Addition of water (acid-catalyzed) mechanism

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Anytime you're trying

00:00:01.247 --> 00:00:03.712
to come up with a
mechanism for a reaction,

00:00:03.712 --> 00:00:06.082
it's worthwhile to study
a little bit of what

00:00:06.082 --> 00:00:08.394
you are starting with
and then thinking about

00:00:08.394 --> 00:00:10.624
what you finish with and
think about what is different.

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So what we're starting
with, we could call this

00:00:12.488 --> 00:00:14.821
one, two, three, four, five,

00:00:17.147 --> 00:00:18.980
so this is, let's see,

00:00:19.900 --> 00:00:21.826
we have methyl group on
the number two carbon,

00:00:21.826 --> 00:00:25.325
it is a pentene, and that
is double-bond between

00:00:25.325 --> 00:00:27.286
the number two and number three carbons,

00:00:27.286 --> 00:00:30.036
so this is two-methyl-pent-2-ene.

00:00:39.965 --> 00:00:41.441
So that's what we start with,

00:00:41.441 --> 00:00:42.274
we're in the presence,

00:00:42.274 --> 00:00:44.189
we're in an acidic environment,

00:00:44.189 --> 00:00:47.702
we've got what's gonna be
catalyzed by our hydronium here,

00:00:47.702 --> 00:00:52.023
and we end up with this,
and how is our product

00:00:52.023 --> 00:00:53.798
different from what we started with?

00:00:53.798 --> 00:00:56.810
Well the double bond is now gone,

00:00:56.810 --> 00:01:00.194
the number three carbon
gains this hydrogen,

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and now the number two carbon
gains a hydroxyl group.

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So one way to think about this is,

00:01:06.406 --> 00:01:09.474
in the presence of an
acid, it's acid-catalyzed,

00:01:09.474 --> 00:01:12.206
we have gained two
hydrogens and an oxygen,

00:01:12.206 --> 00:01:13.634
which is what we've gained,

00:01:13.634 --> 00:01:16.415
what could be used to make a water.

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And this is actually called an

00:01:17.742 --> 00:01:20.413
acid-catalyzed addition of water.

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The water isn't sitting on
one part of the molecule,

00:01:22.646 --> 00:01:24.042
but if you take the hydrogen we added,

00:01:24.042 --> 00:01:26.054
and the hydroxyl we added,
if you combine them,

00:01:26.054 --> 00:01:29.220
that's what you need to make a water.

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So let's think about how we can,

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how this actually happens in
the presence of our hydronium.

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So let me redraw this
molecule right over here.

00:01:38.837 --> 00:01:43.004
So we copy and paste it, so
that's not exactly it yet,

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that is just with the single bond.

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So let me draw, woops, wrong tool.

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Let me draw the double bond there.

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And now let me put it in the
presence of some hydronium.

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Alright, so we have an oxygen bonded, two.

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So this would just be water,
as oxygen has two lone pairs,

00:02:06.541 --> 00:02:08.597
but hydronium is a situation where oxygen

00:02:08.597 --> 00:02:11.722
is sharing one of those lone
pairs with a hydrogen proton,

00:02:11.722 --> 00:02:14.179
thus making the entire molecule positive,

00:02:14.179 --> 00:02:17.536
because the hydrogen proton is positive.

00:02:17.536 --> 00:02:21.792
So there you go, this now
has a positive charge.

00:02:21.792 --> 00:02:24.909
And this can be pretty reactive,
'cause we know that oxygen

00:02:24.909 --> 00:02:28.848
is quite electronegative, it
lives to keep its electrons.

00:02:28.848 --> 00:02:32.315
So what is there was a
way, what if there's a way

00:02:32.315 --> 00:02:36.750
for the oxygen to take back
the electrons in this bond

00:02:36.750 --> 00:02:40.106
right over here, the two
electrons in this bond.

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Well what if one of these carbons,

00:02:42.079 --> 00:02:44.602
especially the ones that
have the double bonds,

00:02:44.602 --> 00:02:47.237
what if some of the electrons
from this double bond

00:02:47.237 --> 00:02:51.032
could be used to snab, to
take that hydrogen proton,

00:02:51.032 --> 00:02:55.237
and then oxygen can hog
its electrons again.

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And you might say, "oh that's reasonable,

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but which of these carbons
would actually do it?"

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And to think about which of
those carbons would do it,

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we have to turn to Markovnikov's rule.

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Markovnikov's rule tells us that, look,

00:03:08.677 --> 00:03:11.625
if you have a reaction like
this, and alkene reaction,

00:03:11.625 --> 00:03:14.498
the carbon that already
has, that's already attached

00:03:14.498 --> 00:03:17.442
to more hydrogens is more
likely to gain more hydrogens,

00:03:17.442 --> 00:03:19.518
the carbon that's attached
to more functional groups

00:03:19.518 --> 00:03:22.232
is more likely to gain
more functional groups.

00:03:22.232 --> 00:03:23.633
Another way to think
about it is think about,

00:03:23.633 --> 00:03:26.486
well that is the order of the carbons?

00:03:26.486 --> 00:03:28.945
Because the higher order of carbon,

00:03:28.945 --> 00:03:33.660
the more stable it will be if
it forms some type of cation.

00:03:33.660 --> 00:03:36.054
So if you look at this
carbon right over here,

00:03:36.054 --> 00:03:39.100
our number two carbon, let me circle it,

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our number two carbon is bonded
to one, two, three carbons.

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So this is a tertiary carbon.

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This one right over here,

00:03:49.003 --> 00:03:50.662
this carbon on the other
side of the double bond

00:03:50.662 --> 00:03:52.971
is only bonded to one, two carbons.

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So this is a secondary carbon.

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So the tertiary carbon is
going to be more stable

00:03:57.621 --> 00:04:00.365
as a carbocation, you can think of it

00:04:00.365 --> 00:04:02.855
as it can spread the charge a little bit.

00:04:02.855 --> 00:04:05.756
So it would be more likely
to lose the electrons

00:04:05.756 --> 00:04:08.461
in one of these, in one of these bonds,

00:04:08.461 --> 00:04:11.795
and so the way that we can
think about this mechanism,

00:04:11.795 --> 00:04:13.198
and it might be a little bit clearer

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when we form the
carbocation, is let's have,

00:04:17.954 --> 00:04:20.709
going to do this in blue.

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Let's have these two
electrons that form this bond,

00:04:24.472 --> 00:04:28.222
well now they form a
bond with that hydrogen,

00:04:29.797 --> 00:04:33.964
and now the oxygen can take
back these two electrons,

00:04:35.993 --> 00:04:40.286
and what are we going,
what is going to result?

00:04:40.286 --> 00:04:41.860
And I'm drawing it in equilibrium,

00:04:41.860 --> 00:04:43.672
remember all of these things
are going back and forth

00:04:43.672 --> 00:04:46.018
depending on how things
bump into each other,

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but what are we left with?

00:04:47.984 --> 00:04:51.041
So let me copy and paste this again

00:04:51.041 --> 00:04:54.011
and I'm copying and pasting
in a way that, just so I,

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this is the backbone, and
I'll add what I need to add.

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So once this happens, we have this carbon,

00:05:02.729 --> 00:05:05.297
the number three carbon, now,

00:05:05.297 --> 00:05:08.384
woops I keep using the wrong tool.

00:05:08.384 --> 00:05:12.551
The number three carbon now
forms a bond with this hydrogen

00:05:15.615 --> 00:05:16.865
just like that.

00:05:17.794 --> 00:05:20.079
This carbon, our number two carbon,

00:05:20.079 --> 00:05:24.942
has lost an electron, it's
no longer sharing this bond.

00:05:24.942 --> 00:05:28.785
And so now it is going to
have a positive charge,

00:05:28.785 --> 00:05:30.629
it is a carbocation and once again

00:05:30.629 --> 00:05:32.533
is a tertiary carbocation,

00:05:32.533 --> 00:05:35.050
it is bonded to one, two, three carbons.

00:05:35.050 --> 00:05:37.067
That is stable, more stable
than if we did it the other way

00:05:37.067 --> 00:05:39.383
around, if this one grabbed
the hydrogen somehow,

00:05:39.383 --> 00:05:41.615
then this would be a
secondary carbocation,

00:05:41.615 --> 00:05:45.433
it'd be harder for it to spread
that positive charge around.

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And what about our, what
about our molecule up here?

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Let's see what it looks like now.

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We have our oxygen bonded
to the two hydrogens,

00:05:54.363 --> 00:05:56.167
it had one of those lone pairs,

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and now the electrons in this bond

00:05:58.367 --> 00:06:00.980
are now going to form another lone pair.

00:06:00.980 --> 00:06:02.868
So it took back an electron,

00:06:02.868 --> 00:06:06.991
or you can think of it, it
gave away a hydrogen proton.

00:06:06.991 --> 00:06:09.673
And so this is now just neutral water,

00:06:09.673 --> 00:06:12.034
and we see that we have a
conservation of charge here,

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this was positive in charge,

00:06:13.481 --> 00:06:17.284
now our original molecule
is positively charged.

00:06:17.284 --> 00:06:19.596
And what feels good about
this is we're getting,

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we're getting close to our end product,

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at least on our number
three carbon, we now have,

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we now have this hydrogen.

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Now we need to think about,

00:06:27.302 --> 00:06:31.573
well how do we get a hydroxyl
group added right over here?

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Well we have all this water,

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we have all this water
floating around, let me,

00:06:37.333 --> 00:06:39.050
I could use this water
molecule but the odds of it

00:06:39.050 --> 00:06:41.537
being the exact same water
molecule, we don't know.

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But there's all sorts of water molecules,

00:06:42.770 --> 00:06:44.177
we're in an aqueous solution,

00:06:44.177 --> 00:06:46.454
so let me draw another
water molecule here.

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So the water molecules are all equivalent,

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but let me draw another
water molecule here.

00:06:53.788 --> 00:06:57.161
And you can imagine, if they
just bump into each other

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in just the right way, this is,

00:07:00.405 --> 00:07:02.955
water is a polar molecule, it has a

00:07:02.955 --> 00:07:06.122
partially negative end near the oxygen

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because the oxygen likes
to hog the electrons,

00:07:09.940 --> 00:07:13.729
and then you have a partial
positive end near the hydrogens

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'cause they get their electrons hogged,

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so you can imagine the
oxygen end might be attracted

00:07:18.343 --> 00:07:23.254
to this tertiary carbocation,
and so just bumping it

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in just the right way,
it might form a bond.

00:07:27.429 --> 00:07:31.291
So let me say these two
electrons right over here,

00:07:31.291 --> 00:07:35.458
let's say they form a bond
with this, with that number two

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carbon, and then what is going to result?

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So let me draw, so what is, what is

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going to result, let me
scroll down a little bit,

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and let me paste, whoops,

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let me copy and paste our
original molecule again.

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So, here we go.

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So what could happen?

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This is the one we constructed actually,

00:08:04.895 --> 00:08:07.800
so we have the hydrogen there.

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We have the hydrogen, now this character,

00:08:11.195 --> 00:08:14.508
so we have the water molecule,

00:08:14.508 --> 00:08:16.901
so oxygen bonded to two hydrogens,

00:08:16.901 --> 00:08:19.249
you have this one lone
pair that isn't reacting,

00:08:19.249 --> 00:08:21.609
but then you have the lone
pair that does do the reacting.

00:08:21.609 --> 00:08:23.859
And so it now forms a bond.

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Woops, let me do it in that orange color.

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It now, it now forms an actual bond.

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And we're really close
to our final product,

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we have our hydrogen on
the number three carbon,

00:08:37.374 --> 00:08:39.912
we have more than we want
on our number two carbon,

00:08:39.912 --> 00:08:41.685
we just want a hydroxyl group,

00:08:41.685 --> 00:08:45.568
now we have a whole water
bonded to the carbon.

00:08:45.568 --> 00:08:48.276
So somehow we have to get
one of these other hydrogens

00:08:48.276 --> 00:08:50.015
swiped off of it, well that could happen

00:08:50.015 --> 00:08:52.236
with just another water molecule.

00:08:52.236 --> 00:08:53.760
So let's draw that.

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So another water molecule someplace,

00:08:56.204 --> 00:08:58.230
I'll do the different color
just to differentiate,

00:08:58.230 --> 00:09:00.926
although as we know water, well
it's hard to see what color

00:09:00.926 --> 00:09:03.286
is water if you're looking
at the molecular scale.

00:09:03.286 --> 00:09:05.638
So here we go, and we're really

00:09:05.638 --> 00:09:08.032
in the home stretch at this point.

00:09:08.032 --> 00:09:10.563
You have another water molecule,

00:09:10.563 --> 00:09:13.487
let's say, let me pick a color.

00:09:13.487 --> 00:09:16.063
So let's say these
electrons right over here,

00:09:16.063 --> 00:09:19.790
maybe they form a bond
with that hydrogen proton,

00:09:19.790 --> 00:09:23.957
and then these, the electrons
in that bond can go back

00:09:25.427 --> 00:09:28.121
to form a lone pair on that oxygen,

00:09:28.121 --> 00:09:29.873
and then what are we left with?

00:09:29.873 --> 00:09:32.013
And this really is the home stretch.

00:09:32.013 --> 00:09:34.513
So we are in equilibrium with,

00:09:36.205 --> 00:09:39.471
so let me draw my five carbons,
so let's see, I have...

