals."
501
00:34:23,632 --> 00:34:26,774
And our (dy/dt) was...
502
00:34:27,163 --> 00:34:30,624
Oh, man. Do I have to reinvent that wheel?
503
00:34:33,995 --> 00:34:35,495
Shoot. I do.
504
00:34:35,992 --> 00:34:41,504
So, (dy/dt) would equal. I'm gonna go
back to this step so I can isolate (dy/dt)
505
00:34:41,504 --> 00:34:45,081
before I've substituted in
numbers for x and for y.
506
00:34:45,081 --> 00:34:47,239
(dy/dt) would be...
507
00:34:48,176 --> 00:34:58,970
Will be -2x times (dx/dt),
and then that would be divided by 2y.
508
00:34:59,851 --> 00:35:00,824
Think I got it.
509
00:35:00,824 --> 00:35:05,944
(dy/dt) would be -2x(dx/dt) when
you subtract this from both sides,
510
00:35:05,944 --> 00:35:10,530
and then to isolate the (dy/dt),
you're dividing both sides by 2y.
511
00:35:10,530 --> 00:35:12,794
So. It looks ugly, but
this is what it looks like.
512
00:35:13,207 --> 00:35:15,313
Now we'll substitute in
our new information.
513
00:35:15,313 --> 00:35:20,789
So, our new x is a 9.
So this is -2 times 9.
514
00:35:20,789 --> 00:35:23,144
(dx/dt) is still 1.
515
00:35:23,634 --> 00:35:27,991
And then 2 times y would be
2 times the square root of 19.
516
00:35:28,590 --> 00:35:31,540
Now, I plugged all that into my
calculator already,
517
00:35:31,540 --> 00:35:38,399
and that was approximately
-2.06 feet per second.
518
00:35:38,399 --> 00:35:42,153
So it sped up. Remember when it was
6 feet away,
519
00:35:42,153 --> 00:35:46,395
the speed at which the top was falling
was -0.75 feet per second.
520
00:35:46,395 --> 00:35:51,128
Now, it sped up to -2.06
feet per second.
521
00:35:51,128 --> 00:35:53,186
[Student] -Professor?
[Instructor] -Yes, go ahead.
522
00:35:53,186 --> 00:35:55,963
[Student] The 9.9, did you round it up?
523
00:35:57,756 --> 00:35:59,263
[Student] The 10? The 10²?
524
00:36:01,757 --> 00:36:05,505
[Student] Know when it says
x + y² = the 10². Is it from the—
525
00:36:05,505 --> 00:36:06,252
[Instructor] Yes.
526
00:36:06,252 --> 00:36:09,509
[Student] But it's a question,
or you just rounded it up?
527
00:36:10,496 --> 00:36:13,260
[Instructor] So this is still going back
to my Pythagorean theorem.
528
00:36:13,260 --> 00:36:14,142
[Student] Oh, okay.
529
00:36:14,142 --> 00:36:17,781
[Instructor] I still have a hypotenuse
of 10 there; the base is 9.
530
00:36:18,750 --> 00:36:20,273
And we were looking for y.
531
00:36:20,715 --> 00:36:21,969
-Gotcha.
-Yeah.
532
00:36:22,769 --> 00:36:25,755
Turned out to be...
the square root of 19.
533
00:36:25,755 --> 00:36:26,907
That fits in there.
534
00:36:26,907 --> 00:36:32,156
So then I would do it again for
9.9, and then for 9.9999999...
535
00:36:32,156 --> 00:36:34,756
I don't have room, so I'm
gonna talk you through it.
536
00:36:35,371 --> 00:36:40,276
So, when you get to 9.9. That
ladder's almost all the way down.
537
00:36:40,796 --> 00:36:45,114
When you go through and calculate
the rate of change of y with respect to t,
538
00:36:45,114 --> 00:36:49,972
when x is 9.9, your rate is then...
539
00:36:50,500 --> 00:36:54,214
Uh, -7 feet per second.
540
00:36:54,214 --> 00:36:58,937
When you go to 9.9999999,
it's approaching infinity.
541
00:36:59,663 --> 00:37:03,760
It is negative, but so large,
it's incredible.
542
00:37:03,760 --> 00:37:08,759
So, as it's slamming the floor, the rate
at which it's slamming the floor?
543
00:37:08,759 --> 00:37:10,762
That rate is approaching infinity.
544
00:37:11,757 --> 00:37:14,610
Can't make this stuff up.
It's really true.
545
00:37:15,733 --> 00:37:17,496
That's why it damages the floor.
546
00:37:18,448 --> 00:37:19,743
It's pretty darn fast.
547
00:37:21,496 --> 00:37:25,492
All right, and that is another
famous sliding-ladder problem.
548
00:37:26,506 --> 00:37:27,785
We'll take that one away.
549
00:37:28,258 --> 00:37:32,213
And now I'm lookin' at
number 10 from the exercises.
550
00:37:32,213 --> 00:37:33,865
This one's comin' up next.
551
00:37:35,378 --> 00:37:39,009
Probably better also check what
time it is. 12:34? We're good.
552
00:37:40,242 --> 00:37:46,786
So exercise 10 says: "A particle
moves along the curve; y = √(1+x³)."
553
00:37:47,756 --> 00:37:53,061
"As it reaches the 0.23, the
y coordinate is increasing at a rate
554
00:37:53,061 --> 00:37:55,373
of 4 centimeters per second."
555
00:37:56,023 --> 00:37:57,865
That's (dy/dt).
556
00:37:58,515 --> 00:38:02,617
"How fast is the x coordinate of
the point changing at that instant?"
557
00:38:02,617 --> 00:38:07,102
Okay. So here, the graph that we draw
is the graph of the function.
558
00:38:07,102 --> 00:38:11,507
So the curve is y = √(1+x³).
559
00:38:11,507 --> 00:38:13,430
That's the graph we want to draw.
560
00:38:13,982 --> 00:38:16,620
So I'm gonna draw my
coordinate system here.
561
00:38:17,100 --> 00:38:18,363
Like so.
562
00:38:18,944 --> 00:38:22,510
And I graphed this on a graphing
calculator earlier to see what it looks like;
563
00:38:22,510 --> 00:38:28,656
and you don't have to be exactly right,
but it looks something like that.
564
00:38:29,363 --> 00:38:34,502
And then this point, I'm gonna
label this point right here at (2,3),
565
00:38:34,502 --> 00:38:37,756
because the particle is moving along,
566
00:38:37,756 --> 00:38:40,519
and at some point, it's
gonna reach that point.
567
00:38:41,250 --> 00:38:45,750
Particle's moving along the curve.
As it reaches the point (2,3),
568
00:38:45,750 --> 00:38:50,909
the y coordinate is increasing at
a rate of 4 centimeters per second.
569
00:38:51,489 --> 00:38:56,161
So we know that (dy/dt) equals 4.
570
00:38:56,548 --> 00:39:00,032
The question is, how fast is
the x coordinate of the point
571
00:39:00,032 --> 00:39:01,767
changing at that instant?
572
00:39:02,247 --> 00:39:07,254
So, what we want is (dx/dt).
573
00:39:07,744 --> 00:39:10,647
We know (dy/dt);
we want (dx/dt).
574
00:39:11,505 --> 00:39:15,509
So if I look at, you know, my little
bullets, and see where I'm at.
575
00:39:15,509 --> 00:39:16,897
I drew a picture.
576
00:39:17,508 --> 00:39:20,067
It says, "Label and assign variables."
