Bacteria

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Kind: captions
Language: en

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I think we've all heard
of the word bacteria.

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And we normally associate
it with negative things.

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You say bacteria, those are
germs. So we normally

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associate those with germs, and
they indeed are germs, and

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they cause a whole set
of negative things.

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Or at least from the standard
point of view, people believe

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that they cause a whole bunch
of negative things.

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So let's just list them all
just to make sure we know

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about them, we're all
on the same page.

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So the bad things they do, they
cause a lot of diseases:

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tuberculosis, Lyme disease.

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I mean, I could go on and on.

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You know, pretty much
any time-- well,

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I'll be careful here.

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Whenever people talk about an
infection, it's often caused

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by a bacteria.

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It can also be caused
by a virus.

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An infection is, in general,
anything entering you and

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taking advantage of your body
to kind of replicate itself,

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and in the process,
making you sick.

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But bacterial infections,
let me write that down.

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And this whole perception of
bacteria being a bad thing is

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probably a good reason why
almost any soap you see now

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will say antibacterial on it.

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Because the makers of the soap
know that in conventional

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thinking, bacteria are viewed
as a negative thing.

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And you're like, OK,
Sal, I know where

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you're going with this.

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Bacteria isn't all bad.

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There are some good traits
of bacteria.

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For example, I could stick some
yogurt in some-- or I

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could stick some bacteria in
some milk and it'll help

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produce some yogurt, sometimes
spelled yoghurt.

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And that's obviously
a good thing.

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It's a delicious thing to eat.

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You say, well, I know I have
bacteria in my gut.

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It helps me digest food.

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And these are all true, but
you're like, look, you know,

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on balance, I still think
bacteria is a bad thing.

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I'm not going to take sides on
that debate, as I tend to

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avoid taking sides on debates
in these science videos.

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Maybe I'll do a whole playlist
where I do nothing but take

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sides on debates, but here I
won't take any sides on that.

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But I'll just point out that you
are to a large degree made

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up of bacteria.

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It's not just your gut.

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It's not just the gut or the
yogurt you might eat or the

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plaque on your teeth, which
is caused by bacteria.

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It's this kind of film that's
created by bacteria that

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eventually causes cavities
and whatever else.

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And it's not just the pimples
on your face.

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Bacteria actually represents
a majority of the

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cells on your body.

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So for every-- and this is kind
of an astounding fact.

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For every one cell on the human
body, every one human

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cell-- so these are all cells
that all have your DNA in them

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and they'll have nucleuses, and
I'll talk about that in a

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second-- you have 20 bacteria.

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Now, your response there is to
say, OK, that's fair enough,

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but these bacteria must be much
smaller than the human

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cell, so it must be a very small
fraction of my mass.

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And you're right.

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It's not like we're mostly
bacteria by mass, although we

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are mostly bacteria
by actual cells.

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But even if you were to take out
all of the water in your

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body, then by mass, bacteria
is going to be

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roughly 10% of your mass.

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So I weigh about 150 pounds,
I've got 15 pounds of bacteria

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walking around with me.

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So we always kind of think of
ourselves as like the bacteria

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is riding on us, but
to a large degree,

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we're kind of in symbiosis.

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We're kind of two creatures, or
not just two creatures, two

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types of creatures living
together, because I don't have

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just one type of
bacteria on me.

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I have thousands of types
of bacteria on me.

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There's a huge amount of
diversity, and we're just

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scratching the surface in terms
of the number and types

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and diversity of bacteria
that exist.

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So I've talked a lot about
bacteria, and hopefully, this

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fact right here will make you
realize that they're super

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important to just our
everyday existence.

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Just to make sure we understand
the magnitude of

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this, in a previous video, and
I looked this up again, we

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have on the order of 10
to a 100 trillion

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cells, human cells.

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So for every one of these we
have 20 bacteria, we're

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talking about having on the
order of 200 to 2,000 trillion

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bacteria on us at any time.

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And I'm a hygienic person.

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I take showers daily,
and that's even me.

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It's not like you can somehow
eliminate them.

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And even more, it's not
like you even want

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to eliminate them.

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But that's fair enough.

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You're probably asking, OK,
Sal, I'm convinced that

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bacteria are important.

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What do they actually
look like?

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And they're these small
unicellular organisms. That's

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my bacteria right there.

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And they're different from the
cells that make up us.

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When I say us, I'll throw in
all plants, animals and

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funguses, fungi.

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And the big difference, or the
one that people noticed first,

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is that all of the Eukarya,
which includes plants, animals

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and fungi, all of their DNA
is inside of a nucleus, a

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cellular nucleus.

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So that's the nucleus
right there.

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And all of our DNA,
it's normally in

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its chromatin form.

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It's all just spread around
something like that.

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In bacteria, which are what
people originally just

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classify it on whether or not
you have a nucleus, in

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bacteria, there is no membrane
surrounding the DNA.

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So what they have is just
a big bundle of DNA.

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They just have this
big bundle of DNA.

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It's sometimes in a
loop all in one

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circle called a nucleoid.

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Now, whenever we look at
something, and we say, oh, we

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have this thing; it doesn't;
there's this assumption that

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somehow we're superior or we're
more advanced beings.

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But the reality is that bacteria
have infiltrated far

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more ecosystems in every part
of the planet than Eukarya

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have, and there's far more
diversity in bacteria than

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there is in Eukarya.