00:09:39.471 --> 00:09:41.971
H3C, carbon, carbon, CH2, CH3,

00:09:48.536 --> 00:09:51.786
I have a CH3, I say H3C instead of CH3,

00:09:54.443 --> 00:09:56.830
I wrote it that way just so
it's clear that the carbons

00:09:56.830 --> 00:09:58.617
are bonded to the carbons,

00:09:58.617 --> 00:10:02.617
you have the original
hydrogen right over there,

00:10:03.885 --> 00:10:05.758
you have the one that we just added

00:10:05.758 --> 00:10:07.925
as part of this mechanism,

00:10:09.459 --> 00:10:13.626
you have this orange bond
to now, this hydroxyl group,

00:10:18.393 --> 00:10:22.424
the hydroxyl group, and it
had one lone pair before,

00:10:22.424 --> 00:10:24.280
it had one lone pair
before, but bound to both

00:10:24.280 --> 00:10:28.235
of the electrons from this
bond to form another lone pair.

00:10:28.235 --> 00:10:31.759
To form another lone pair,
which I am depicting in pink,

00:10:31.759 --> 00:10:34.800
and then this water is now,

00:10:34.800 --> 00:10:37.929
this water molecule is
now a hydronium molecule.

00:10:37.929 --> 00:10:41.316
So let me draw that, so this is now,

00:10:41.316 --> 00:10:45.235
oxygen, hydrogen, had one
lone pair that didn't react,

00:10:45.235 --> 00:10:46.914
and it had one lone pair that I put

00:10:46.914 --> 00:10:51.081
in blue that is reacting
with this hydrogen proton.

00:10:52.259 --> 00:10:55.176
With that hydrogen, just like this,

00:10:56.737 --> 00:11:00.072
and so since it got the
hydrogen proton it's giving its,

00:11:00.072 --> 00:11:03.677
sharing its electrons now,
now this has a positive,

00:11:03.677 --> 00:11:07.260
this has a positive
charge, just like that.

00:11:08.316 --> 00:11:10.483
I have to be very careful,

00:11:11.475 --> 00:11:14.451
in the last step I forgot
to draw the positive charge!

00:11:14.451 --> 00:11:15.861
We always wanna make sure that your charge

00:11:15.861 --> 00:11:18.499
is being conserved, we started
off with the positive charge

00:11:18.499 --> 00:11:21.011
on the hydronium, then we
have the positive charge

00:11:21.011 --> 00:11:22.515
on the tertiary carbocation

00:11:22.515 --> 00:11:24.534
right over here on our number two carbon,

00:11:24.534 --> 00:11:28.001
and now we have the positive charge,

00:11:28.001 --> 00:11:32.005
would be right over here,
because that oxygen,

00:11:32.005 --> 00:11:33.505
what we saw before that oxygen,

00:11:33.505 --> 00:11:36.769
which this water molecule was neutral,

00:11:36.769 --> 00:11:38.431
but you could say, you could view it is

00:11:38.431 --> 00:11:40.635
"well now it's going to be,

00:11:40.635 --> 00:11:42.068
it's now sharing these two electrons

00:11:42.068 --> 00:11:43.163
instead of keeping them,"

00:11:43.163 --> 00:11:47.428
so you can view it is maybe
it's giving away an electron,

00:11:47.428 --> 00:11:50.133
and so now it becomes positive.

00:11:50.133 --> 00:11:53.178
And then, and then the positive charge

00:11:53.178 --> 00:11:56.824
finally gets transferred to
that other water molecule

00:11:56.824 --> 00:11:59.090
when it becomes hydronium.

00:11:59.090 --> 00:12:01.177
But just like that we are done.

00:12:01.177 --> 00:12:04.655
We have added a hydroxyl
group and a hydrogen combined,

00:12:04.655 --> 00:12:08.488
that's a water, so that's why
we call it addition of water,

00:12:08.488 --> 00:12:10.068
and it was catalyzed by acid,

00:12:10.068 --> 00:12:13.735
so it's an acid-catalyzed
addition of water.


Limits at infinity of quotients with square roots (even power)

WEBVTT
Kind: captions
Language: en

00:00:00.173 --> 00:00:01.406
- [Voiceover] Let's see
if we can find the limit

00:00:01.406 --> 00:00:03.494
as x approaches negative infinity

00:00:03.494 --> 00:00:06.915
of the square root of
4x to the fourth minus x

00:00:06.915 --> 00:00:09.165
over 2x squared plus three.

00:00:10.474 --> 00:00:11.574
And like always, pause this video

00:00:11.574 --> 00:00:13.857
and see if you can figure it out.

00:00:13.857 --> 00:00:15.785
Well, whenever we're trying to find limits

00:00:15.785 --> 00:00:17.763
at either positive or negative infinity

00:00:17.763 --> 00:00:19.837
of rational expressions like this,

00:00:19.837 --> 00:00:23.264
it's useful to look at what
is the highest degree term

00:00:23.264 --> 00:00:26.075
in the numerator or in the denominator,

00:00:26.075 --> 00:00:28.525
or, actually in the numerator
and the denominator,

00:00:28.525 --> 00:00:30.989
and then divide the
numerator and the denominator

00:00:30.989 --> 00:00:34.694
by that highest degree,
by x to that degree.

00:00:34.694 --> 00:00:36.818
Because if we do that,
then we're going to end up

00:00:36.818 --> 00:00:38.575
with some constants and some other things

00:00:38.575 --> 00:00:40.618
that will go to zero
as we approach positive

00:00:40.618 --> 00:00:42.329
or negative infinity,
and we should be able

00:00:42.329 --> 00:00:43.838
to find this limit.

00:00:43.838 --> 00:00:46.099
So what I'm talking about,
let's divide the numerator

00:00:46.099 --> 00:00:50.266
by one over x squared and
let's divide the denominator

00:00:51.113 --> 00:00:53.770
by one over x squared.

00:00:53.770 --> 00:00:54.744
Now, you might be saying, "Wait, wait,

00:00:54.744 --> 00:00:56.006
"I see an x to the fourth here.

00:00:56.006 --> 00:00:56.972
"That's a higher degree."

00:00:56.972 --> 00:00:59.281
But remember, it's under the radical here.

00:00:59.281 --> 00:01:01.251
So if you wanna look at
it at a very high level,

00:01:01.251 --> 00:01:03.103
you're saying, okay, well x
to the fourth, but it's under,

00:01:03.103 --> 00:01:05.943
you're gonna take the square
root of this entire expression,

00:01:05.943 --> 00:01:08.774
so you can really view this
as a second degree term.

00:01:08.774 --> 00:01:10.610
So the highest degree
is really second degree,

00:01:10.610 --> 00:01:11.590
so let's divide the numerator

00:01:11.590 --> 00:01:13.884
and the denominator by x squared.

00:01:13.884 --> 00:01:15.491
And if we do that, dividing,

00:01:15.491 --> 00:01:18.387
so this is going to be the same thing as,

00:01:18.387 --> 00:01:21.137
so this is going to be the limit,

00:01:23.309 --> 00:01:27.476
the limit as x approaches
negative infinity of,

00:01:28.466 --> 00:01:30.756
so let me just do a
little bit of a side here.

00:01:30.756 --> 00:01:32.409
So if I have,

00:01:32.409 --> 00:01:35.076
if I have one over x squared,

00:01:36.614 --> 00:01:37.498
all right, let me write it.

00:01:37.498 --> 00:01:39.652
Let me just, one over x
squared times the square root

00:01:39.652 --> 00:01:41.779
of 4x to the fourth minus x,

00:01:41.779 --> 00:01:43.711
like we have in the numerator here.

00:01:43.711 --> 00:01:46.203
This is equal to, this is the same thing

00:01:46.203 --> 00:01:50.515
as one over the square
root of x to the fourth

00:01:50.515 --> 00:01:54.682
times the square root of
4x to the fourth minus x.

00:01:56.882 --> 00:01:59.311
And so this is equal to the square root

00:01:59.311 --> 00:02:01.561
of 4x to the fourth minus x

00:02:03.129 --> 00:02:07.296
over x to the fourth, which is
equal to the square root of,

00:02:08.251 --> 00:02:10.922
and all I did is I brought
the radical in here.

00:02:10.922 --> 00:02:12.351
You could view this as the
square root of all this

00:02:12.351 --> 00:02:13.582
divided by the square root of this,

00:02:13.582 --> 00:02:15.609
which is equal to, just
using our exponent rules,

00:02:15.609 --> 00:02:18.152
the square root of 4x
to the fourth minus x

00:02:18.152 --> 00:02:19.638
over x to the fourth.

00:02:19.638 --> 00:02:23.482
And then this is the
same thing as four minus,

00:02:23.482 --> 00:02:27.219
x over x to the fourth is
one over x to the third.

00:02:27.219 --> 00:02:30.667
So this numerator is going to be,

00:02:30.667 --> 00:02:32.216
the numerator's going
to be the square root

00:02:32.216 --> 00:02:35.799
of four minus one, x to the third power.

00:02:37.602 --> 00:02:39.602
And then the denominator

00:02:40.448 --> 00:02:42.583
is going to be equal to,

00:02:42.583 --> 00:02:45.725
well, you divide 2x squared by x squared.

00:02:45.725 --> 00:02:47.503
You're just going to be left with two.

00:02:47.503 --> 00:02:49.924
And then three divided
by x squared is gonna be

00:02:49.924 --> 00:02:51.924
three over x squared.

00:02:53.380 --> 00:02:54.479
Now, let's think about the limit

00:02:54.479 --> 00:02:56.835
as we approach negative infinity.

00:02:56.835 --> 00:02:59.706
As we approach negative infinity,

00:02:59.706 --> 00:03:02.706
this is going to approach zero.

00:03:02.706 --> 00:03:05.842
One divided by things that are becoming

00:03:05.842 --> 00:03:07.805
more and more and more and
more and more negative,

00:03:07.805 --> 00:03:08.988
their magnitude is getting larger,

00:03:08.988 --> 00:03:12.008
so this is going to approach zero.

00:03:12.008 --> 00:03:14.280
This over here is also going to be,

00:03:14.280 --> 00:03:16.546
this thing is also going
to be approaching zero.

00:03:16.546 --> 00:03:20.128
We're dividing by larger and
larger and larger values.

00:03:20.128 --> 00:03:22.152
And so what this is going to result in

00:03:22.152 --> 00:03:25.532
is the square root of four,
the principal root of four,

00:03:25.532 --> 00:03:27.945
over two, which is the same thing

00:03:27.945 --> 00:03:29.855
as two over two,

00:03:29.855 --> 00:03:31.688
which is equal to one.

00:03:33.223 --> 00:03:34.556
And we are done.


Analyzing unbounded limits: mixed function

WEBVTT
Kind: captions
Language: en

00:00:00.628 --> 00:00:02.404
- [Voiceover] So we're told that f of x

00:00:02.404 --> 00:00:07.063
is equal to x over one
minus cosine of x minus two,

00:00:07.063 --> 00:00:09.195
and they ask us to select
the correct description

00:00:09.195 --> 00:00:13.312
of the one-sided limits
of f at x equals two.

00:00:13.312 --> 00:00:15.498
And we see that right at x equals two,

00:00:15.498 --> 00:00:18.623
if we try to evaluate f of two,

00:00:18.623 --> 00:00:22.018
we get two over one minus cosine

00:00:22.018 --> 00:00:23.435
of two minus two,

00:00:24.383 --> 00:00:28.478
which is the same thing as cosine of zero,

00:00:28.478 --> 00:00:31.601
and cosine of zero is just one,

00:00:31.601 --> 00:00:34.018
and so one minus one is zero,

00:00:34.910 --> 00:00:38.833
and so the function is not
defined at x equals two,

00:00:38.833 --> 00:00:41.364
and that's why it might be
interesting to find the limit

00:00:41.364 --> 00:00:44.177
as x approaches two, and
especially the one-sided limit.

00:00:44.177 --> 00:00:45.241
And the one-sided limit.

00:00:45.241 --> 00:00:46.666
Well, we'll obviously leave it at that

00:00:46.666 --> 00:00:48.568
so let's try to approach this.

00:00:48.568 --> 00:00:50.872
So there's actually a couple
of ways you could do it.

00:00:50.872 --> 00:00:52.994
There is one way you could
do this without a calculator

00:00:52.994 --> 00:00:55.618
by just inspecting what's going on here,

00:00:55.618 --> 00:00:58.098
and thinking about the properties
of the cosine function,

00:00:58.098 --> 00:01:00.526
and if that inspires you,
pause the video and work it out,

00:01:00.526 --> 00:01:02.503
and I will do that at
the end of this video.

00:01:02.503 --> 00:01:06.356
The other way, if you have a calculator

00:01:06.356 --> 00:01:08.035
is to do it with a little bit of a table

00:01:08.035 --> 00:01:09.824
like we've done in other example problems.

00:01:09.824 --> 00:01:13.235
So if we think about x approaching two

00:01:13.235 --> 00:01:16.954
from the positive direction, well then.

00:01:16.954 --> 00:01:18.778
We can make a little table here

00:01:18.778 --> 00:01:22.587
where you have x, and
then you have f of x.

00:01:22.587 --> 00:01:26.478
And so for approaching two
from values greater than two,

00:01:26.478 --> 00:01:28.061
you could have 2.1,

00:01:29.770 --> 00:01:30.603
2.01.

00:01:32.312 --> 00:01:33.629
Now the reason why I said calculators,

00:01:33.629 --> 00:01:35.662
these aren't trivial to evaluate

00:01:35.662 --> 00:01:38.245
because this would be what, 2.1

00:01:39.638 --> 00:01:41.555
over 1 minus cosine of,

00:01:42.428 --> 00:01:44.178
2.1 minus two is 0.1.