577
00:39:20,497 --> 00:39:23,369
Well, I guess I kind of did.
I've got the point labeled,
578
00:39:23,369 --> 00:39:26,998
and I wrote down
what (dy/dt) is, and...
579
00:39:26,998 --> 00:39:30,273
I wrote down what I don't
know, which is (dx/dt).
580
00:39:30,873 --> 00:39:34,758
So then I find an equation or formula
that relates all of these variables.
581
00:39:34,758 --> 00:39:41,490
Well, that equation or formula
is the y = √(1+x³).
582
00:39:41,490 --> 00:39:43,763
That's relating x and y.
583
00:39:44,503 --> 00:39:49,296
We wanna use implicit differentiation
now to differentiate with respect to time.
584
00:39:49,296 --> 00:39:54,753
And then, we'll substitute in what we
know; solve for what we don't know.
585
00:39:55,900 --> 00:40:00,619
So now I need to find the
derivative with respect to t.
586
00:40:00,619 --> 00:40:05,024
So I want derivative with respect
to t of the left-hand side.
587
00:40:05,024 --> 00:40:09,101
I want derivative with respect
to t of the right-hand side,
588
00:40:09,101 --> 00:40:14,510
which I'm going to rewrite
as (1+x³) to the ½ power.
589
00:40:14,510 --> 00:40:16,725
Just makes it easier
for me to differentiate.
590
00:40:17,499 --> 00:40:21,511
So now on the left, it is just (dy/dt).
591
00:40:22,245 --> 00:40:26,961
And then on the right, derivative
of that (1+x³) to the ½.
592
00:40:26,961 --> 00:40:29,383
So bring my ½ down in front.
593
00:40:29,383 --> 00:40:33,736
(1+x³) to the -½ power.
594
00:40:33,736 --> 00:40:36,760
Now multiply by the
derivative of the inside.
595
00:40:36,760 --> 00:40:40,681
Okay, now. Your inside is this (1+x³).
596
00:40:40,681 --> 00:40:44,102
Derivative of (1+x³) is...
597
00:40:45,177 --> 00:40:46,258
3x².
598
00:40:46,888 --> 00:40:50,669
But now, chain rule says,
"Do it again"; it's a double chain.
599
00:40:51,275 --> 00:40:55,525
Now we need to multiply by the
derivative of x with respect to t.
600
00:40:55,525 --> 00:40:58,763
Because x was the wrong letter.
601
00:40:58,763 --> 00:41:00,567
t's the right letter;
x is the wrong letter,
602
00:41:00,567 --> 00:41:03,457
so I've gotta follow it
with that (dx/dt).
603
00:41:04,260 --> 00:41:10,276
Now I've differentiated implicitly;
now it's time to sub in what I know.
604
00:41:10,746 --> 00:41:14,393
So, I do know that (dy/dt) is 4.
605
00:41:15,503 --> 00:41:21,394
That's 4 equals ½ times 1 plus...
What's x at this point?
606
00:41:23,059 --> 00:41:23,791
[Student] -2.
[Instructor] -2.
607
00:41:24,327 --> 00:41:31,772
So that's a 2³, to the -½.
And that's times 3 times a 2².
608
00:41:31,772 --> 00:41:34,717
And then that's times (dx/dt).
609
00:41:34,717 --> 00:41:38,403
(dx/dt) is the unknown;
that's what I need to solve for.
610
00:41:38,946 --> 00:41:41,085
All right, so this is 4 equals.
611
00:41:41,647 --> 00:41:46,499
Ummm, 2³ is 8.
8 plus 1 is 9.
612
00:41:46,499 --> 00:41:54,272
9 to the -½, so it's like 1 over
the square root of 9, is... 3, I think.
613
00:41:54,716 --> 00:41:58,561
So this would be 1 over, 2 times...
614
00:41:59,111 --> 00:42:02,266
8+1 is 9; square root of that is 3.
615
00:42:02,266 --> 00:42:04,355
So that's 1 over 6.
616
00:42:04,355 --> 00:42:08,524
And then 3 times 2²;
that's 4 times 3; that's 12.
617
00:42:09,032 --> 00:42:10,640
(dx/dt).
618
00:42:11,258 --> 00:42:16,511
So, this is 4 equals. 2 goes into 12
six times; 6 over 3—
619
00:42:16,511 --> 00:42:20,150
that's just a 2 times (dx/dt).
620
00:42:20,150 --> 00:42:25,502
I think I'm ready to isolate my (dx/dt)
by dividing both sides by 2.
621
00:42:26,136 --> 00:42:30,618
And (dx/dt) is 4 over 2, which is 2.
622
00:42:31,210 --> 00:42:34,890
And the rate is in centimeters per second.
623
00:42:36,989 --> 00:42:40,498
Okay. So sometimes, I guess, solving
the equation after you substitute in
624
00:42:40,498 --> 00:42:43,498
your known values can get
a little tricky, but you know;
625
00:42:43,498 --> 00:42:46,759
just take it one step at a time,
and you'll get there.
626
00:42:47,403 --> 00:42:49,271
So let me know how that one went.
627
00:42:52,381 --> 00:42:55,035
[Student] Can you just go over
what happened to, uh, 12?
628
00:42:56,105 --> 00:42:57,246
[Instructor] Yeah, sure.
629
00:42:57,246 --> 00:43:01,497
So, the ½, times the 12? Is 6.
630
00:43:02,747 --> 00:43:06,263
So I just canceled the 2 with
the 12, leaving me a 6 on top;
631
00:43:06,263 --> 00:43:09,269
but 6 over 3 is 2.
632
00:43:14,907 --> 00:43:15,805
Good?
633
00:43:17,757 --> 00:43:18,519
[Student] Yeah.
634
00:43:20,667 --> 00:43:25,853
All right. So, you guys are so quiet.
I don't—I don't like that about Zoom;
635
00:43:25,853 --> 00:43:28,770
it's different than being in a classroom;
in a classroom, you know...
636
00:43:28,770 --> 00:43:31,948
We can see each other's eyeballs,
and you can just ask a question;
637
00:43:31,948 --> 00:43:34,333
or sometimes I'll look at you,
and I know you have a question,
638
00:43:34,333 --> 00:43:35,666
and I'll say "What's up."
639
00:43:36,006 --> 00:43:39,270
Um, jump in there; really. Stop me
any time you wanna stop me.
640
00:43:39,270 --> 00:43:43,259
Don't be shy or embarrassed about it.
Stop me, and ask your question.
641
00:43:43,259 --> 00:43:46,275
Because the most important thing
is that you guys continue to learn.
642
00:43:47,498 --> 00:43:51,244
Exercise 4 says, "The length of a
rectangle is increasing at a rate
643
00:43:51,244 --> 00:43:54,646
of 8 centimeters per second."
Got a rectangle.
644
00:43:54,646 --> 00:43:58,982
"And its width is increasing at a rate
of 3 centimeters per second."
645
00:43:58,982 --> 00:44:02,497
"When the length is 20,
and the width is 10,
646
00:44:02,497 --> 00:44:06,743
how fast is the area of
the rectangle increasing?"
647
00:44:06,743 --> 00:44:10,763
Okay. So the first thing we're
gonna do? Draw a picture.
648
00:44:11,073 --> 00:44:13,258
So, I've got a rectangle here.
649
00:44:14,502 --> 00:44:17,396
Here we go. And I'm gonna
label this thing.
650
00:44:17,888 --> 00:44:20,506
So it says, "The length of the rectangle
is increasing at a rate
651
00:44:20,506 --> 00:44:22,501
of 8 centimeters per second;
652
00:44:22,501 --> 00:44:26,036
width is increasing at a rate of
3 centimeters per second."
653
00:44:26,036 --> 00:44:30,206
"When the length is 20,
and the width is 10,
654
00:44:30,384 --> 00:44:34,393
how fast is the area of
the rectangle increasing."
655
00:44:34,685 --> 00:44:41,114
So, like, right now, the area is
20 times 10, or 200, but.
656
00:44:41,114 --> 00:44:43,502
We're gonna be increasing the length
and the width,
657
00:44:43,502 --> 00:44:46,771
and looking at how fast
that area is changing.
658
00:44:46,771 --> 00:44:50,263
So I'm gonna write down the things
that I know. I've given a lot in this problem.
659
00:44:50,263 --> 00:44:56,104
It says the length is increasing at a rate
of 8 centimeters per second.
660
00:44:56,104 --> 00:45:03,313
So, that would be the derivative
of l, with respect to time.
661
00:45:03,916 --> 00:45:05,454
That is 8.
662
00:45:05,753 --> 00:45:09,984
It says, the width is increasing at a
rate of 3 centimeters, so.
663
00:45:10,390 --> 00:45:16,032
dw, the change in width,
with respect to time. That one is 3.
664
00:45:16,401 --> 00:45:24,948
We know that we're kind of stopping this
when l is 20, and when w is 10.
665
00:45:25,217 --> 00:45:27,259
So, there are four things that I know.
666
00:45:27,259 --> 00:45:30,089
What do I not know? What do I need.
667
00:45:30,611 --> 00:45:34,372
I need, or want to know...
how fast the area—
668
00:45:34,372 --> 00:45:37,474
[Student] -(da/dt)?
[Instructor] Yeah. How fast the area.
669
00:45:37,905 --> 00:45:40,624
Derivative of area with respect to time.
670
00:45:40,624 --> 00:45:44,515
I need the rate of change of
the area with respect to time.
671
00:45:45,104 --> 00:45:48,372
So, if I'm looking for the
rate of change of area,
672
00:45:48,372 --> 00:45:50,915
then I want to use
the area formula here.
673
00:45:51,286 --> 00:45:53,005
Area of a rectangle?
674
00:45:58,649 --> 00:46:00,014
Length times width.
675
00:46:00,014 --> 00:46:03,025
So there's my formula
relating all of my variables;
676
00:46:03,025 --> 00:46:05,630
it's time to differentiate implicitly.
677
00:46:06,145 --> 00:46:12,076
So now we'll get the derivative
with respect to t of the left-hand side.
678
00:46:12,416 --> 00:46:16,808
And the derivative with respect
to t of the right-hand side.
679
00:46:17,248 --> 00:46:21,019
Now on the left, there's your (da/dt).
680
00:46:21,019 --> 00:46:23,128
This is the very thing we're lookin' for.
681
00:46:23,128 --> 00:46:26,986
So then on the right, we need to get
the derivative of length times width.
682
00:46:26,986 --> 00:46:31,903
So I said it: length times width.
This is a...?
683
00:46:32,449 --> 00:46:34,332
[Student] -Product rule.
[Instructor] -Product rule.
684
00:46:34,332 --> 00:46:40,078
So we want the first function, l. Times
the derivative of the second function.
685
00:46:40,078 --> 00:46:44,606
Okay now, remember: t's the right letter.
Everything else is the wrong letter.
686
00:46:44,606 --> 00:46:47,507
So when I do first times
the derivative of the second,
687
00:46:47,507 --> 00:46:50,258
I don't know what the
derivative of the second is,
688
00:46:50,258 --> 00:46:53,370
so I have to write (dw/dt).
689
00:46:54,130 --> 00:46:56,974
So, l times (dw/dt).
690
00:46:57,505 --> 00:47:03,367
And then plus the second, which is w,
times the derivative of the first,
691
00:47:03,367 --> 00:47:06,818
which has to be (dl /dt).
692
00:47:07,622 --> 00:47:08,953
So now I'm gonna go back up here,
693
00:47:08,953 --> 00:47:11,765
where I was given these
four pieces of information.
694
00:47:11,765 --> 00:47:13,920
I'm gonna substitute them in.
695
00:47:14,610 --> 00:47:19,757
(da/dt) = l, at this moment
in time, is 20.
696
00:47:20,422 --> 00:47:23,274
(dw/dt) is 3.
697
00:47:24,259 --> 00:47:27,612
Plus w at this moment is 10.
698
00:47:27,612 --> 00:47:30,423
And (dl/dt) is 8.
699
00:47:30,423 --> 00:47:32,092
Okay, this one's gonna be easy.
700
00:47:32,982 --> 00:47:36,509
Don't have to isolate anything;
just multiply and add.
701
00:47:36,509 --> 00:47:39,764
So, that's gonna be 60 plus 80,
702
00:47:39,764 --> 00:47:43,529
and 80 plus 60 would be 140.
703
00:47:43,948 --> 00:47:46,936
Now, units. What are the
units for the area?
704
00:47:49,389 --> 00:47:51,482
[Student] Centimeters
squared per second?
705
00:47:51,840 --> 00:47:52,658
[Instructor] Mhm.
706
00:47:53,254 --> 00:47:56,511
So, units for area are
centimeters squared.
707
00:47:56,511 --> 00:48:00,279
The units for time are second.
So it says that,
708
00:48:00,279 --> 00:48:04,901
at this point in time, when our
rectangle is this big, and it's increasing?
709
00:48:04,901 --> 00:48:10,871
That the rate of change in the area is
140 square centimeters for every second.
710
00:48:11,662 --> 00:48:13,965
Okay. Almost done.
That was an easy one.
711
00:48:15,015 --> 00:48:16,805
Maybe we should've done that one first.
712
00:48:21,519 --> 00:48:28,624
Okay. The last one on this handout,
I believe. Exercise 32. Exercise 32.
713
00:48:29,631 --> 00:48:31,236
Oh, but this is a good one.
714
00:48:32,034 --> 00:48:38,061
So, exercise 32 says, "Two sides of
a triangle have lengths 12 meters
715
00:48:38,061 --> 00:48:41,638
and 15 meters."
Two sides of a triangle.
716
00:48:41,638 --> 00:48:46,004
It didn't say a right triangle.
Just said "a triangle."
717
00:48:46,004 --> 00:48:52,753
"The angle between them is increasing
at a rate of 2 degrees per minute."
718
00:48:52,753 --> 00:48:55,763
"How fast is the length of
the third side increasing
719
00:48:55,763 --> 00:49:00,125
when the angle between
the sides of fixed length is 60?"
720
00:49:00,617 --> 00:49:01,953
[Exaggerated shriek]
721
00:49:01,953 --> 00:49:05,757
If it were a right triangle, this
would be so much easier to draw!
722
00:49:05,757 --> 00:49:09,515
But it didn't say that; and it's not;
and it's changing; so, man!
723
00:49:09,935 --> 00:49:11,259
Let me just go for it.
724
00:49:11,259 --> 00:49:13,270
So I'm gonna draw a triangle.
725
00:49:14,079 --> 00:49:17,256
Maybe something like—I'm gonna
make this pretty big. [Chuckle]
726
00:49:17,256 --> 00:49:18,497
Something like that.
727
00:49:19,255 --> 00:49:22,279
And your triangle doesn't have to
look exactly like mine, but.