00:01:46.838 --> 00:01:50.083
I do not know what cosine of
0.1 is without a calculator.

00:01:50.083 --> 00:01:53.083
I do know that cosine of zero is one

00:01:54.576 --> 00:01:58.318
so this is very, very close
to one without getting to one,

00:01:58.318 --> 00:02:00.023
and it's going to be less than one.

00:02:00.023 --> 00:02:03.140
Cosine is never going
to be greater than one.

00:02:03.140 --> 00:02:05.616
The cosine function is
bounded between negative one

00:02:05.616 --> 00:02:07.699
is less than cosine of x.

00:02:09.069 --> 00:02:11.516
I'll just write the x, then
I don't need the parenthesis.

00:02:11.516 --> 00:02:13.766
Which is less than one.

00:02:13.766 --> 00:02:17.760
The cosine function just
oscillates between these two values

00:02:17.760 --> 00:02:20.880
so this, this thing is
gonna be approaching one

00:02:20.880 --> 00:02:22.467
but it's going to be less than one.

00:02:22.467 --> 00:02:25.081
It definitely cannot be greater than one,

00:02:25.081 --> 00:02:25.957
and that's actually a good hint

00:02:25.957 --> 00:02:28.154
for how you can just
explore the structure here,

00:02:28.154 --> 00:02:30.525
and then you could say, "All right, 2.01.

00:02:30.525 --> 00:02:32.373
"Well, that's going to be 2.01

00:02:32.373 --> 00:02:33.623
"over one minus

00:02:34.629 --> 00:02:36.046
"cosine of 0.01."

00:02:39.874 --> 00:02:42.969
And this is going to even closer to one

00:02:42.969 --> 00:02:47.314
without being one but it's
going to be less than one.

00:02:47.314 --> 00:02:48.873
No matter what, cosine of anything

00:02:48.873 --> 00:02:52.034
is going to be between
negative one and one,

00:02:52.034 --> 00:02:54.237
and it could even be
including those things

00:02:54.237 --> 00:02:56.817
but as we approach two,

00:02:56.817 --> 00:03:00.439
this thing is going to approach one,

00:03:00.439 --> 00:03:03.208
I guess you could say
approach one from below.

00:03:03.208 --> 00:03:05.119
And so you can start to
make some intuitions here.

00:03:05.119 --> 00:03:09.005
If it's approaching from
below, this thing over here,

00:03:09.005 --> 00:03:12.642
this whole expression
is going to be positive,

00:03:12.642 --> 00:03:15.346
and as we approach x equals two.

00:03:15.346 --> 00:03:16.555
Well, the numerator is positive.

00:03:16.555 --> 00:03:17.611
It's approaching two.

00:03:17.611 --> 00:03:20.539
The denominator is positive
so this whole thing

00:03:20.539 --> 00:03:23.049
has to be approaching a positive value

00:03:23.049 --> 00:03:24.387
or it could become unbounded

00:03:24.387 --> 00:03:26.008
in the positive direction as we'll see,

00:03:26.008 --> 00:03:29.008
this is unbounded because this thing

00:03:30.138 --> 00:03:32.379
is even closer to one than this thing,

00:03:32.379 --> 00:03:34.301
and you would see that
if you have a calculator

00:03:34.301 --> 00:03:36.715
but needless to say, this
is going to be unbounded

00:03:36.715 --> 00:03:39.554
in the positive direction
so we're going to be going

00:03:39.554 --> 00:03:43.426
towards positive infinity so
these two choices have that,

00:03:43.426 --> 00:03:47.531
and we can make the exact
same argument as we approach x

00:03:47.531 --> 00:03:50.305
in the negative or from below,

00:03:50.305 --> 00:03:53.126
as we approach two from
below, I should say.

00:03:53.126 --> 00:03:56.194
So that's x, and that is f of x,

00:03:56.194 --> 00:03:58.354
and once again, I don't have
a calculator in front of me.

00:03:58.354 --> 00:04:00.338
You could evaluate these
things at a calculator,

00:04:00.338 --> 00:04:02.287
and become very clear
that these are positive,

00:04:02.287 --> 00:04:03.624
and as we get closer to,

00:04:03.624 --> 00:04:06.632
they become even larger
and larger positive values,

00:04:06.632 --> 00:04:10.549
and the same thing would
happen if you did 1.9,

00:04:12.986 --> 00:04:14.569
and if you did 1.99

00:04:16.020 --> 00:04:20.103
because here, you'd be
1.9 over one minus cosine.

00:04:21.179 --> 00:04:23.500
Now here, you'd have 1.9 minus two

00:04:23.500 --> 00:04:26.000
so this would be negative 0.1.

00:04:27.610 --> 00:04:29.215
Let me scroll over a little bit.

00:04:29.215 --> 00:04:32.018
The second one would be 1.99

00:04:32.018 --> 00:04:33.768
over one minus cosine

00:04:34.680 --> 00:04:36.097
of negative 0.01.

00:04:39.559 --> 00:04:42.043
And cosine of negative 0.1

00:04:42.043 --> 00:04:44.088
is the same thing as cosine of 0.1.

00:04:44.088 --> 00:04:46.005
Cosine of negative 0.01

00:04:48.072 --> 00:04:51.111
is the same thing as cosine of 0.01.

00:04:51.111 --> 00:04:54.385
So these two things, this is
going to be equal to that.

00:04:54.385 --> 00:04:56.269
That is going to be equal to that.

00:04:56.269 --> 00:04:59.102
And once again, we're gonna be
approaching positive infinity

00:04:59.102 --> 00:05:01.227
so the only choice where
all of that is true

00:05:01.227 --> 00:05:03.517
is this first one.

00:05:03.517 --> 00:05:05.547
Whether we approach two
from the right-hand side

00:05:05.547 --> 00:05:08.510
or the left-hand side, we're
approaching positive infinity

00:05:08.510 --> 00:05:10.132
but the other way you could
have deduced that is say,

00:05:10.132 --> 00:05:11.902
"Okay, as we approach two,

00:05:11.902 --> 00:05:13.800
the numerator is going to be positive

00:05:13.800 --> 00:05:17.429
'cause two is positive,
and then over here,

00:05:17.429 --> 00:05:18.612
as we approach two,

00:05:18.612 --> 00:05:21.843
cosine of anything can
never be greater than one.

00:05:21.843 --> 00:05:24.617
It's going to approach
one but be less than one

00:05:24.617 --> 00:05:28.534
so if this is less than
as x approaches two.

00:05:29.675 --> 00:05:32.410
It becomes one when x is equal to two.

00:05:32.410 --> 00:05:34.173
Well then, this right over here,

00:05:34.173 --> 00:05:37.633
one minus something less than
one is going to be positive

00:05:37.633 --> 00:05:39.351
so you have a positive
divided by a positive

00:05:39.351 --> 00:05:41.857
so you're definitely going
to get positive values

00:05:41.857 --> 00:05:43.305
as you approach two.

00:05:43.305 --> 00:05:45.870
And we know, and they've already told us

00:05:45.870 --> 00:05:48.477
that these are going to be
unbounded based on the choices

00:05:48.477 --> 00:05:50.481
so you would also pick that

00:05:50.481 --> 00:05:51.943
but you should also feel good about it,

00:05:51.943 --> 00:05:55.068
that the closer that we get to two,

00:05:55.068 --> 00:05:59.235
the closer that this value
right over here gets to zero.

00:06:00.618 --> 00:06:02.335
And the closer that
this value gets to zero,

00:06:02.335 --> 00:06:03.842
the closer we get to one.

00:06:03.842 --> 00:06:07.883
The closer we get to one, the
smaller the denominator gets.

00:06:07.883 --> 00:06:10.077
And then you divide by smaller
and smaller denominators,

00:06:10.077 --> 00:06:12.908
you're going to become
unbounded towards infinity,

00:06:12.908 --> 00:06:15.857
which is exactly what we
see in that first choice.


Analyzing unbounded limits: rational function

WEBVTT
Kind: captions
Language: en

00:00:00.447 --> 00:00:03.990
- [Voiceover] Let f of x
be equal to negative one

00:00:03.990 --> 00:00:06.514
over x minus one squared.

00:00:06.514 --> 00:00:09.404
Select the correct description
of the one-sided limits

00:00:09.404 --> 00:00:11.370
of f at x equals one.

00:00:11.370 --> 00:00:13.460
And so we can see, we
have a bunch of choices

00:00:13.460 --> 00:00:15.697
where we're approaching x
from the right-hand side

00:00:15.697 --> 00:00:18.003
and we're approaching x
from the left-hand side.

00:00:18.003 --> 00:00:20.347
And we're trying to figure
out do we get unbounded

00:00:20.347 --> 00:00:22.966
on either of those, in the positive,

00:00:22.966 --> 00:00:25.802
towards positive infinity
or negative infinity.

00:00:25.802 --> 00:00:27.556
And there's a couple of ways to tackle it.

00:00:27.556 --> 00:00:28.984
The most straightforward, well,

00:00:28.984 --> 00:00:30.746
let's just consider each
of these separately.

00:00:30.746 --> 00:00:33.067
So we can think about the limit of f of x

00:00:33.067 --> 00:00:37.990
as x approaches one from
the positive direction

00:00:37.990 --> 00:00:41.823
and limit of f of x as x approaches one,

00:00:45.408 --> 00:00:48.413
as x approaches one
from the left-hand side.

00:00:48.413 --> 00:00:50.074
This is from the right-hand side.

00:00:50.074 --> 00:00:52.371
This is from the left-hand side.

00:00:52.371 --> 00:00:54.444
So I'm just gonna make a
table and try out some values

00:00:54.444 --> 00:00:58.966
as we approach, as we approach
one from the different sides,

00:00:58.966 --> 00:01:02.966
x, f of x, and I'll do
the same thing over here.

00:01:04.035 --> 00:01:08.202
So, we are going to have
our x and have our f of x

00:01:10.525 --> 00:01:14.163
and if we approach one from
the right-hand side here,

00:01:14.163 --> 00:01:16.319
that would be approaching one from above,

00:01:16.319 --> 00:01:19.569
so we could try 1.1, we could try 1.01.

00:01:21.825 --> 00:01:25.992
Now f of 1.1 is negative one
over 1.1 minus one squared.

00:01:30.688 --> 00:01:34.275
So see this denominator here
is going to be .1 squared.

00:01:34.275 --> 00:01:38.358
So this is going to be,
this is going to be 0.01,

00:01:40.758 --> 00:01:44.091
and so this is going to be negative 100.

00:01:45.290 --> 00:01:46.424
So let me just write that down.

00:01:46.424 --> 00:01:48.184
That's going to be negative 100.

00:01:48.184 --> 00:01:51.721
So if x is 1.01, well, this is going to be

00:01:51.721 --> 00:01:55.138
negative one over 1.01 minus one squared.

00:01:58.208 --> 00:02:01.114
Well, then this denominator
this is going to be,

00:02:01.114 --> 00:02:04.364
this is the same thing as 0.01 squared,

00:02:05.949 --> 00:02:10.116
which is the same thing
as 0.0001, 1/10000.

00:02:13.031 --> 00:02:16.170
And so the negative one 1/10000

00:02:16.170 --> 00:02:18.433
is going to be negative 10,000.

00:02:18.433 --> 00:02:22.433
So, let's just write that
down, negative 10,000.

00:02:23.628 --> 00:02:25.817
And so this looks like, as we get closer,

00:02:25.817 --> 00:02:28.733
'cause notice, as I'm going
here I am approaching one

00:02:28.733 --> 00:02:30.002
from the positive direction,

00:02:30.002 --> 00:02:31.756
I'm getting closer and
closer to one from above

00:02:31.756 --> 00:02:35.303
and I'm going unbounded
towards negative infinity.

00:02:35.303 --> 00:02:37.911
So this looks like it
is negative infinity.

00:02:37.911 --> 00:02:40.686
Now we can do the same thing
from the left-hand side.

00:02:40.686 --> 00:02:43.353
I could do 0.9, I could do 0.99.

00:02:46.258 --> 00:02:50.912
Now 0.9 is actually also
going to get me negative 100

00:02:50.912 --> 00:02:54.829
'cause 0.9 minus one is
going to be negative .1

00:02:58.322 --> 00:02:59.926
but then when you square
it the negative goes away

00:02:59.926 --> 00:03:03.259
so you get a .01 and then
one divided by that is 100

00:03:03.259 --> 00:03:06.556
but you have the negative,
so this is also negative 100.

00:03:06.556 --> 00:03:08.661
And if you don't follow those
calculations, I'll do it,

00:03:08.661 --> 00:03:11.243
let me do it one more time
just so you see it clearly.

00:03:11.243 --> 00:03:13.933
This is going to be negative one over,

00:03:13.933 --> 00:03:16.788
so now I'm doing x is equal to 0.99,

00:03:16.788 --> 00:03:18.311
so I'm getting even closer to one,

00:03:18.311 --> 00:03:20.971
but I'm approaching from
below from the left-hand side.

00:03:20.971 --> 00:03:24.804
So this is going to be
0.99 minus one squared.

00:03:27.358 --> 00:03:31.525
Well, 0.99 minus one is, is
going to be negative 1/100,

00:03:33.276 --> 00:03:37.026
so this is going to be
negative 0.01 squared.

00:03:39.615 --> 00:03:41.338
When you square it the negative goes away

00:03:41.338 --> 00:03:43.649
and you're left with 1/10000.