728
00:49:22,990 --> 00:49:25,075
I'll be danged if that doesn't
look like a right triangle.
729
00:49:25,075 --> 00:49:27,771
That looks like a right angle right
there. I just couldn't help myself.
730
00:49:27,771 --> 00:49:29,998
It's not. Not a right triangle.
731
00:49:29,998 --> 00:49:31,494
So I'm gonna label my sides.
732
00:49:31,494 --> 00:49:35,779
I'm gonna call that one 12, and
this one 15, because it looks longer.
733
00:49:36,509 --> 00:49:42,261
And then there's an angle between them,
and that angle between them we'll call θ.
734
00:49:42,261 --> 00:49:46,374
So, if that's θ, and here
are the two sides.
735
00:49:46,374 --> 00:49:49,163
What's happening is,
that is opening up.
736
00:49:49,163 --> 00:49:54,265
So as that opens up, we're looking to
see how that third side is changing.
737
00:49:54,265 --> 00:49:57,643
It's obviously growing; it's getting
longer. We're looking for that.
738
00:49:58,223 --> 00:50:01,273
"How fast is the length of
the third side increasing
739
00:50:01,273 --> 00:50:06,092
when the angle between the sides
of fixed length is 60 degrees."
740
00:50:06,560 --> 00:50:09,624
So guess let's start writing down
the things that we know here.
741
00:50:10,504 --> 00:50:17,096
So, we know that two sides of the
triangle are 12 and 15... OK, got that.
742
00:50:17,096 --> 00:50:21,511
The angle between them is increasing
at a rate. Ah. This is a rate that we know.
743
00:50:21,511 --> 00:50:25,901
And it's the rate of change of that
angle with respect to time.
744
00:50:25,901 --> 00:50:32,503
So, we know (dθ/dt). (dθ/dt).
745
00:50:32,503 --> 00:50:35,374
And that is a rate of 2 degrees
per minute.
746
00:50:35,374 --> 00:50:38,030
So (dθ/dt) is 2.
747
00:50:38,612 --> 00:50:41,508
"How fast is the length of
the third side increasing
748
00:50:41,508 --> 00:50:45,258
when the angle between the
sides of fixed length is 60?
749
00:50:45,924 --> 00:50:48,753
So, it's telling us that we're kind of
stopping this,
750
00:50:48,753 --> 00:50:52,622
looking at when that angle
is 60 degrees right then,
751
00:50:52,622 --> 00:50:54,808
how fast is the third side changing.
752
00:50:55,247 --> 00:50:57,993
Well, we need to give a name
to that third side.
753
00:50:58,633 --> 00:51:02,251
Hmm, I don't—what do you wanna
call that third side? Anybody?
754
00:51:03,749 --> 00:51:04,940
Any variable?
755
00:51:06,377 --> 00:51:08,656
[Student] -x.
[Instructor] -Why not?
756
00:51:08,656 --> 00:51:12,185
So we'll call that third side x.
Works for me.
757
00:51:12,822 --> 00:51:14,506
Now we need a formula.
758
00:51:14,506 --> 00:51:17,260
We need a formula that relates
what's going on here.
759
00:51:17,260 --> 00:51:21,504
So, look at your picture.
Your knowns; your unknowns.
760
00:51:21,504 --> 00:51:23,950
Does a formula come to mind—
761
00:51:23,950 --> 00:51:27,800
and it cannot be Pythagoras,
because this is not a right triangle.
762
00:51:28,767 --> 00:51:31,877
[Student] This is the
double-angle thing? I mean...
763
00:51:31,877 --> 00:51:36,404
It's sine over hypotenuse
equals sine over hypotenuse?
764
00:51:36,404 --> 00:51:37,649
[Instructor] Not that one.
765
00:51:38,633 --> 00:51:42,410
[Student #2] Is this a sine-angle-sine
problem? Or a side-angle-side problem?
766
00:51:42,410 --> 00:51:45,514
[Student #3] -Is it the law of sines?
[Instructor] -Yes. Yes, it's SAS.
767
00:51:48,993 --> 00:51:51,150
So, think trigonometry.
768
00:51:53,380 --> 00:51:55,133
[Student #3] It's not the
law of sines or anything, is it?
769
00:51:56,045 --> 00:51:58,136
[Instructor] Keep thinkin'. You're close.
770
00:52:05,508 --> 00:52:06,501
[Student] Law of cosine?
771
00:52:06,501 --> 00:52:08,947
[Instructor] Yeah, that might
help if I write that in there.
772
00:52:08,947 --> 00:52:14,203
So when you know two sides and the
included angle, that's a law of cosines.
773
00:52:14,203 --> 00:52:19,622
And we know two sides. And we know
the included angle at this moment is 60°.
774
00:52:19,622 --> 00:52:22,614
So definitely a law of cosines.
775
00:52:23,219 --> 00:52:25,752
Yayyy, I love the law of cosines!
776
00:52:25,752 --> 00:52:30,248
That part of trig was so fun; solving
for the triangles using the law of sines
777
00:52:30,248 --> 00:52:32,498
and the law of cosines.
I loved doing those problems.
778
00:52:32,498 --> 00:52:35,997
Remember the vector problems?
They were great.
779
00:52:36,622 --> 00:52:39,493
Okay, now, what does the
law of cosines say?
780
00:52:39,493 --> 00:52:42,342
Well, the law of cosines says this:
781
00:52:42,342 --> 00:52:46,251
That your side opposite,
which we're calling x.
782
00:52:46,251 --> 00:52:49,628
We're gonna square it.
x² is equal to.
783
00:52:49,628 --> 00:52:53,509
And it's the sum of the squares
of the other two sides;
784
00:52:53,509 --> 00:52:56,914
so it starts out kind of looking
like the Pythagorean theorem.
785
00:52:56,914 --> 00:53:04,615
But then it's minus 2 times a
times b times the cosine of θ.
786
00:53:04,615 --> 00:53:06,785
This is the law of cosines.
787
00:53:07,275 --> 00:53:10,035
So that's our formula relating everything.
788
00:53:10,514 --> 00:53:12,780
Ummmm, what do we next?
789
00:53:14,331 --> 00:53:16,027
Implicit differentiation.
790
00:53:16,875 --> 00:53:20,288
So, we want derivative
with respect to t.
791
00:53:21,357 --> 00:53:24,263
Of the left-hand side, which is x².
792
00:53:24,263 --> 00:53:28,245
And the derivative with respect
to t of the right-hand side,
793
00:53:28,245 --> 00:53:34,309
which is (a² + b² − 2abcosθ).
794
00:53:34,763 --> 00:53:39,775
Well, I say before we get this derivative,
maybe we substitute in what we know,
795
00:53:39,775 --> 00:53:42,587
with the sides 12 and 15?
796
00:53:42,587 --> 00:53:44,927
Ummm, I can do that.
797
00:53:44,927 --> 00:53:49,021
Or not; I don't have to.
I can live with it like this.
798
00:53:49,021 --> 00:53:50,150
Y'all, give me a preference.
799
00:53:50,150 --> 00:53:54,089
Do you want me to substitute in those
numbers now, or get the derivative first?
800
00:53:54,089 --> 00:53:57,635
If I get the derivative first, you know,
these will just be zeros,
801
00:53:57,635 --> 00:53:59,748
because there are only constants.
802
00:53:59,748 --> 00:54:01,998
Weigh in with your preference here.
803
00:54:05,421 --> 00:54:06,583
[Student] Put the numbers?
804
00:54:07,093 --> 00:54:09,737
[Instructor] -Put the numbers in?
[Student] -Yes.
805
00:54:11,497 --> 00:54:17,373
So, derivative with respect to t of x²
equals the derivative with respect to t;
806
00:54:17,373 --> 00:54:19,414
and we'll put those numbers in.
807
00:54:19,414 --> 00:54:24,261
So, the a? I guess I'll just
call a the base; 15.
808
00:54:24,261 --> 00:54:29,828
That would be 15². Plus the other
side squared; so that's 12².
809
00:54:29,828 --> 00:54:36,504
Minus 2 times 15 times 12.
Times the cosine of θ.
810
00:54:36,504 --> 00:54:38,642
And then we can clean that up a bit.
811
00:54:38,642 --> 00:54:41,937
This is the derivative
with respect to t of x².
812
00:54:41,937 --> 00:54:45,610
Notice I'm not taking the derivative
yet; I'm just cleaning this up a bit.
813
00:54:45,610 --> 00:54:48,966
Equals derivative with respect to t of.
814
00:54:48,966 --> 00:54:52,671
If I do 15² + 12².
815
00:54:52,671 --> 00:54:55,002
Go into my calculator here.
816
00:54:56,106 --> 00:54:59,988
15² plus 12².
817
00:54:59,988 --> 00:55:02,654
Okay. That is 369.
818
00:55:03,244 --> 00:55:06,756
So that would be 369 minus.
819
00:55:06,756 --> 00:55:10,155
Now, the 2 times 15 times 12?
820
00:55:12,032 --> 00:55:14,518
That is 360.
821
00:55:16,003 --> 00:55:18,369
Sitting in front of the cosθ.
822
00:55:18,369 --> 00:55:21,973
Okay. Now, let's differentiate.
Let's do it now.
823
00:55:21,973 --> 00:55:26,256
So then on the left-hand side,
remember that x is the wrong letter.
824
00:55:26,256 --> 00:55:32,171
So when I get the derivative of x²,
it's 2x, but follow it by...?
825
00:55:36,627 --> 00:55:37,895
(dx/dt).
826
00:55:39,047 --> 00:55:40,242
On the right-hand side.
827
00:55:40,242 --> 00:55:45,609
The derivative of 369 is just 0,
so we won't worry about that.
828
00:55:45,609 --> 00:55:50,419
So now let's look at the
derivative of -360cosθ.
829
00:55:51,170 --> 00:55:54,181
Well, that constant in front
just hangs out.
830
00:55:54,531 --> 00:55:56,263
What's the derivative of cosine?
831
00:55:57,508 --> 00:55:58,754
[Student] Negative-sine.
832
00:55:58,754 --> 00:56:01,767
[Instructor] So since it's
negative-sine, then we can do...
833
00:56:02,317 --> 00:56:03,161
That.
834
00:56:03,511 --> 00:56:06,257
So 360sinθ.
835
00:56:06,257 --> 00:56:11,024
Now, θ is the wrong variable,
so what do we follow this by?
836
00:56:15,877 --> 00:56:17,079
(dθ/dt).
837
00:56:17,079 --> 00:56:18,648
And that's the chain rule.
838
00:56:18,962 --> 00:56:24,759
So if you have a cosθ,
derivative is -sinθ (dθ/dt).
839
00:56:24,759 --> 00:56:26,510
That's the derivative of the inside.
840
00:56:27,250 --> 00:56:28,266
Okay, great.
841
00:56:28,266 --> 00:56:31,248
So now we're ready to substitute in
things that we know;
842
00:56:31,248 --> 00:56:33,672
and we're solving for...
843
00:56:33,672 --> 00:56:36,824
What are we solving for? I didn't
write down what we needed to know.
844
00:56:38,714 --> 00:56:39,756
We need...
845
00:56:40,346 --> 00:56:44,208
And it says, "How fast is the length
of the third side increasing?"
846
00:56:44,781 --> 00:56:50,244
We need (dx/dt), the rate of
change of x with respect to t.
847
00:56:50,244 --> 00:56:52,954
Okay, got it. So I'm solving for (dx/dt).
848
00:56:53,384 --> 00:56:56,746
Well, then on the left-hand side,
I'll have 2 times x.
849
00:56:57,260 --> 00:57:00,168
Ummm. How are we gonna find x here?
850
00:57:06,354 --> 00:57:08,247
How we gonna find x.
851
00:57:09,509 --> 00:57:12,536
I'm bringing my picture right down
in front of your face, there.
852
00:57:16,000 --> 00:57:17,834
If I'm looking for this side...
853
00:57:18,479 --> 00:57:20,256
[Student] Is it a [inaudible] equation?
854
00:57:20,256 --> 00:57:23,014
[Instructor] We're gonna plug it
into the law of cosines
855
00:57:23,014 --> 00:57:25,375
to find out what this
third side would be
856
00:57:25,375 --> 00:57:31,821
when the two sides are 12 and 15,
and, at this moment, that angle is 60°.
857
00:57:31,821 --> 00:57:36,354
So we're going to go back to the law of
cosines just to determine this unknown.
858
00:57:36,734 --> 00:57:40,255
Remember, we had to do this
before on one of the ladder problems.
859
00:57:41,283 --> 00:57:44,675
Okay. So then, using a law
of cosines, it would say...
860
00:57:44,675 --> 00:57:46,931
I'll try to do this over here on the side.
861
00:57:46,931 --> 00:57:51,848
It would say that x²
is equal to a² + b².
862
00:57:51,848 --> 00:57:55,373
So that's 15² + 12² again.
863
00:57:55,373 --> 00:58:03,736
Minus 15 times 12 times the 2;
times the cosine of 60°.
864
00:58:04,630 --> 00:58:09,170
So our x² equals.
That 15² +12² ?
865
00:58:09,170 --> 00:58:12,650
That was the 369.
866
00:58:12,650 --> 00:58:20,951
And then 15 times 12 times 2,
that was the -360cos60°.
867
00:58:20,951 --> 00:58:26,306
So x² is 369 minus 360 times.
868
00:58:26,306 --> 00:58:29,761
And the cosine of 60°
is one that we know.
869
00:58:29,761 --> 00:58:31,626
[Student] -One-half?
[Instructor] -Is what?
870
00:58:31,626 --> 00:58:33,014
[Student] -One-half, I think?
[Instructor] -One over two.
871
00:58:33,014 --> 00:58:34,352
One-half is right.
872
00:58:34,352 --> 00:58:36,056
So this is ½.
873
00:58:36,056 --> 00:58:41,279
x² is 369 minus...
I guess that'd be 180?
874
00:58:41,694 --> 00:58:45,884
And then 369 minus 180 is—
[goofy voice] I unno.
875
00:58:51,843 --> 00:58:53,261
[Instructor] -I got—
[Student] -189.
876
00:58:53,261 --> 00:58:57,993
[Instructor] 189. So x would be
the square root of that.
877
00:58:57,993 --> 00:59:03,150
Which is not real pretty;
it's 13.7-ish.
878
00:59:03,150 --> 00:59:06,504
So I'm just gonna leave it at 13.7.
879
00:59:09,257 --> 00:59:11,775
More decimal places would be better;
880
00:59:11,775 --> 00:59:15,614
but I kinda messed myself up by not
giving myself very much room to write
881
00:59:15,614 --> 00:59:19,520
any number in here at all, sooo,
I'm gonna have to just round it off.