00:03:43.649 --> 00:03:46.862
So this is going to be 0.0001

00:03:46.862 --> 00:03:50.751
and so when you evaluate
this you get 10,000.

00:03:50.751 --> 00:03:54.615
So that, or sorry, you
get negative 10,000.

00:03:54.615 --> 00:03:56.684
So in either case,
regardless of which direction

00:03:56.684 --> 00:04:00.851
we approach from, we are
approaching negative infinity.

00:04:01.741 --> 00:04:05.539
So that is this choice right over here.

00:04:05.539 --> 00:04:07.258
Now there's other ways you
could have tackled this

00:04:07.258 --> 00:04:08.615
if you just look at, kind of,

00:04:08.615 --> 00:04:10.775
the structure of this expression here,

00:04:10.775 --> 00:04:12.105
the numerator is a constant,

00:04:12.105 --> 00:04:14.037
so that's clearly always
going to be positive.

00:04:14.037 --> 00:04:16.243
Let's ignore this negative
for the time being.

00:04:16.243 --> 00:04:17.978
That negative's out front.

00:04:17.978 --> 00:04:20.636
This numerator, this one is
always going to be positive.

00:04:20.636 --> 00:04:23.984
Down here, we're taking at x equals one,

00:04:23.984 --> 00:04:26.254
while this becomes zero
and the whole expression

00:04:26.254 --> 00:04:29.513
becomes undefined, but as we approach one,

00:04:29.513 --> 00:04:32.561
x minus one could be positive or negative

00:04:32.561 --> 00:04:35.503
as we see over here, but
then when we square it,

00:04:35.503 --> 00:04:37.984
this is going to become positive as well.

00:04:37.984 --> 00:04:39.607
So the denominator is going to be positive

00:04:39.607 --> 00:04:42.602
for any x other than one.

00:04:42.602 --> 00:04:44.607
So positive divided by
positive is gonna be positive

00:04:44.607 --> 00:04:46.064
but then we have a negative out front.

00:04:46.064 --> 00:04:48.015
So this thing is going to be negative

00:04:48.015 --> 00:04:50.196
for any x other than one,

00:04:50.196 --> 00:04:52.485
and it's actually not
defined at x equals one.

00:04:52.485 --> 00:04:54.585
And so you could, from
that, you could deduce,

00:04:54.585 --> 00:04:56.707
well, okay then, we can
only go to negative infinity

00:04:56.707 --> 00:04:59.403
there's actually no way
to get positive values

00:04:59.403 --> 00:05:00.903
for this function.


Example of one sides unbounded limits

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.664
- [Voiceover] We're asked to
select the correct description.

00:00:02.664 --> 00:00:04.462
It looks like all the
descriptions deal with

00:00:04.462 --> 00:00:06.864
what is the limit of f
of x as we approach six

00:00:06.864 --> 00:00:10.816
from either the right hand side
or from the left hand side,

00:00:10.816 --> 00:00:12.538
so let's think about that.

00:00:12.538 --> 00:00:14.842
So first let me just
do the left hand side.

00:00:14.842 --> 00:00:16.675
So the limit of f of x

00:00:18.089 --> 00:00:22.058
as we approach six from
the left hand side,

00:00:22.058 --> 00:00:24.357
what is this going to be equal to?

00:00:24.357 --> 00:00:26.593
So as we approach from the left hand side

00:00:26.593 --> 00:00:29.011
we can see f of four
is a little under two,

00:00:29.011 --> 00:00:31.633
f of five looks like it's around three,

00:00:31.633 --> 00:00:34.653
f of 5.5 looks like it's a lot higher,

00:00:34.653 --> 00:00:38.700
f of 5.75 even is just
going off the charts,

00:00:38.700 --> 00:00:40.689
so it looks like this is going unbounded

00:00:40.689 --> 00:00:42.660
in the positive direction, so we could say

00:00:42.660 --> 00:00:46.323
that this right over here
is positive infinity,

00:00:46.323 --> 00:00:50.406
and if we were to think
about the limit of f of x

00:00:51.311 --> 00:00:55.319
as we approach six from the right side,

00:00:55.319 --> 00:00:56.961
what is this going to be?

00:00:56.961 --> 00:00:59.541
Well here, f of seven, it's negative,

00:00:59.541 --> 00:01:02.451
f of 7.5 is even more negative,

00:01:02.451 --> 00:01:05.034
f of 6.5 is even more negative,

00:01:09.120 --> 00:01:12.069
f of 6.1 is way more negative than that,

00:01:12.069 --> 00:01:15.495
f of 6.01 would be even
more negative than that.

00:01:15.495 --> 00:01:17.324
So it looks like this is unbounded

00:01:17.324 --> 00:01:19.784
in the negative direction,
so this is negative infinity.

00:01:19.784 --> 00:01:22.335
so let's see which of our
choices match up to that.

00:01:22.335 --> 00:01:26.299
so the way they listed
it, they listed the limit

00:01:26.299 --> 00:01:29.780
from the right direction
first, so that's this one,

00:01:29.780 --> 00:01:32.877
so the limit of f of x as x approaches six

00:01:32.877 --> 00:01:36.749
from values greater than six,
we have negative infinity,

00:01:36.749 --> 00:01:40.713
so that is these two choices,
these two choices say that,

00:01:40.713 --> 00:01:43.589
so we're gonna rule out those two choices.

00:01:43.589 --> 00:01:46.980
And then we think about as
we approach six from the left

00:01:46.980 --> 00:01:49.644
hand side, we see that we
go to positive infinity,

00:01:49.644 --> 00:01:51.869
and that is this choice right over here,

00:01:51.869 --> 00:01:56.036
so we rule out that one, and
that is what we will pick.


Intermediate value theorem example

WEBVTT
Kind: captions
Language: en

00:00:00.877 --> 00:00:05.027
- [Voiceover] Let F be
a continuous function

00:00:05.027 --> 00:00:08.250
on the closed interval
from negative two to one

00:00:08.250 --> 00:00:11.238
where F of negative two is equal to three

00:00:11.238 --> 00:00:13.424
and F of one is equal to six.

00:00:13.424 --> 00:00:15.560
Which of the following is guaranteed

00:00:15.560 --> 00:00:18.097
by the Intermediate Value Theorem?

00:00:18.097 --> 00:00:19.486
So before I even look at this,

00:00:19.486 --> 00:00:22.022
what do we know about the
Intermediate Value Theorem?

00:00:22.022 --> 00:00:23.384
Well it applies here,

00:00:23.384 --> 00:00:25.654
it's a continuous function
on this closed interval.

00:00:25.654 --> 00:00:29.232
We know what the value of the
function is at negative two.

00:00:29.232 --> 00:00:31.555
It's three so let me write that.

00:00:31.555 --> 00:00:34.599
F of negative two is equal to three

00:00:34.599 --> 00:00:38.098
and F of one, they tell
us right over here,

00:00:38.098 --> 00:00:39.673
is equal to six

00:00:39.673 --> 00:00:41.863
and all the Intermediate
Value Theorem tells us

00:00:41.863 --> 00:00:43.465
and if this is completely
unfamiliar to you,

00:00:43.465 --> 00:00:45.036
I encourage you to watch the video

00:00:45.036 --> 00:00:46.852
on the Intermediate Value Theorem,

00:00:46.852 --> 00:00:48.561
is that if we have a continuous function

00:00:48.561 --> 00:00:50.596
on some closed interval,

00:00:50.596 --> 00:00:54.401
then the function must take on every value

00:00:54.401 --> 00:00:58.674
between the values at the
endpoints of the interval

00:00:58.674 --> 00:01:01.344
or another way to say it is

00:01:01.344 --> 00:01:02.177
for any L

00:01:06.311 --> 00:01:08.144
between three and six,

00:01:11.455 --> 00:01:12.622
three and six,

00:01:14.562 --> 00:01:17.068
there is at least one C,

00:01:17.068 --> 00:01:19.068
there is at least one C,

00:01:25.296 --> 00:01:26.872
one C,

00:01:26.872 --> 00:01:31.518
between, or I could say
once C in the interval

00:01:31.518 --> 00:01:36.058
from negative two to
one, the closed interval,

00:01:36.058 --> 00:01:38.641
such that F of C is equal to L.

00:01:42.253 --> 00:01:45.266
This comes straight out of
the Intermediate Value Theorem

00:01:45.266 --> 00:01:46.762
and just saying it in everyday language

00:01:46.762 --> 00:01:48.497
is look this is a continuous function.

00:01:48.497 --> 00:01:51.096
Actually I'll draw it
visually in a few seconds.

00:01:51.096 --> 00:01:54.989
But it makes sense that
if it's continuous,

00:01:54.989 --> 00:01:57.292
if I were to draw the graph,
I can't pick up my pencil,

00:01:57.292 --> 00:01:58.984
well that it makes sense
that I would have to

00:01:58.984 --> 00:02:01.734
take on every value between three and six

00:02:01.734 --> 00:02:03.951
or there's at least one
point in this interval

00:02:03.951 --> 00:02:07.662
where I take on any given
value between three and six.

00:02:07.662 --> 00:02:10.333
So let's see which of these
answers are consistent with that

00:02:10.333 --> 00:02:11.774
and we only pick one.

00:02:11.774 --> 00:02:13.607
So F of C equals four.

00:02:15.273 --> 00:02:18.242
So that would be a case
where L is equal to four.

00:02:18.242 --> 00:02:20.325
So there's at least one C

00:02:22.822 --> 00:02:25.973
in this interval such that
F of C is equal to four.

00:02:25.973 --> 00:02:27.628
We could say that.

00:02:27.628 --> 00:02:29.930
But that's not exactly
what they're saying here.

00:02:29.930 --> 00:02:33.263
F of C could be four for at least one C,

00:02:34.151 --> 00:02:38.584
not in this interval,
remember the C is our X.

00:02:38.584 --> 00:02:40.827
This is our X right over here.

00:02:40.827 --> 00:02:43.738
So the C is going to be in this interval

00:02:43.738 --> 00:02:45.634
and I'll take a look at
it visually in a second

00:02:45.634 --> 00:02:47.743
so that we can validate that.

00:02:47.743 --> 00:02:49.078
We're not saying for at least one C

00:02:49.078 --> 00:02:52.069
between three and six F
of C is equal to four,

00:02:52.069 --> 00:02:56.329
we're saying for at least
one C in this interval

00:02:56.329 --> 00:02:58.898
F of C is going to be equal to four.

00:02:58.898 --> 00:03:01.942
It's important that four
is between three and six

00:03:01.942 --> 00:03:04.105
because that's the value of our function

00:03:04.105 --> 00:03:07.843
and the C needs to be
in our closed interval

00:03:07.843 --> 00:03:09.792
along the x-axis.

00:03:09.792 --> 00:03:11.555
So I'm gonna rule this out.

00:03:11.555 --> 00:03:13.451
They're trying to confuse us.

00:03:13.451 --> 00:03:14.759
Alright.

00:03:14.759 --> 00:03:17.842
F of C equals zero for at least one C

00:03:18.925 --> 00:03:20.401
between negative two and one.

00:03:20.401 --> 00:03:23.926
Well here they got the interval
along the x-axis right,

00:03:23.926 --> 00:03:25.635
that's where the C would be between,

00:03:25.635 --> 00:03:28.065
but it's not guaranteed by
the Intermediate Value Theorem

00:03:28.065 --> 00:03:30.361
that F of C is going to be equal to zero

00:03:30.361 --> 00:03:33.806
because zero is not between three and six.

00:03:33.806 --> 00:03:36.530
So I'm gonna rule that one out.

00:03:36.530 --> 00:03:38.128
I'm going to rule this one out,

00:03:38.128 --> 00:03:40.317
it's saying F of C equals zero,

00:03:40.317 --> 00:03:41.706
and let's see, we're
only left with this one

00:03:41.706 --> 00:03:42.828
so I hope it works.

00:03:42.828 --> 00:03:44.456
So F of C is equal to four,

00:03:44.456 --> 00:03:45.658
well that seems reasonable because

00:03:45.658 --> 00:03:47.954
four is between three and six,

00:03:47.954 --> 00:03:51.239
for at least one C between
negative two and one.

00:03:51.239 --> 00:03:54.363
Well yeah because that's in
this interval right over here.

00:03:54.363 --> 00:03:57.176
So I am feeling good about that

00:03:57.176 --> 00:03:59.193
and we could think about
this visually as well.

00:03:59.193 --> 00:04:00.181
The Intermediate Value Theorem

00:04:00.181 --> 00:04:03.385
when you think about it
visually makes a lot of sense.

00:04:03.385 --> 00:04:06.718
So let me draw the x-axis first actually

00:04:07.631 --> 00:04:10.595
and then let me draw my y-axis

00:04:10.595 --> 00:04:12.006
and I'm gonna draw them
at different scales

00:04:12.006 --> 00:04:14.276
'cause my y-axis, well let's see.

00:04:14.276 --> 00:04:16.776
If this is six, this is three.

00:04:19.429 --> 00:04:21.325
That's my y-axis.

00:04:21.325 --> 00:04:24.158
This is one, this is negative one,

00:04:25.411 --> 00:04:27.078
this is negative two

00:04:28.482 --> 00:04:30.484
and so we're continuous
on the closed interval

00:04:30.484 --> 00:04:32.193
from negative two to one

00:04:32.193 --> 00:04:34.677
and F of negative two is equal to three.

00:04:34.677 --> 00:04:35.665
So let me plot that.

00:04:35.665 --> 00:04:38.224
F of negative two is equal to three.