882
00:59:19,520 --> 00:59:20,942
So 13.7.
883
00:59:20,942 --> 00:59:25,492
And that equals the 360
times the sine of θ.
884
00:59:25,492 --> 00:59:34,607
Oh, but the sine of θ is the sine of...
60°, times (dθ/dt), which was 2.
885
00:59:36,036 --> 00:59:37,736
Hold on. I can fix this.
886
00:59:43,998 --> 00:59:49,869
My Calc 2 student told me on Wednesday,
"So why don't you just use a pencil?"
887
00:59:49,869 --> 00:59:52,508
"Then if you mess up, it's no big deal!"
888
00:59:53,231 --> 00:59:54,036
Well.
889
00:59:55,020 --> 00:59:57,873
I've got a... Wite-Out tape here.
890
01:00:02,090 --> 01:00:03,640
So let's fix all that.
891
01:00:04,850 --> 01:00:07,300
Like, really? You're not gonna work?
892
01:00:10,249 --> 01:00:12,779
[Cries] Why is my life so hard?!
893
01:00:13,498 --> 01:00:15,059
All right. So I'll just rewrite it.
894
01:00:15,760 --> 01:00:20,384
2 times 13.7, times (dx/dt).
895
01:00:20,384 --> 01:00:25,009
Equals 360 times the sine of 60°;
896
01:00:25,009 --> 01:00:28,766
times (dθ/dt), which was 2.
897
01:00:31,489 --> 01:00:32,752
There we go.
898
01:00:32,752 --> 01:00:36,685
Now, 2 times 13.7—aw, heck.
You know what I'm gonna do?
899
01:00:37,125 --> 01:00:43,746
Say (dx/dt) is equal to.
2 times 360 would be...
900
01:00:44,516 --> 01:00:50,059
720. Sine of 60°.
I know that one, too.
901
01:00:51,119 --> 01:00:52,148
That would be...
902
01:00:52,148 --> 01:00:55,508
[Student] -Square root of 3 over 2.
[Instructor] -Square root of 3 over 2.
903
01:00:55,508 --> 01:01:00,430
And then, let's divide that
by 2 times 13.7.
904
01:01:01,767 --> 01:01:04,760
All right, I'm going to my
calculator to figure this one out.
905
01:01:05,070 --> 01:01:09,337
720. Times the square root of 3.
906
01:01:18,265 --> 01:01:19,785
Divided by 2.
907
01:01:19,785 --> 01:01:24,954
And then that divided by
2 times 13.7.
908
01:01:24,954 --> 01:01:32,277
Hey y'all; if I didn't fat-finger this,
I got approximately 22.7.
909
01:01:32,951 --> 01:01:34,438
And now I need a unit for that.
910
01:01:34,438 --> 01:01:38,605
This was a rate of change
of x with respect to time.
911
01:01:38,605 --> 01:01:44,914
And x was measured in meters,
and time was measured in minutes.
912
01:01:44,914 --> 01:01:50,266
So, 22.7 meters per minute.
913
01:01:53,044 --> 01:01:54,043
Okay.
914
01:01:54,043 --> 01:01:55,263
[Student] Umm.
915
01:01:56,222 --> 01:01:56,992
[Instructor] Yes?
916
01:01:58,425 --> 01:02:01,886
[Student] -Oh, nothing; I just said "Wow."
[Instructor] -Oh, okay. Wow!
917
01:02:03,232 --> 01:02:07,636
So, yeah. This was a pretty
challenging section; but also doable.
918
01:02:07,636 --> 01:02:11,502
So if you look at those bullets,
draw the picture; label;
919
01:02:11,502 --> 01:02:14,964
write down what you know and what you
don't know; what you need to know.
920
01:02:14,964 --> 01:02:17,257
Find a formula that relates everything.
921
01:02:17,257 --> 01:02:20,761
If you try to go through that
step-by-step, I think you'll be just fine.
922
01:02:20,761 --> 01:02:24,022
And I tried to pick problems—
most of them—
923
01:02:24,022 --> 01:02:29,170
are like the ones that we did...
in class today? On Zoom today?
924
01:02:29,170 --> 01:02:33,530
And so, hopefully you'll have an example
for almost everything in the homework.
925
01:02:34,010 --> 01:02:38,003
But definitely stop by
during office hours; um...
926
01:02:38,003 --> 01:02:43,054
I've got that all figured out now,
and I'm in Blackboard from 10 to 11;
927
01:02:43,054 --> 01:02:46,991
so, you know, before your
class for an hour, stop in.
928
01:02:46,991 --> 01:02:52,040
Or this afternoon from 3:45 to 4:45. I'm
on Blackboard again, and Collaborate.
929
01:02:52,040 --> 01:02:56,419
So, there's a link in your Blackboard.
Just go to it; and I'll be there,
930
01:02:56,419 --> 01:02:58,998
and I can help you with
homework problems, so.
931
01:02:58,998 --> 01:03:03,147
Especially some of you who were used to
coming by when we were still at Hays.
932
01:03:03,147 --> 01:03:06,774
Come on by! I wanna still be able
to help you, even though it's not...
933
01:03:06,774 --> 01:03:10,521
quite as effective this way?
It's still better than nothing.
934
01:03:11,080 --> 01:03:14,320
Umm. And then the tutoring labs,
the learning labs,
935
01:03:14,320 --> 01:03:17,371
have gone online using Brainfuse.
936
01:03:18,082 --> 01:03:20,987
They were supposed to send out
an email about that.
937
01:03:20,987 --> 01:03:24,510
I never got one. I'm hoping
that the students did.
938
01:03:24,510 --> 01:03:27,388
Somebody let me know if
you've got anything about that?
939
01:03:33,098 --> 01:03:35,291
[Student #1] -I didn't.
[Instructor] -Ahh.
940
01:03:35,291 --> 01:03:38,252
[Student #2] I think I remember seeing
something about [inaudible]...
941
01:03:40,983 --> 01:03:43,743
[Student #3] So, I've been
talking to a tutor. Um.
942
01:03:43,743 --> 01:03:48,623
And we've been meeting on Zoom.
But the way that she sees it
943
01:03:48,623 --> 01:03:52,711
is that it's almost like a ticketing system,
kinda like how Highland works now;
944
01:03:52,711 --> 01:03:55,398
where you send in a ticket
for a singular question,
945
01:03:55,398 --> 01:03:57,575
and then they can reach
out to you and help?
946
01:03:57,575 --> 01:04:00,075
[Instructor] Do you do it
from the website?
947
01:04:01,245 --> 01:04:03,004
[Student #3] Yeah, I believe so.
948
01:04:03,004 --> 01:04:06,746
There was a post on it on the front
page; I don't know if it's still there.
949
01:04:06,746 --> 01:04:10,675
[Instructor] Oh, okay. So, maybe
just go to austincc.edu.
950
01:04:11,207 --> 01:04:13,269
And if there's not anything
on the front page,
951
01:04:13,269 --> 01:04:15,271
maybe do a search for "learning lab,"
952
01:04:15,271 --> 01:04:18,242
and then hopefully their page
will come up with information.
953
01:04:18,242 --> 01:04:20,187
I did not recieve anything about it.
954
01:04:20,187 --> 01:04:24,758
I just—I heard from someone who works
there that they were gonna do Brainfuse.
955
01:04:25,728 --> 01:04:28,333
So yeah, the ticketing system,
that would be okay, I guess;
956
01:04:28,333 --> 01:04:30,879
and just kinda wait until it's your turn.
957
01:04:33,685 --> 01:04:35,957
[Student #3] What problem set
are we working on?
958
01:04:35,957 --> 01:04:37,061
[Instructor] Sorry?
959
01:04:37,061 --> 01:04:40,622
[Student #3] What problem set are we
going to be working on, for next class?
960
01:04:40,622 --> 01:04:45,629
[Instructor] Oh, um. So this is
Section... uhh, what is this? 4.1.
961
01:04:47,223 --> 01:04:49,653
So you'll have homework from 4.1.
962
01:04:50,626 --> 01:04:53,738
And then on next class,
we're gonna try to do...
963
01:04:53,738 --> 01:04:56,232
2.8, which will go fast.
964
01:04:56,232 --> 01:04:58,503
And then, 4.2.
965
01:04:58,503 --> 01:05:00,762
We'll try to do two sections. We'll see.
966
01:05:01,913 --> 01:05:04,661
[Student] So the homework
is 4.1, right? And 4.2?
967
01:05:05,389 --> 01:05:07,713
[Instructor] -Yeah.
[Student] -Okay.
968
01:05:09,384 --> 01:05:10,881
[Student #3] Wait, so um...
969
01:05:10,881 --> 01:05:14,629
We're doing the homework
for 4.1 and 4.2 for next class?
970
01:05:14,629 --> 01:05:16,650
-We're not doing the—
-[Instructor] No-no-no-no-no, no-no.
971
01:05:16,650 --> 01:05:19,756
So, all you need to be working
on right now is 4.1.
972
01:05:21,136 --> 01:05:22,828
[Student] Okay. So we'll just
do the normal homework.
973
01:05:22,828 --> 01:05:25,089
So we're not doing problem
sets... between Mondays
974
01:05:25,089 --> 01:05:26,872
-and Wednesdays anymore?
-Oh, oh; I see what you're saying.
975
01:05:26,872 --> 01:05:31,085
So I'm not giving you a problem set
this week, because you just had a test.
976
01:05:31,085 --> 01:05:34,118
So I don't really have anything
to problem-set you over.
977
01:05:35,238 --> 01:05:37,402
-I appreciate that.
-[laughs] You're welcome.
978
01:05:37,402 --> 01:05:40,283
But we will next Monday. Next
Monday, you'll get a problem set.
979
01:05:41,725 --> 01:05:45,627
Um, I was gonna say that in my
Calc 2 class, some of those students
980
01:05:45,627 --> 01:05:49,040
are also meeting on Zoom,
to work on homework together.
981
01:05:49,040 --> 01:05:53,096
So, they had study groups going, and
they're just keeping those going on Zoom.
982
01:05:53,616 --> 01:05:57,407
So, I'm gonna throw that out there.
If any of you guys had study groups.
983
01:05:57,407 --> 01:05:58,984
You know, continue to do that.
984
01:06:03,204 --> 01:06:04,240
[Student] Um, before we go.
985
01:06:04,240 --> 01:06:08,274
Is this meeting going to be
posted in Recorded Meetings?
986
01:06:08,274 --> 01:06:10,883
[Instructor] Yes, it is.
It just takes a while.
987
01:06:10,883 --> 01:06:15,522
So, once I'm finished, it has to convert
it, or something? I don't know.
988
01:06:15,522 --> 01:06:17,366
And that can take hours.
989
01:06:17,366 --> 01:06:21,361
So, hopefully... hopefully by
tonight I'll have it posted?
990
01:06:21,361 --> 01:06:24,245
But in the morning, as a last resort.
991
01:06:25,658 --> 01:06:26,745
[Student] Okay. Heard.
992
01:06:30,875 --> 01:06:32,656
Okay. Well, so, we are early.
993
01:06:32,656 --> 01:06:38,006
And I'm gonna let you go; so if you wanna
go, just go ahead and exit the meeting.
994
01:06:38,006 --> 01:06:39,611
I'm gonna just stay here for a minute,
995
01:06:39,611 --> 01:06:42,526
in case anybody wants to
talk or ask me a question.
996
01:06:45,889 --> 01:06:46,607
And if you're leaving, bye—
997
01:06:46,607 --> 01:06:50,517
[Student] -I actually have a question.
[Instructor] -Sure. You can hang out.
998
01:06:50,517 --> 01:06:56,753
So, I actually was struggling with the
particle moves along a curve equation.
999
01:06:56,753 --> 01:06:57,753
[Instructor] Okay.
1000
01:06:57,753 --> 01:07:01,674
I would just like you to break down
a little bit more what you did,
1001
01:07:01,674 --> 01:07:04,148
um, a couple steps through it.
1002
01:07:04,148 --> 01:07:06,622
[Instructor] Sure. Let me turn
my screen-sharing back on.
1003
01:07:28,504 --> 01:07:30,223
It's really taking forever.
1004
01:07:45,338 --> 01:07:48,031
[Student] Uh, you said this is
already recorded already, right?
1005
01:07:48,031 --> 01:07:49,188
[Instructor] Mhmm.
1006
01:07:49,188 --> 01:07:52,335
[Student] How do we go to view it?
Do we just go on Blackboard, then...
1007
01:07:52,335 --> 01:07:58,508
[Instructor] Yeah; so. Once it's ready
to post, then you'll see a link.
1008
01:07:58,508 --> 01:08:01,520
I think the link's already
there, right? That says...
1009
01:08:02,260 --> 01:08:05,643
Zoom Recordings, or Recorded
Meetings. I forgot what I called it,
1010
01:08:05,643 --> 01:08:07,606
but. "Recorded" is in it.
1011
01:08:07,606 --> 01:08:10,019
So you'll just click there,
and then you'll see them.
1012
01:08:10,758 --> 01:08:12,499
[Student] -All right. Thank you.
[Instructor] -Sure.
1013
01:08:13,894 --> 01:08:16,641
Okay. So here's my particle problem, again.
1014
01:08:17,493 --> 01:08:19,633
So the deal was, um.
1015
01:08:19,633 --> 01:08:23,259
Little dude is moving.
Little particle is moving.
1016
01:08:23,978 --> 01:08:28,245
And the curve, the y = √(1+x³)?
1017
01:08:28,245 --> 01:08:32,859
That is the formula that relates
your variables, x and y.
1018
01:08:33,632 --> 01:08:35,267
And then, let's see.
1019
01:08:36,007 --> 01:08:40,510
The y coordinate was increasing
at a rate of 4; so as it's moving,
1020
01:08:40,510 --> 01:08:43,556
the rate of change of
the y coordinate is 4.
1021
01:08:43,556 --> 01:08:46,750
And what we wanted here was the
rate of change of dx coordinates.
1022
01:08:46,750 --> 01:08:48,611
So we want (dx/dt).
1023
01:08:49,995 --> 01:08:54,503
Okay, so my formula was y = √(1+x³)?
1024
01:08:55,363 --> 01:08:57,391
And one thing that's nice
about these problems is
1025
01:08:57,391 --> 01:09:00,010
you don't have to find the formula,
or think about it,
1026
01:09:00,010 --> 01:09:02,627
or figure out what it is, because
it's just handed to you.
1027
01:09:02,627 --> 01:09:05,864
It's whatever the equation is.
It's kinda nice, really.
1028
01:09:06,644 --> 01:09:09,246
So we're always differentiating
with respect to t;
1029
01:09:09,246 --> 01:09:12,515
so I wrote (d/dt) of both sides.