00:04:38.224 --> 00:04:40.601
So that's right over there

00:04:40.601 --> 00:04:43.744
and F of one is equal to six.

00:04:43.744 --> 00:04:45.533
So that's right over there

00:04:45.533 --> 00:04:49.325
and so let's try to draw
a continuous function.

00:04:49.325 --> 00:04:53.304
So a continuous function
includes these points

00:04:53.304 --> 00:04:55.787
and it's continuous so an
intuitive way to think about it

00:04:55.787 --> 00:04:57.515
is I can't pick up my
pencil if I'm drawing

00:04:57.515 --> 00:05:00.319
the graph of the function,
which contains these two points.

00:05:00.319 --> 00:05:01.902
So I can't do that.

00:05:02.802 --> 00:05:04.378
That would be picking up my pencil.

00:05:04.378 --> 00:05:06.834
So it is a continuous function.

00:05:06.834 --> 00:05:08.730
So it takes on every value.

00:05:08.730 --> 00:05:10.386
As we can see, it definitely does that.

00:05:10.386 --> 00:05:13.323
It takes on every value
between three and six.

00:05:13.323 --> 00:05:15.709
It might take on other
values, but we know for sure

00:05:15.709 --> 00:05:19.568
it has to take on every
value between three and six

00:05:19.568 --> 00:05:23.735
and so if we think about
four, four is right over here.

00:05:24.695 --> 00:05:27.098
The way I drew it, it looks
like it's almost taking on

00:05:27.098 --> 00:05:30.022
that value right at the y-axis.

00:05:30.022 --> 00:05:32.906
I forgot to label my x-axis here.

00:05:32.906 --> 00:05:34.771
But you can see it took on that value

00:05:34.771 --> 00:05:39.362
in for a C in this case
between negative two and one

00:05:39.362 --> 00:05:41.595
and I could have drawn that
graph multiple different ways.

00:05:41.595 --> 00:05:44.381
I could have drawn it something
like I could have done it

00:05:44.381 --> 00:05:45.223
and actually it takes on,

00:05:45.223 --> 00:05:48.587
there's multiple times it
takes on the value four here.

00:05:48.587 --> 00:05:50.990
So this could be our C, but once again

00:05:50.990 --> 00:05:53.884
it's between the interval
negative two and one.

00:05:53.884 --> 00:05:55.820
This could be our C once again

00:05:55.820 --> 00:05:58.063
in the interval between
negative two and one

00:05:58.063 --> 00:05:59.799
or this could be our C in between

00:05:59.799 --> 00:06:00.973
the interval of negative two and one

00:06:00.973 --> 00:06:02.869
and that's just the way
I happen to draw it.

00:06:02.869 --> 00:06:06.020
I could have drawn this thing
as just a straight line.

00:06:06.020 --> 00:06:07.863
I could have drawn it like this

00:06:07.863 --> 00:06:11.227
and then it looks like
it's taking on for only one

00:06:11.227 --> 00:06:14.592
and it's doing it right around there.

00:06:14.592 --> 00:06:18.794
This isn't necessarily
true that you take on,

00:06:18.794 --> 00:06:20.489
that you become four for at least one C

00:06:20.489 --> 00:06:21.691
between three and six.

00:06:21.691 --> 00:06:24.228
Three and six aren't
even on our graph here.

00:06:24.228 --> 00:06:28.153
I would have to go all
the way to two, three.

00:06:28.153 --> 00:06:33.013
There's not guarantee that
our function takes on four

00:06:33.013 --> 00:06:35.283
for one C between three and six.

00:06:35.283 --> 00:06:37.152
We don't even know what the function does

00:06:37.152 --> 00:06:39.819
when X is between three and six.


Inductor kickback 2 of 2

WEBVTT
Kind: captions
Language: en

00:00:02.481 --> 00:00:04.731
- [Voiceover] So, the problem
with allowing this spark

00:00:04.731 --> 00:00:08.353
to happen across here is if
this is not a mechanical switch,

00:00:08.353 --> 00:00:11.710
we can build switches out of
electronic devices, as well.

00:00:11.710 --> 00:00:13.894
But this is what we use transistors for.

00:00:13.894 --> 00:00:17.389
And a transistor is a rather
small, delicate device.

00:00:17.389 --> 00:00:20.694
And so, if I go in here
and I somehow cause

00:00:20.694 --> 00:00:25.189
a huge spike in voltage to
occur, very, very likely

00:00:25.189 --> 00:00:27.810
I'm gonna destroy that
transistor that's used

00:00:27.810 --> 00:00:32.303
in this position right here
as an electronic switch.

00:00:32.303 --> 00:00:35.339
So, the way to deal with
this is to provide a place

00:00:35.339 --> 00:00:36.397
for this current to go.

00:00:36.397 --> 00:00:39.641
Here's this current that's
flowing through this conductor,

00:00:39.641 --> 00:00:42.885
and the magnetic energy that's
stored around this inductor

00:00:42.885 --> 00:00:45.201
is gonna force that current to flow,

00:00:45.201 --> 00:00:47.007
even if we open this switch.

00:00:47.007 --> 00:00:49.384
And in the case here we saw when we opened

00:00:49.384 --> 00:00:52.562
the mechanical switch, it
sparks right across there,

00:00:52.562 --> 00:00:55.176
so the inductor wins,
the inductor current wins

00:00:55.176 --> 00:00:56.161
in that case.

00:00:56.161 --> 00:00:58.167
And it can be very damaging to whatever

00:00:58.167 --> 00:01:00.667
is in this position down here.

00:01:02.863 --> 00:01:07.030
The best way to solve this is
with a device called a diode.

00:01:09.207 --> 00:01:11.149
And this is another semiconductor device,

00:01:11.149 --> 00:01:14.066
it's not a switch, but a diode has,

00:01:15.829 --> 00:01:17.766
a diode has an IV curve.

00:01:17.766 --> 00:01:19.349
Here's V, here's I.

00:01:21.319 --> 00:01:23.236
And it looks like this.

00:01:25.811 --> 00:01:28.309
The current, I'm just gonna sketch this,

00:01:28.309 --> 00:01:30.933
the current is zero when
the voltage on the diode

00:01:30.933 --> 00:01:31.933
is negative.

00:01:33.240 --> 00:01:37.473
And it's somewhere around
here that the current goes up

00:01:37.473 --> 00:01:38.306
like that.

00:01:39.617 --> 00:01:44.045
So, when we have a diode, this
is the symbol for a diode,

00:01:44.045 --> 00:01:47.782
like that, and that's plus and minus V.

00:01:47.782 --> 00:01:50.215
And this is the current through the diode.

00:01:50.215 --> 00:01:53.144
So, positive current and
positive voltage is in this part

00:01:53.144 --> 00:01:56.896
of the graph over here, and
when I have a positive voltage,

00:01:56.896 --> 00:02:00.823
this is about .6 volts,
something like that,

00:02:00.823 --> 00:02:04.990
could be .5, could be .7, when
you have a positive voltage

00:02:06.249 --> 00:02:08.939
on this, then the current starts to flow

00:02:08.939 --> 00:02:10.117
freely through this.

00:02:10.117 --> 00:02:12.178
If this voltage goes
negative, so that means that

00:02:12.178 --> 00:02:15.176
this voltage is higher than this voltage,

00:02:15.176 --> 00:02:17.670
then the current actually
goes pretty much to zero,

00:02:17.670 --> 00:02:19.587
very, very small value.

00:02:21.351 --> 00:02:23.503
So, I'm gonna take advantage
of this diode device

00:02:23.503 --> 00:02:26.939
to help me with this
problem I have over here

00:02:26.939 --> 00:02:30.557
with the current, and the
way we do that is we do this.

00:02:30.557 --> 00:02:34.724
We hook up a diode pointing
in this direction here.

00:02:36.423 --> 00:02:37.735
Pretty distinctive.

00:02:37.735 --> 00:02:40.983
But anytime you have a circuit
that has an inductor in it,

00:02:40.983 --> 00:02:44.381
so it's a solenoid, or
a motor, or a relay,

00:02:44.381 --> 00:02:46.905
this is a way you can protect the devices

00:02:46.905 --> 00:02:48.405
driving your coil.

00:02:51.089 --> 00:02:53.217
So, let's look at what
happens here when we push this

00:02:53.217 --> 00:02:55.954
push button down and let it go again.

00:02:55.954 --> 00:02:58.394
When we started out, we
had no current flowing,

00:02:58.394 --> 00:03:00.574
the push button was open,
so there's no current

00:03:00.574 --> 00:03:02.674
going through here, and
there's no voltage difference

00:03:02.674 --> 00:03:05.988
across here, both these
points are at three volts.

00:03:05.988 --> 00:03:08.664
And so, there's no
voltage across the diode.

00:03:08.664 --> 00:03:12.168
And that's that point right
there, no current, no voltage.

00:03:12.168 --> 00:03:14.914
So, all the currents are
zero, and the voltage across

00:03:14.914 --> 00:03:18.092
the diode is zero, so there's no current.

00:03:18.092 --> 00:03:19.341
Nothing's happening.

00:03:19.341 --> 00:03:20.965
Now we push the button down.

00:03:20.965 --> 00:03:23.959
And a current starts to
flow through the inductor,

00:03:23.959 --> 00:03:26.393
and there's three volts
across the inductor.

00:03:26.393 --> 00:03:30.200
So, let's look at that,
there's plus three volts here,

00:03:30.200 --> 00:03:32.950
and there's zero volts down here.

00:03:34.752 --> 00:03:38.996
So, there's three volts
on this side of the diode,

00:03:38.996 --> 00:03:41.616
this is the side where it doesn't conduct.

00:03:41.616 --> 00:03:44.236
That three volts represents
negative three volts

00:03:44.236 --> 00:03:45.653
across the diode.

00:03:47.795 --> 00:03:49.791
Let me mark the diode voltage on here.

00:03:49.791 --> 00:03:52.124
Here's minus, plus, V diode.

00:03:55.035 --> 00:03:57.368
And we'll call this V diode.

00:03:58.903 --> 00:04:02.461
So, this is a V diode of negative three,

00:04:02.461 --> 00:04:06.628
so we're operating over here
out at negative three volts

00:04:07.820 --> 00:04:11.568
on the diode, and the current
through the diode is zero,

00:04:11.568 --> 00:04:13.258
there's no current flowing over here.

00:04:13.258 --> 00:04:14.876
All of this current is going down

00:04:14.876 --> 00:04:17.432
and going through the
inductor, and doing whatever

00:04:17.432 --> 00:04:20.099
this inductor is supposed to do.

00:04:21.676 --> 00:04:24.093
Okay, now we open the switch.

00:04:25.668 --> 00:04:27.297
Now we open the switch.

00:04:27.297 --> 00:04:30.483
And as you recall a minute
ago, what happened was

00:04:30.483 --> 00:04:34.400
this voltage right here,
this voltage went big.

00:04:37.442 --> 00:04:39.818
This went plus big, and we had like,

00:04:39.818 --> 00:04:42.562
100,000 volts or something.

00:04:42.562 --> 00:04:44.060
That's where it was headed.

00:04:44.060 --> 00:04:45.616
Well, we're not gonna let it get there,

00:04:45.616 --> 00:04:47.949
so I'm gonna take that away.

00:04:50.112 --> 00:04:54.223
So, this voltage here, as
soon as this switch opens,

00:04:54.223 --> 00:04:57.639
this voltage right
here, this is headed up.

00:04:57.639 --> 00:04:59.516
This is going up, hard.

00:04:59.516 --> 00:05:01.505
But we're not gonna let it get very high,

00:05:01.505 --> 00:05:02.941
and that's the job of this diode.

00:05:02.941 --> 00:05:07.562
So, as soon as this
voltage here gets to about

00:05:07.562 --> 00:05:10.992
6/10ths of a volt higher than right here.

00:05:10.992 --> 00:05:14.325
So, when this voltage gets to 3.6 volts,

00:05:16.114 --> 00:05:19.281
what that means is that the V diode...

00:05:20.542 --> 00:05:23.292
Now equals about .6 volts, right?

00:05:24.845 --> 00:05:26.599
This is at three volts.

00:05:26.599 --> 00:05:30.349
This is at 3.6 volts,
so V diode is .6 volts.

00:05:31.461 --> 00:05:34.878
And that puts us about here on the curve.

00:05:36.329 --> 00:05:40.699
And what's gonna happen is
this current that we had here,

00:05:40.699 --> 00:05:43.382
this current that was flowing
down through this inductor

00:05:43.382 --> 00:05:44.374
is gonna do what?

00:05:44.374 --> 00:05:46.870
It's gonna go this way.

00:05:46.870 --> 00:05:49.932
It's gonna flow this
way through the diode,

00:05:49.932 --> 00:05:53.985
and that's this portion
of the curve right here.

00:05:53.985 --> 00:05:57.108
And it'll keep going like that.

00:05:57.108 --> 00:05:59.420
Look at that, we've
provided a current path

00:05:59.420 --> 00:06:04.283
for this inductor that's not
gonna destroy what's down here.

00:06:04.283 --> 00:06:06.533
The highest voltage this will get to

00:06:06.533 --> 00:06:10.700
is whatever voltage it climbs
to on this diode curve here,

00:06:11.769 --> 00:06:15.936
which is gonna be between
.6 and point, say .8 volts.

00:06:17.384 --> 00:06:20.318
So, the highest this
is gonna get is point,

00:06:20.318 --> 00:06:21.735
3.6 to 3.8 volts.

00:06:24.748 --> 00:06:27.244
And typically, and typically
a transistor down here

00:06:27.244 --> 00:06:30.862
will easily withstand
this kind of a voltage

00:06:30.862 --> 00:06:32.279
on its terminals.

00:06:34.109 --> 00:06:36.733
So, this is called, the
effect of having this voltage

00:06:36.733 --> 00:06:39.400
go way north is called kickback.

00:06:43.719 --> 00:06:45.593
That's kickback from an inductor.

00:06:45.593 --> 00:06:47.403
This diode is a protection diode,

00:06:47.403 --> 00:06:50.519
and basically what it does
is it supplies a pathway

00:06:50.519 --> 00:06:54.080
for that inductor current
when this voltage goes

00:06:54.080 --> 00:06:56.913
above the top end of the inductor.

00:06:59.134 --> 00:07:01.876
So, whenever you build something
that has an inductor in it,

00:07:01.876 --> 00:07:04.186
and it could be, you could build a circuit

00:07:04.186 --> 00:07:06.808
that drives a motor, or an actuator,

00:07:06.808 --> 00:07:09.114
or a solenoid, or something like that,

00:07:09.114 --> 00:07:11.423
or part of a robot, this
is one of the circuits

00:07:11.423 --> 00:07:12.611
you wanna keep in mind.

00:07:12.611 --> 00:07:14.399
Whenever you have an inductor
and you're gonna switch it

00:07:14.399 --> 00:07:17.565
on and off, you basically,
you wanna design in

00:07:17.565 --> 00:07:21.398
a diode like this to protect
your circuit from those

00:07:21.398 --> 00:07:23.898
unstoppable inductor currents.


Inductor kickback 1 of 2

WEBVTT
Kind: captions
Language: en

00:00:00.623 --> 00:00:01.456
- [Voiceover] I want to talk about

00:00:01.456 --> 00:00:03.933
a new example of an inductor circuit,

00:00:03.933 --> 00:00:05.446
and we have one shown here,

00:00:05.446 --> 00:00:08.618
where this inductor is now
controlled by a switch.

00:00:08.618 --> 00:00:12.814
This is a push button switch
that we move in and out,

00:00:12.814 --> 00:00:16.126
and this metal plate here
will touch these two contacts

00:00:16.126 --> 00:00:18.864
and complete this circuit here.

00:00:18.864 --> 00:00:21.545
This is a lot like the
circuit we looked at before

00:00:21.545 --> 00:00:24.112
without the switch,
where the voltage source

00:00:24.112 --> 00:00:26.322
had a pulse going up and down.

00:00:26.322 --> 00:00:28.475
This circuit has a real
interesting side effect

00:00:28.475 --> 00:00:30.940
when this switch opens up.

00:00:30.940 --> 00:00:33.083
So, let me give you a
couple of examples first,

00:00:33.083 --> 00:00:35.140
some practical examples of where

00:00:35.140 --> 00:00:37.367
this kind of an inductor will appear,

00:00:37.367 --> 00:00:41.083
connected to a voltage source
and controlled by a switch.

00:00:41.083 --> 00:00:42.511
And there's a couple of examples.

00:00:42.511 --> 00:00:45.261
One of them is called a solenoid.

00:00:48.985 --> 00:00:51.050
And a solenoid is a coil of wire

00:00:51.050 --> 00:00:52.800
that looks like this,

00:00:54.666 --> 00:00:55.916
and has a metal

00:00:57.555 --> 00:00:59.305
rod going through it.

00:01:00.236 --> 00:01:01.950
When you put a pulse of current into this,

00:01:01.950 --> 00:01:04.533
what happens is, the rod moves.

00:01:06.952 --> 00:01:08.912
The rod will move back and forth,

00:01:08.912 --> 00:01:11.089
so this is a way, for
instance, a doorbell.

00:01:11.089 --> 00:01:14.339
A doorbell has this kind of a actuator.

00:01:18.349 --> 00:01:21.197
Another kind of device
that has an inductor in it,

00:01:21.197 --> 00:01:23.614
or a coil, is called a relay.

00:01:24.873 --> 00:01:28.373
And that one, again, has a coil like this.

00:01:30.969 --> 00:01:31.802
And,

00:01:34.348 --> 00:01:36.657
these are all forms of an electromagnet,

00:01:36.657 --> 00:01:39.928
and this one actually has a switch,

00:01:39.928 --> 00:01:42.269
some sort of a piece of metal

00:01:42.269 --> 00:01:45.289
that will move back and forth.

00:01:45.289 --> 00:01:47.211
When the coil is energized,

00:01:47.211 --> 00:01:50.667
this piece of metal will
actually tip over here,

00:01:50.667 --> 00:01:52.402
and when the current's stopped,

00:01:52.402 --> 00:01:55.091
it'll move back to its original state.

00:01:55.091 --> 00:01:59.319
So, for example, if there
is a connection point here,

00:01:59.319 --> 00:02:02.071
and a connection point here,

00:02:02.071 --> 00:02:03.205
this piece of metal will go

00:02:03.205 --> 00:02:05.441
from making a short across there,

00:02:05.441 --> 00:02:06.597
and move away,

00:02:06.597 --> 00:02:09.976
and open and close this bigger switch.

00:02:09.976 --> 00:02:13.059
So, a relay is some sort of a switch.

00:02:14.268 --> 00:02:16.148
And you might find this in a car,

00:02:16.148 --> 00:02:19.379
where this relay is controlling, like,

00:02:19.379 --> 00:02:21.276
the windshield wipers of a car,

00:02:21.276 --> 00:02:22.655
where the current for the windshield wiper

00:02:22.655 --> 00:02:24.085
goes through this switch,

00:02:24.085 --> 00:02:26.429
and there's a smaller current over here

00:02:26.429 --> 00:02:28.374
that pulls this switch open and closed

00:02:28.374 --> 00:02:32.541
to start and stop the
windshield wipers or the motor.

00:02:33.805 --> 00:02:36.516
And that's just two
examples of where inductors,

00:02:36.516 --> 00:02:41.051
or windings of wire can show
up in real applications,

00:02:41.051 --> 00:02:42.390
and there's some fairly large currents

00:02:42.390 --> 00:02:44.541
that flow through these windings.

00:02:44.541 --> 00:02:45.512
So, we want to look what happens

00:02:45.512 --> 00:02:47.338
when we switch this current on and off,

00:02:47.338 --> 00:02:49.754
and this is the example
circuit that we'll use.

00:02:49.754 --> 00:02:53.342
So, won't assume we have a
inductor of 10 milliHenries,

00:02:53.342 --> 00:02:54.252
and what we're gonna do is

00:02:54.252 --> 00:02:56.498
we're gonna push this
push button switch down

00:02:56.498 --> 00:03:00.665
and connect it up, and
then we're gonna let it go.

00:03:02.308 --> 00:03:06.472
And we're gonna look at a couple
of different voltages here.

00:03:06.472 --> 00:03:10.683
This is gonna be, we'll
call this voltage V,

00:03:10.683 --> 00:03:14.606
and that's gonna be measured
with respect to ground.

00:03:14.606 --> 00:03:17.071
And then, there's another voltage here

00:03:17.071 --> 00:03:20.167
that's interesting too, which is VL,

00:03:20.167 --> 00:03:22.674
and that's the voltage
across the inductor.

00:03:22.674 --> 00:03:26.841
So V plus VL is equal to
three volts, all the time.

00:03:28.597 --> 00:03:29.516
And now, we look at what happens

00:03:29.516 --> 00:03:31.506
when we push the button down,

00:03:31.506 --> 00:03:32.839
and then, after that,
we'll look at what happens

00:03:32.839 --> 00:03:34.697
when we let go of the button.

00:03:34.697 --> 00:03:36.112
Something interesting's gonna happen

00:03:36.112 --> 00:03:38.355
when we let go of this button.

00:03:38.355 --> 00:03:39.942
So, as we look at our circuit here,

00:03:39.942 --> 00:03:42.203
we see there's an open circuit,

00:03:42.203 --> 00:03:45.370
so there's no current flowing in here.

00:03:47.508 --> 00:03:49.591
That means I equals zero.

00:03:51.449 --> 00:03:53.999
We'll call that I zero because
it's the initial current

00:03:53.999 --> 00:03:56.082
when we close our switch.

00:03:57.318 --> 00:03:58.959
So now, let's close our switch.

00:03:58.959 --> 00:04:00.959
We'll do that like this.

00:04:05.372 --> 00:04:09.387
And we just closed our switch
at time, T equals zero.

00:04:09.387 --> 00:04:10.433
And let's look at what happens.

00:04:10.433 --> 00:04:12.200
All of a sudden now,

00:04:12.200 --> 00:04:16.022
our inductor has a voltage across it,

00:04:16.022 --> 00:04:17.681
and it's a voltage of,

00:04:17.681 --> 00:04:19.598
this node is at three volts,

00:04:19.598 --> 00:04:21.308
this node is at zero volts,

00:04:21.308 --> 00:04:25.343
so all of a sudden, we have
three volts across our inductor.

00:04:25.343 --> 00:04:28.536
So, let's use the integral
form of the inductor equation

00:04:28.536 --> 00:04:32.817
to solve for the current
that's gonna happen here.

00:04:32.817 --> 00:04:34.400
I equals one over L

00:04:36.321 --> 00:04:39.071
times the integral from zero to T

00:04:40.742 --> 00:04:41.575
of

00:04:42.678 --> 00:04:43.511
V d Tau,

00:04:46.070 --> 00:04:47.070
plus I zero.

00:04:50.270 --> 00:04:53.291
Let's fill in what we know.

00:04:53.291 --> 00:04:56.124
I equals one over 10 milliHenries,

00:04:58.539 --> 00:05:03.341
and V is a constant, V is
a constant three volts,

00:05:03.341 --> 00:05:04.341
times three,

00:05:05.640 --> 00:05:08.223
and it's the integral of d Tau,

00:05:10.112 --> 00:05:12.169
from zero to T,

00:05:12.169 --> 00:05:13.752
and I zero is zero.

00:05:15.790 --> 00:05:17.934
And then, we get the final form,

00:05:17.934 --> 00:05:21.684
which is I equals three
over 10 milliHenries,

00:05:23.995 --> 00:05:27.548
times the integral from
zero to T of d Tau,

00:05:27.548 --> 00:05:28.381
is just T,

00:05:30.692 --> 00:05:31.525
and that's the answer,

00:05:31.525 --> 00:05:33.557
so what we have here,
just like we did before,

00:05:33.557 --> 00:05:36.622
when we had the switching power supply,

00:05:36.622 --> 00:05:39.282
just like we had before,
we're gonna get a current

00:05:39.282 --> 00:05:42.470
that has a ramp that looks like that.

00:05:42.470 --> 00:05:44.735
The slope of that is three

00:05:44.735 --> 00:05:46.485
over 10 milliHenries,

00:05:48.403 --> 00:05:49.403
which equals

00:05:51.387 --> 00:05:53.054
300 amps per second,

00:05:57.063 --> 00:05:58.436
so, oh my goodness.

00:05:58.436 --> 00:06:00.427
This current is going up really fast.

00:06:00.427 --> 00:06:02.769
That's the slope of that
current right there.

00:06:02.769 --> 00:06:05.555
This is going up really fast,
and that's what it does.

00:06:05.555 --> 00:06:06.465
Now, in a real circuit,

00:06:06.465 --> 00:06:09.304
there'll be real resistances in here,

00:06:09.304 --> 00:06:11.072
but it uh, and so,

00:06:11.072 --> 00:06:12.863
there'll be a limit to
the amount of current

00:06:12.863 --> 00:06:14.964
that will be determined by the resistor,

00:06:14.964 --> 00:06:17.017
but for the purposes of showing you

00:06:17.017 --> 00:06:20.116
just how this inductor equation works,

00:06:20.116 --> 00:06:21.882
that's the kind of slope you would see

00:06:21.882 --> 00:06:25.344
at the initial closing of the switch.

00:06:25.344 --> 00:06:27.348
Okay, I'm gonna clean off the screen here,

00:06:27.348 --> 00:06:29.848
so I can keep my same circuit.

00:06:33.015 --> 00:06:35.421
And now, we're gonna
look at what happens here

00:06:35.421 --> 00:06:36.995
when we open this switch.

00:06:36.995 --> 00:06:38.912
Let me open the switch.

00:06:49.869 --> 00:06:51.515
And now, we've opened our switch back up.

00:06:51.515 --> 00:06:52.932
It went that way.

00:06:54.984 --> 00:06:56.748
All right, so,

00:06:56.748 --> 00:06:58.998
we have an initial current,

00:07:00.241 --> 00:07:02.459
and it's gonna be some value,

00:07:02.459 --> 00:07:05.303
depending on how long
we held the switch down,

00:07:05.303 --> 00:07:08.368
so it's gonna be some current,

00:07:08.368 --> 00:07:10.362
and that's flowing in the inductor.

00:07:10.362 --> 00:07:12.358
Now, let's look at what happens

00:07:12.358 --> 00:07:14.774
when we open the switch,
and all of a sudden,

00:07:14.774 --> 00:07:17.774
I goes directly and sharply to zero.

00:07:19.852 --> 00:07:21.333
That's what the switch does.

00:07:21.333 --> 00:07:23.371
When we open this switch contact,

00:07:23.371 --> 00:07:25.003
one moment it's touching,

00:07:25.003 --> 00:07:27.653
and the other moment, it's not touching.

00:07:27.653 --> 00:07:32.322
So, if we look at, what's
delta I, or what's dI.