1030
01:09:12,515 --> 01:09:15,505
But what I did over here was,
I just rewrote it in the form of
1031
01:09:15,505 --> 01:09:19,511
a rational exponent, because it makes
it easier for me to differentiate.
1032
01:09:20,762 --> 01:09:22,757
So my left is (dy/dt).
1033
01:09:22,757 --> 01:09:25,630
And... here's differentiating
on the right-hand side.
1034
01:09:25,630 --> 01:09:29,500
½ down in front. Rewrite 1 +x³.
1035
01:09:29,500 --> 01:09:31,996
Decrease by 1, so -½.
1036
01:09:32,496 --> 01:09:35,267
This is the derivative of the inside.
1037
01:09:35,267 --> 01:09:38,270
The inside is the (1 +x³).
1038
01:09:38,270 --> 01:09:45,214
But the derivative of (1 +x³)
is 3x²(dx/dt).
1039
01:09:45,719 --> 01:09:50,882
Any time it's the wrong letter,
gotta follow it by that (dx/dt).
1040
01:09:51,752 --> 01:09:54,500
And then I just substituted
all of the stuff in.
1041
01:09:54,500 --> 01:09:57,245
(dy/dt) was 4.
1042
01:09:57,245 --> 01:09:59,506
½ times 1 plus.
1043
01:09:59,506 --> 01:10:03,758
x was 2, because that's the
point we're kind of lookin' at here.
1044
01:10:05,658 --> 01:10:10,753
Here's 3 times 2²;
and then (dx/dt).
1045
01:10:10,753 --> 01:10:12,759
And then I did a bunch
of stuff in my head.
1046
01:10:12,759 --> 01:10:14,135
Maybe that's where...
1047
01:10:14,135 --> 01:10:16,058
[Student] That's the part
I had a problem with. Yeah.
1048
01:10:16,058 --> 01:10:16,910
[Instructor] Sorry about that.
1049
01:10:16,910 --> 01:10:21,041
I could see that I didn't have much
room left; and so, that's why I did that.
1050
01:10:21,796 --> 01:10:26,327
So the 2³ plus 1 is 9.
But it's 9 to the -½.
1051
01:10:26,327 --> 01:10:30,104
That's 1 over the square root of 9.
1052
01:10:31,614 --> 01:10:32,898
So that's ⅓.
1053
01:10:34,387 --> 01:10:37,624
And that's why I wrote this 3—
oh, it paused on me—
1054
01:10:37,624 --> 01:10:39,635
That's why I wrote this 3 down here.
1055
01:10:40,635 --> 01:10:43,630
So my ½ is one over two.
1056
01:10:43,630 --> 01:10:46,534
This, when I bring it downstairs, is 3.
1057
01:10:47,255 --> 01:10:51,071
And then, 2² is 4. Times 3;
there's the 12.
1058
01:10:51,991 --> 01:10:53,412
And then (dx/dt).
1059
01:10:53,412 --> 01:10:54,456
Did that help?
1060
01:10:56,764 --> 01:10:59,018
[Student] -Uh, just give me a moment.
[Instructor] -Sure.
1061
01:11:05,236 --> 01:11:09,746
So you're multiplying ½ by -⅑ ?
1062
01:11:09,746 --> 01:11:13,254
So, it's more like this. I'm gonna show
y'all the steps out here to the side.
1063
01:11:13,254 --> 01:11:15,947
The 1 + 2³, that's a 9.
1064
01:11:15,947 --> 01:11:18,621
But it's a 9 to the -½.
1065
01:11:18,621 --> 01:11:23,014
Well, that's the same as
⅑ to the positive ½.
1066
01:11:25,964 --> 01:11:28,753
And then times 3 times 4.
1067
01:11:29,946 --> 01:11:36,956
So that's ½ times ⅓; 'cause 9 to the
½ is square root of 9, and that's 3.
1068
01:11:37,411 --> 01:11:39,190
And that's times 12.
1069
01:11:42,380 --> 01:11:46,042
So, this is 12 over 6, which is 2.
1070
01:11:48,078 --> 01:11:49,762
And that's where that comes from.
1071
01:11:51,906 --> 01:11:52,871
-Thank you.
-Sure.
1072
01:11:52,871 --> 01:11:54,993
-I really needed to see that.
-Good.
1073
01:11:56,115 --> 01:11:57,229
Glad to help.
1074
01:11:58,286 --> 01:12:01,204
Anybody else still in the room,
you have a question; go ahead.
1075
01:12:25,007 --> 01:12:28,497
[Student] So, is this considered as
a multivariable calculus, or...
1076
01:12:28,497 --> 01:12:30,036
This is just single-variable?
1077
01:12:30,036 --> 01:12:33,395
[Instructor] Uh, yeah; actually,
that's a good question.
1078
01:12:34,138 --> 01:12:39,495
Uh, no. It's not considered
multi-variable calculus. It's not.
1079
01:12:41,145 --> 01:12:41,999
[Student, softly] Okay.
1080
01:12:46,759 --> 01:12:49,748
There you'll get into, you know...
1081
01:12:49,748 --> 01:12:55,620
xyz, and all this cool stuff
called partial differentiation,
1082
01:12:55,620 --> 01:12:59,872
and you'll have double and
triple integrals. It's good.
1083
01:12:59,872 --> 01:13:02,140
So, no; this is not that.
1084
01:13:06,494 --> 01:13:08,608
-Can't wait to learn that.
-Yeah, thank you for today; I'm leaving.
1085
01:13:08,608 --> 01:13:09,874
Oh, you're welcome. Bye-bye.
1086
01:13:11,487 --> 01:13:14,181
-Manon, you said you can't wait?
-Yeah, I can't wait to learn that.
1087
01:13:14,181 --> 01:13:17,071
Yeah, it's beautiful stuff.
You'll love it.
1088
01:13:20,116 --> 01:13:23,504
I was looking at one of the
hardest challenging problems
1089
01:13:23,504 --> 01:13:27,957
in the mathematics right now,
which is Riemann's data function,
1090
01:13:27,957 --> 01:13:30,211
-that they can't prove it.
-They did? Oh wow.
1091
01:13:31,564 --> 01:13:35,135
Yeah; they can't, um.
There's no proof of it.
1092
01:13:35,135 --> 01:13:38,490
We know the answer, like...
1093
01:13:39,250 --> 01:13:42,265
The numbers, but. We can't
prove it, basically, right now.
1094
01:13:43,515 --> 01:13:46,785
Yeah. So, if you can't prove it,
then you don't know it.
1095
01:13:48,386 --> 01:13:52,919
We can graph it, and look at the values,
and we know where it's approaching.
1096
01:13:53,499 --> 01:13:54,835
The answer, but.
1097
01:13:55,991 --> 01:13:59,000
Yeah, there's a $1,000,000 prize
on it, if someone solves it.
1098
01:13:59,000 --> 01:14:01,546
-Oh ho-ho, nice.
-Yeah.
1099
01:14:07,495 --> 01:14:08,963
Anybody else?
1100
01:14:08,963 --> 01:14:10,904
[Student] All right, I'm gonna leave.
Thank you for the lecture.
1101
01:14:10,904 --> 01:14:12,237
Okay. Bye, Manon!
1102
01:14:12,237 --> 01:14:15,487
-Yeah. Stay safe.
-You, too.
1103
01:14:20,740 --> 01:14:24,211
All right, everybody;
I'll end the meeting, and um...
1104
01:14:25,505 --> 01:14:28,116
I'll see ya Wednesday. Bye!