00:07:32.322 --> 00:07:34.806
It's the ending current

00:07:34.806 --> 00:07:37.806
minus the starting current, minus I.

00:07:38.857 --> 00:07:41.300
And if we look at what's
the time involved.

00:07:41.300 --> 00:07:44.629
What's the change in time
involved in opening the switch?

00:07:44.629 --> 00:07:48.009
Well, it's something like zero.

00:07:48.009 --> 00:07:50.072
The switch was closed, then it was open,

00:07:50.072 --> 00:07:52.081
and that's how much, that took,

00:07:52.081 --> 00:07:54.550
let's say that took zero time to happen.

00:07:54.550 --> 00:07:56.972
Now, here's something that
happens with inductors

00:07:56.972 --> 00:07:58.972
that is kind of strange.

00:08:00.249 --> 00:08:02.473
Let's calculate the
voltage on the inductor

00:08:02.473 --> 00:08:04.012
right when this happens.

00:08:04.012 --> 00:08:06.322
And we know that V equals

00:08:06.322 --> 00:08:07.155
L times dI

00:08:09.296 --> 00:08:10.129
dT.

00:08:11.348 --> 00:08:12.520
And plug in some numbers here,

00:08:12.520 --> 00:08:14.270
so we have V equals L

00:08:15.240 --> 00:08:16.895
times what, dI,

00:08:16.895 --> 00:08:17.812
is minus I,

00:08:19.783 --> 00:08:20.910
over what?

00:08:20.910 --> 00:08:21.743
Over zero.

00:08:23.575 --> 00:08:24.408
And that equals what?

00:08:24.408 --> 00:08:26.908
That equals negative infinity.

00:08:30.306 --> 00:08:31.607
What the heck is going on here?

00:08:31.607 --> 00:08:33.024
Is that possible?

00:08:34.514 --> 00:08:35.597
No, it's not.

00:08:37.228 --> 00:08:39.129
Well lets, okay, let's take a second.

00:08:39.129 --> 00:08:42.635
Let's say this switch
didn't open in zero time.

00:08:42.635 --> 00:08:44.077
Let's say this switch,

00:08:44.077 --> 00:08:45.494
let's say dT was,

00:08:47.798 --> 00:08:49.848
let's say it was one nanosecond.

00:08:49.848 --> 00:08:53.164
Let's give it some time to open up.

00:08:53.164 --> 00:08:54.526
All right.

00:08:54.526 --> 00:08:55.607
Maybe that'll save us.

00:08:55.607 --> 00:08:57.045
Maybe that'll make more sense.

00:08:57.045 --> 00:08:57.878
Okay, let's do that.

00:08:57.878 --> 00:08:58.711
Let's go

00:08:58.711 --> 00:08:59.544
V equals L

00:09:01.585 --> 00:09:03.888
times dI dT, and dI,

00:09:03.888 --> 00:09:06.544
we decided was minus I

00:09:06.544 --> 00:09:10.820
over, it took one nanosecond
to open the switch.

00:09:10.820 --> 00:09:14.903
One nanosecond is one
times 10 to the minus nine.

00:09:16.667 --> 00:09:19.121
Okay, so what is that equal to?

00:09:19.121 --> 00:09:20.288
That equals L,

00:09:21.862 --> 00:09:23.362
negative L times I

00:09:24.602 --> 00:09:26.685
times 10 to the holy cow.

00:09:30.416 --> 00:09:32.083
10 to the plus nine.

00:09:32.930 --> 00:09:34.594
Okay, let's plug in some real values

00:09:34.594 --> 00:09:36.677
and see what we get here.

00:09:40.114 --> 00:09:42.096
Let's say V equals

00:09:42.096 --> 00:09:44.763
let's say L was 10 milliHenries,

00:09:46.144 --> 00:09:48.083
and let's say I was 10 milliamps

00:09:48.083 --> 00:09:50.727
flowing down through the,

00:09:50.727 --> 00:09:53.477
so delta I is minus 10 milliamps,

00:09:55.909 --> 00:09:56.742
and

00:09:57.781 --> 00:09:59.158
the time involved is

00:09:59.158 --> 00:10:01.575
10 to the minus nine seconds.

00:10:03.984 --> 00:10:05.848
What does that work out to?

00:10:05.848 --> 00:10:08.479
10 milliHenries, that's minus three

00:10:08.479 --> 00:10:09.812
and minus three,

00:10:11.064 --> 00:10:12.814
so it's gonna be 100,

00:10:15.209 --> 00:10:19.900
minus three minus three is
minus six over minus nine

00:10:19.900 --> 00:10:21.733
times 10 to the third.

00:10:23.914 --> 00:10:25.247
Okay, that is...

00:10:30.348 --> 00:10:32.527
And there's a minus sign here.

00:10:32.527 --> 00:10:33.953
This says that V,

00:10:33.953 --> 00:10:35.791
this point right here,

00:10:35.791 --> 00:10:38.335
or the voltage across the inductor.

00:10:38.335 --> 00:10:41.174
Our calculation just
said it's gonna be 100,

00:10:41.174 --> 00:10:43.032
minus 100,000 volts.

00:10:43.032 --> 00:10:45.500
Now, minus 100,000 volts means

00:10:45.500 --> 00:10:47.530
that the negative terminal

00:10:47.530 --> 00:10:51.280
is 100,000 volts above
the positive terminal,

00:10:52.365 --> 00:10:56.282
so this voltage V is
actually at 100,003 volts.

00:10:57.742 --> 00:10:59.075
How can this be?

00:11:00.172 --> 00:11:01.500
This is a puzzle about inductors

00:11:01.500 --> 00:11:03.315
that we actually have to solve by

00:11:03.315 --> 00:11:05.406
actually looking right up close

00:11:05.406 --> 00:11:08.514
at what's going on here, right
in this switch area here.

00:11:08.514 --> 00:11:11.262
So, this is where reality
comes in and saves us

00:11:11.262 --> 00:11:12.702
from our, our,

00:11:12.702 --> 00:11:14.692
the crazy results we're getting from our

00:11:14.692 --> 00:11:16.442
ideal equations here.

00:11:17.778 --> 00:11:20.317
So, let's pretend, here's the uh,

00:11:20.317 --> 00:11:22.622
here's the terminal of
the switch, right here.

00:11:22.622 --> 00:11:25.180
Let's do a blow up, and here's,

00:11:25.180 --> 00:11:27.263
here's that switch plate.

00:11:31.200 --> 00:11:33.137
As soon as this switch moves away,

00:11:33.137 --> 00:11:35.905
there's an air gap created here.

00:11:35.905 --> 00:11:39.809
In our ideal world, this air
gap was a perfect insulator,

00:11:39.809 --> 00:11:41.173
but what's gonna happen,

00:11:41.173 --> 00:11:43.137
because of these extreme numbers here,

00:11:43.137 --> 00:11:44.872
because of this is happening,

00:11:44.872 --> 00:11:47.452
this air gap is actually
not an ideal conductor,

00:11:47.452 --> 00:11:48.714
and what's gonna happen

00:11:48.714 --> 00:11:51.024
is we're gonna get a bright spark

00:11:51.024 --> 00:11:52.941
goes right across here.

00:11:56.762 --> 00:11:58.512
This really happens in real life.

00:11:58.512 --> 00:12:00.415
There is actually, when you open a switch,

00:12:00.415 --> 00:12:03.098
there can be a little
spark that goes between,

00:12:03.098 --> 00:12:05.180
and that gives this equation here,

00:12:05.180 --> 00:12:06.623
this dI dT,

00:12:06.623 --> 00:12:09.248
enough time to release that energy

00:12:09.248 --> 00:12:11.128
and that current from the inductor

00:12:11.128 --> 00:12:15.000
continues to flow and go
over into this switch.

00:12:15.000 --> 00:12:17.039
That current will flow, and it does it

00:12:17.039 --> 00:12:19.346
by breaking down the air molecules,

00:12:19.346 --> 00:12:20.208
and when you do that,

00:12:20.208 --> 00:12:22.762
that voltage there, for air.

00:12:22.762 --> 00:12:24.179
If this gap here,

00:12:25.398 --> 00:12:27.898
if this gap is one millimeter,

00:12:29.326 --> 00:12:31.659
for air, that's 3,000 volts.

00:12:33.644 --> 00:12:37.337
3,000 volts will cause
that spark to happen there.

00:12:37.337 --> 00:12:39.187
So, this is what actually happens,

00:12:39.187 --> 00:12:40.738
and you can build switches

00:12:40.738 --> 00:12:43.646
that will take this spark
and work for many years,

00:12:43.646 --> 00:12:47.813
but it's not always a good
idea to let this happen.


Inductor equations

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.601
- [Voiceover] Now we're gonna
talk about the two forms

00:00:02.601 --> 00:00:04.279
of the Inductor Equation and get

00:00:04.279 --> 00:00:06.161
familiar with these things.

00:00:06.161 --> 00:00:08.236
I'm gonna do some examples to show you how

00:00:08.236 --> 00:00:10.561
the Inductor Equations work.

00:00:10.561 --> 00:00:13.199
So, we know that the
Inductor Equation is the

00:00:13.199 --> 00:00:15.616
voltage across an inductor is

00:00:16.738 --> 00:00:20.738
a factor called L, the
inductance, times di, dt.

00:00:23.193 --> 00:00:25.340
So the voltage is proportional

00:00:25.340 --> 00:00:29.173
to the slope or the rate
of change of current.

00:00:30.037 --> 00:00:32.416
Let me do a quick review
of the two letters

00:00:32.416 --> 00:00:35.029
that are used as variables for inductors.

00:00:35.029 --> 00:00:37.029
When we have a resistor,

00:00:38.177 --> 00:00:39.662
we have, we have a,

00:00:39.662 --> 00:00:40.999
we call it a R,

00:00:40.999 --> 00:00:42.692
and it's in units of ohms,

00:00:42.692 --> 00:00:45.692
and that's named after Ohm, himself.

00:00:47.194 --> 00:00:50.027
And when we have a capacitor,

00:00:50.027 --> 00:00:52.793
the units are c, for capacitance,

00:00:52.793 --> 00:00:54.876
and f for faraday, farad,

00:00:56.314 --> 00:00:59.231
that's named after Michael Faraday.

00:01:00.642 --> 00:01:01.927
For the inductor, we have two

00:01:01.927 --> 00:01:04.427
other letters we use, L and H,

00:01:05.934 --> 00:01:07.671
and L is the name of the component,

00:01:07.671 --> 00:01:10.690
and that's named after
Lenz, Heinrich Lenz,

00:01:10.690 --> 00:01:14.027
and he did some research in induction,

00:01:14.027 --> 00:01:16.860
and the unit is named after Henry,

00:01:19.237 --> 00:01:22.879
and so that's the
corresponding R, c, and L

00:01:22.879 --> 00:01:24.243
are the names of the components,

00:01:24.243 --> 00:01:27.910
and ohms, farads, and
henries are the units.

00:01:29.396 --> 00:01:31.641
Okay, let's get back to our inductor.

00:01:31.641 --> 00:01:34.381
There's another form of
the inductor equation

00:01:34.381 --> 00:01:36.795
where we write it in terms of just i,

00:01:36.795 --> 00:01:38.388
in terms of v,

00:01:38.388 --> 00:01:39.671
and to get to that point,

00:01:39.671 --> 00:01:42.088
we need to get rid of
this derivative here.

00:01:42.088 --> 00:01:43.838
So, what we, what I'm gonna do

00:01:43.838 --> 00:01:47.051
is take the integral of
both sides of this equation.

00:01:47.051 --> 00:01:49.617
So, we take the integral of v,

00:01:49.617 --> 00:01:51.367
with respect to time,

00:01:53.067 --> 00:01:54.984
and that equals L times

00:01:56.828 --> 00:01:59.078
take the integral of di, dt

00:02:02.720 --> 00:02:04.470
with respect to time,

00:02:07.168 --> 00:02:09.190
and these two terms go away,

00:02:09.190 --> 00:02:11.425
and have the integral of di,

00:02:11.425 --> 00:02:14.323
the antiderivative of di is, is

00:02:14.323 --> 00:02:18.001
what, what function
has a derivative of di?

00:02:18.001 --> 00:02:20.171
And that would be just i.

00:02:20.171 --> 00:02:23.185
So, let me flip this all around here now.

00:02:23.185 --> 00:02:26.473
I'm gonna write i, on the left side

00:02:26.473 --> 00:02:27.890
equals one over L

00:02:28.727 --> 00:02:31.060
times the integral of v, dt.

00:02:35.231 --> 00:02:37.163
Now we always put limits on this integral,

00:02:37.163 --> 00:02:39.923
so the limits in this case are

00:02:39.923 --> 00:02:42.490
time is goes from minus infinity

00:02:42.490 --> 00:02:46.312
till some time now, and
this tells us that the, ah,

00:02:46.312 --> 00:02:48.287
that the current through an inductor

00:02:48.287 --> 00:02:51.240
depends on the voltage
across that inductor

00:02:51.240 --> 00:02:52.771
for the entire life

00:02:52.771 --> 00:02:55.737
all the way back to time
equals minus infinity, and

00:02:55.737 --> 00:02:57.402
that's not so convenient.

00:02:57.402 --> 00:02:58.710
We don't want to track this,

00:02:58.710 --> 00:03:01.480
the inductor's voltage
back that far in time.

00:03:01.480 --> 00:03:03.701
So what we'll do instead is we'll,

00:03:03.701 --> 00:03:07.320
we'll say, let's say at t equals zero,

00:03:07.320 --> 00:03:09.794
that we know the vol,
that we know the current,

00:03:09.794 --> 00:03:10.870
that we know some,

00:03:10.870 --> 00:03:12.787
we know some i of zero.

00:03:14.653 --> 00:03:17.159
And then we can change the limits on our,

00:03:17.159 --> 00:03:20.742
our integral here to
be zero on the bottom.

00:03:22.359 --> 00:03:26.359
One over L, integral from
time is zero until now

00:03:27.350 --> 00:03:28.517
time t, big T,

00:03:29.837 --> 00:03:31.420
of v of T times dt.

00:03:35.201 --> 00:03:36.856
In order to account for
everything that's happened

00:03:36.856 --> 00:03:38.710
before time equals zero,

00:03:38.710 --> 00:03:40.321
we'll just say, what was the current

00:03:40.321 --> 00:03:41.820
at time equals zero?

00:03:41.820 --> 00:03:43.487
So that's i of zero.

00:03:45.106 --> 00:03:47.422
And I'll do one more little change.

00:03:47.422 --> 00:03:50.350
I, I, I want this variable up here to be

00:03:50.350 --> 00:03:53.347
small t, and that means
I have to replace this t

00:03:53.347 --> 00:03:56.219
inside here with some other symbol as a,

00:03:56.219 --> 00:03:59.229
as a dummy variable,
and so now I'm gonna get

00:03:59.229 --> 00:04:00.562
i of t, little t

00:04:02.228 --> 00:04:04.728
one over L, integral from zero

00:04:06.548 --> 00:04:08.048
to little t of vd,

00:04:11.271 --> 00:04:13.188
let's just call it Tao,

00:04:14.405 --> 00:04:15.655
plus i of zero.

00:04:18.158 --> 00:04:20.878
So this is the integral form.

00:04:20.878 --> 00:04:23.962
There's the integral form
of the inductor equation

00:04:23.962 --> 00:04:26.770
and here is the derivative form.

00:04:26.770 --> 00:04:27.617
And we'll use both these.

00:04:27.617 --> 00:04:28.710
What I'm gonna do now,

00:04:28.710 --> 00:04:30.255
is we're gonna do an exercise

00:04:30.255 --> 00:04:32.922
where we use this one down here.

00:04:34.275 --> 00:04:36.887
So, in our circuit we
have a voltage source

00:04:36.887 --> 00:04:39.720
with a time pattern voltage in it,

00:04:43.958 --> 00:04:45.821
and we have a current
going through this thing.

00:04:45.821 --> 00:04:50.109
And we wanna find out,
what's this current?

00:04:50.109 --> 00:04:52.315
So, we're connected to a inductor.

00:04:52.315 --> 00:04:54.709
It's a 10 millihenry inductor

00:04:54.709 --> 00:04:56.440
and the voltage waveform is,

00:04:56.440 --> 00:04:57.858
it starts at zero,

00:04:57.858 --> 00:04:59.507
at time equals zero,

00:04:59.507 --> 00:05:02.033
the, the voltage goes up to two volts,

00:05:02.033 --> 00:05:05.385
then after two milliseconds
it goes to minus two volts.

00:05:05.385 --> 00:05:07.177
After two more milliseconds,

00:05:07.177 --> 00:05:11.303
it goes to plus two volts and on and on.

00:05:11.303 --> 00:05:12.783
So, we know v of t,

00:05:12.783 --> 00:05:14.487
and what we're going to
figure out is, what's,

00:05:14.487 --> 00:05:16.757
what's i through the, in the inductor?

00:05:16.757 --> 00:05:17.590
And to do that,

00:05:17.590 --> 00:05:19.343
we're gonna use our, our

00:05:19.343 --> 00:05:21.121
inductor equation here, the integral

00:05:21.121 --> 00:05:23.621
form of the inductor equation.

00:05:26.035 --> 00:05:29.202
So I actually need some
room on the screen here, so

00:05:29.202 --> 00:05:33.035
let me take a second and
clean out some stuff.

00:05:38.424 --> 00:05:40.424
Okay, we're ready to go.

00:05:41.391 --> 00:05:42.910
So let's look at what's going on here

00:05:42.910 --> 00:05:44.317
before time equals zero.

00:05:44.317 --> 00:05:45.681
So, in the before state,

00:05:45.681 --> 00:05:46.836
we're gonna make an assumption.

00:05:46.836 --> 00:05:48.363
We're gonna assume that

00:05:48.363 --> 00:05:50.530
i is zero a long time ago,

00:05:52.199 --> 00:05:54.072
and if we look at our chart, here,

00:05:54.072 --> 00:05:56.739
it says that v is zero, as well.

00:05:57.733 --> 00:05:59.332
And if I take this fact

00:05:59.332 --> 00:06:02.239
and I look at this equation right here,

00:06:02.239 --> 00:06:03.788
if v is zero,

00:06:03.788 --> 00:06:05.955
L is some positive number,

00:06:08.665 --> 00:06:09.741
so we can say that,

00:06:09.741 --> 00:06:10.682
so then we know that

00:06:10.682 --> 00:06:12.265
di, dt equals zero.

00:06:14.880 --> 00:06:17.648
So this circuit is
pretty much doing nothing

00:06:17.648 --> 00:06:20.648
before this waveform starts working.

00:06:23.763 --> 00:06:27.760
Over here I'm gonna
make a little time chart

00:06:27.760 --> 00:06:30.737
of what, what we discover.

00:06:30.737 --> 00:06:34.654
That's time, and this'll
be, ah, current, here.

00:06:37.713 --> 00:06:40.666
So before zero, before time is zero,

00:06:40.666 --> 00:06:43.047
we can say that the current is zero.

00:06:43.047 --> 00:06:45.473
Now, we've plotted in
part of our, our curve.

00:06:45.473 --> 00:06:46.557
Okay, now something happens.

00:06:46.557 --> 00:06:49.574
Now the voltage goes up two volts,

00:06:49.574 --> 00:06:51.079
and we're gonna use this equation,

00:06:51.079 --> 00:06:52.270
the integral form of the,

00:06:52.270 --> 00:06:55.429
the inductor equation to
figure out what happens

00:06:55.429 --> 00:06:58.679
to i of t now that voltage has changed.

00:07:00.268 --> 00:07:02.018
So, we can say that i

00:07:02.930 --> 00:07:04.347
equals one over L

00:07:05.965 --> 00:07:08.532
times the integral from zero,

00:07:08.532 --> 00:07:11.699
times zero is our starting point here,

00:07:14.012 --> 00:07:16.345
to t, some time t of v of t.

00:07:19.187 --> 00:07:22.600
Now, v, in this little
region of time here,

00:07:22.600 --> 00:07:24.350
is two volts plus two

00:07:26.851 --> 00:07:28.851
times d Tao plus i zero,

00:07:32.847 --> 00:07:34.498
and we decided i zero was the current

00:07:34.498 --> 00:07:36.804
that it started at right there;

00:07:36.804 --> 00:07:38.054
i zero is zero.

00:07:40.870 --> 00:07:42.356
Let's keep working on this.

00:07:42.356 --> 00:07:43.773
We can say that i

00:07:44.700 --> 00:07:47.283
equals one over 10 millihenries

00:07:50.151 --> 00:07:53.546
times the two comes out of the integral,

00:07:53.546 --> 00:07:54.858
and now it's the integral

00:07:54.858 --> 00:07:56.858
from zero to t of d Tao.

00:07:59.666 --> 00:08:01.583
And the zero goes away,

00:08:03.312 --> 00:08:07.312
that equals two over 10
millihenries times what?

00:08:10.241 --> 00:08:12.214
What's this integral evaluate to?

00:08:12.214 --> 00:08:14.327
It evaluates to t.

00:08:14.327 --> 00:08:17.951
Integral of d Tao between zero and t

00:08:17.951 --> 00:08:20.618
is t minus zero or just plain t.

00:08:22.883 --> 00:08:24.183
So looking at this, I see that

00:08:24.183 --> 00:08:26.516
i equals a constant times t,

00:08:27.435 --> 00:08:30.298
and that's the equation of a line.

00:08:30.298 --> 00:08:31.504
That's the equation of a line

00:08:31.504 --> 00:08:33.504
if I plot it on t and i,

00:08:35.253 --> 00:08:38.773
so two over 10 millihenries is the slope,

00:08:38.773 --> 00:08:40.160
and to plot that line,

00:08:40.160 --> 00:08:42.561
what I need is two points on the line.

00:08:42.561 --> 00:08:44.103
I know one of the points already.

00:08:44.103 --> 00:08:44.936
I know one of the points is gonna be

00:08:44.936 --> 00:08:47.019
right there at zero zero.

00:08:48.499 --> 00:08:51.412
What is the current at the
end of this pulse here?

00:08:51.412 --> 00:08:52.635
So, at two milliseconds,

00:08:52.635 --> 00:08:54.629
let's figure out what the current is,

00:08:54.629 --> 00:08:57.938
and that equals, at two milliseconds it's

00:08:57.938 --> 00:09:00.105
two times two milliseconds

00:09:02.561 --> 00:09:04.811
divided by 10 millihenries.

00:09:07.343 --> 00:09:09.794
The millis cancel themselves out

00:09:09.794 --> 00:09:11.961
and I get four over 10, or

00:09:13.110 --> 00:09:15.693
zero point four, what, Amperes.

00:09:18.378 --> 00:09:20.378
So, at two milliseconds,

00:09:24.342 --> 00:09:26.592
the value of the current is

00:09:29.000 --> 00:09:33.167
point four amperes, and I can
draw a straight between there.

00:09:35.639 --> 00:09:37.832
So now we have the value of the current

00:09:37.832 --> 00:09:40.972
during this first excursion,

00:09:40.972 --> 00:09:44.484
this first part of the voltage waveform.

00:09:44.484 --> 00:09:45.636
Now, we're gonna switch again.

00:09:45.636 --> 00:09:47.112
Now we have a new i zero.

00:09:47.112 --> 00:09:50.195
Let's, let's go to a new point, here.

00:09:51.330 --> 00:09:55.709
Let's go to a new point,
at, now we have i,

00:09:55.709 --> 00:09:57.792
i zero equals point four,

00:10:01.075 --> 00:10:03.242
and we have v equals what?

00:10:05.245 --> 00:10:08.575
Now it goes to minus two volts,

00:10:08.575 --> 00:10:12.345
minus two volts, so let's use our inductor

00:10:12.345 --> 00:10:14.039
equation one more time

00:10:14.039 --> 00:10:18.141
with these, with these,
ah, initial conditions

00:10:18.141 --> 00:10:20.891
that tells us i equals one over L

00:10:22.923 --> 00:10:25.923
integral from, what's the new point?

00:10:27.101 --> 00:10:28.851
Two milliseconds to t

00:10:31.930 --> 00:10:34.930
and v of t after two milliseconds is

00:10:36.161 --> 00:10:38.828
minus two volts, minus two d Tao

00:10:43.066 --> 00:10:46.664
plus, now we have a starting current.

00:10:46.664 --> 00:10:50.081
Our starting current was point four amps.

00:10:51.328 --> 00:10:53.078
Zero point four amps.

00:10:56.134 --> 00:10:57.538
And let's continue on here.

00:10:57.538 --> 00:11:00.871
Let's, ah, maybe can squeeze it in here;

00:11:03.302 --> 00:11:06.052
i equals one over 10 millihenries

00:11:09.223 --> 00:11:12.140
times minus two times the integral,

00:11:15.909 --> 00:11:19.909
the integral from two
milliseconds to t of d Tao

00:11:24.805 --> 00:11:26.555
plus zero point four;

00:11:31.434 --> 00:11:34.184
i equals minus two over 10 millis

00:11:36.783 --> 00:11:38.459
times, what is this?

00:11:38.459 --> 00:11:40.542
It's again, it's t minus,

00:11:42.800 --> 00:11:46.217
t minus two milliseconds plus point four,

00:11:52.806 --> 00:11:55.373
and again, we have the equation of a line.

00:11:55.373 --> 00:11:57.764
This is the equation of a line, i,

00:11:57.764 --> 00:12:00.470
there's a constant times t,

00:12:00.470 --> 00:12:02.649
there's a con, a,

00:12:02.649 --> 00:12:04.181
followed by a constant.

00:12:04.181 --> 00:12:06.798
And this is basically a sloping line.

00:12:06.798 --> 00:12:08.388
It has a negative slope,

00:12:08.388 --> 00:12:11.635
and the slope is negative
two over 10 millis,

00:12:11.635 --> 00:12:15.141
which is just the opposite
of what we had over here.

00:12:15.141 --> 00:12:17.588
We had two over 10 millihenries,

00:12:17.588 --> 00:12:20.110
so it's the same looking kind of line.

00:12:20.110 --> 00:12:22.110
So let me mark out here.

00:12:22.979 --> 00:12:26.019
Here's four milliseconds and it's gonna

00:12:26.019 --> 00:12:28.602
be a line that looks like this.

00:12:29.780 --> 00:12:32.151
So, the current's gonna ramp up,

00:12:32.151 --> 00:12:35.460
it's gonna turn around and
ramp straight back down

00:12:35.460 --> 00:12:37.255
because the, the slope up

00:12:37.255 --> 00:12:38.486
and the time up is the same as

00:12:38.486 --> 00:12:41.068
the slope down and the time down,

00:12:41.068 --> 00:12:43.182
just with a negative slope.

00:12:43.182 --> 00:12:44.616
So there's an example of the

00:12:44.616 --> 00:12:46.858
inductor equation in action.