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So when you really think about
it, these are the more

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successful organisms. If a comet
were to hit the Earth--

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God forbid-- the organisms more
likely to survive are

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going to be the bacteria than
the Eukarya, than the ones

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with the larger-- not always
larger, but the organisms that

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do have this nucleus and have
membrane-bound organelles like

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mitochondria and all that.

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We'll talk more about
it in the future.

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Bacteria, for the
most part, are

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just big bags of cytoplasm.

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They have their DNA there.

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They do have ribosomes because
they have to code for proteins

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just like the rest of us do.

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Some of those proteins, they'll
make some from--

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bacteria, they'll make these
flagella, which are tails that

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allow them to move around.

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They also have these
things called pili.

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Pili is plural for pilus or
pee-lus, so these pili.

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And we'll see in a second that
the pili are kind of how the

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bacteria are able to do one form
of introducing genetic

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variation into their
populations.

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Actually, I'll take a little
side note here.

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I'm pointing out bacteria as
not having a cell wall.

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There's actually another class
that used to be categorized as

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type of a bacteria, and they're
called Archaea.

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I should give them a little
bit of justice.

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They're always kind
of the stepchild.

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They used to be called Archaea
bacteria, but now people

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realize, they've actually looked
at the DNA, because

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when they originally looked at
these, they said, OK, these

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guys also have no nucleus and a
bunch of DNA running around.

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These must be a form
of bacteria.

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But now that we've actually been
able to look into the DNA

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of the things, we've
seen that they're

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actually quite different.

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But all of these, both bacteria
and Archaea, are

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considered prokaryotes.

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And this just means
no nucleus.

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No nucleus, and more generally,
this is what most

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people refer to, but more
generally, they don't have

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these membrane-bound organelles
that our cells

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have.

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Now, the next question you might
say is, well, how do

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these bacteria reproduce?

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And for the most part, they do
something not completely

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different from mitosis,
although I

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want to call it mitosis.

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We call it binary fission.

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I'm not going to go into the
deep mechanism here, but the

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idea is fairly simple.

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I have a bacteria right here.

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It replicates its DNA, so
it'll have two of these

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nucleoids here, and then the
cytoplasm essentially splits,

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or it's kind of a form of
cleavage right there.

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It splits and then you
have two of them.

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You have two of them then.

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And then each of them, they
can code for the proteins

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necessary to produce all of
their extra appendages, the

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flagellum, which is this
long tail-like thing

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that can help it move.

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And it's actually fascinating
because it's operating at such

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a small scale, but you can still
kind of get this motor

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movement going on.

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Even at this very, very small
scale, using very primitive--

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I won't say primitive, because
that's making a value judgment

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on these things, but using-- you
know, these flagellum are

00:10:21.066 --> 00:10:23.650
on the order of several
nanometers, on the order of

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tens of nanometers wide.

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So you don't have a lot of atoms
to deal with, but you're

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still able to get this kind of
wave-like motion that can move

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the bacteria around.

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Now, you're saying, hey, Sal,
in that first video on

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evolution, you told me that we
see evolution every day and

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bacteria is one example.

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When we use antibiotics, we
think it'll help eliminate

00:10:45.100 --> 00:10:47.800
bacteria, but that one bacteria
that has some type of

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resistance, it'll survive,
so it is more fit.

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How did these guys
get variation?

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Well, the one way, and this is
the way everything can get

00:10:55.010 --> 00:10:58.730
variation, is they can
get mutations.

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And bacteria replicate so
quickly, they reproduce so

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quickly that even if you have a
mutation that's one in every

00:11:05.810 --> 00:11:07.705
thousand times, by the time you
have a million bacteria,

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you'll have a thousand
mutations.

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So they have mutations, but
they also have this form.

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I don't want to call it sexual
reproduction, because it's not

00:11:15.050 --> 00:11:15.850
sexual reproduction.

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They don't form gametes and the
gametes don't fertilize

00:11:18.170 --> 00:11:20.090
each other and then
produces a zygote.

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But two bacteria can get near
each other and then one of

00:11:29.840 --> 00:11:35.720
their piluses-- I'll
do that right here.

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So the piluses are these little
structures on the side

00:11:38.830 --> 00:11:39.690
of the bacteria.

00:11:39.690 --> 00:11:42.250
They're these little
tubes, really.

00:11:42.250 --> 00:11:46.340
One of the piluses can connect
from one bacteria to another,

00:11:46.340 --> 00:11:49.710
and then essentially you have a
mixing of what's inside one

00:11:49.710 --> 00:11:50.790
bacteria with another.

00:11:50.790 --> 00:11:52.530
So let me draw their
nucleoids.

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And then they have these other
pieces of just DNA that hangs

00:11:59.570 --> 00:12:01.310
out called plasmids.

00:12:01.310 --> 00:12:04.710
These are just circular
pieces of DNA.

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Maybe this guy has got this
extra neat plasmid.

00:12:07.460 --> 00:12:10.540
He got it from someplace, and
it's making him able to do

00:12:10.540 --> 00:12:11.790
things that this guy
couldn't do.

00:12:11.790 --> 00:12:17.040
Maybe this is the R plasmid,
which is known for making a

00:12:17.040 --> 00:12:19.530
bacteria resistant to a
lot of antibiotics.

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And what happens is, that
bacteria-- and actually,

00:12:21.270 --> 00:12:23.530
there's mechanisms where the
bacteria know that, hey, this

00:12:23.530 --> 00:12:24.850
guy doesn't have
the R plasmid.

00:12:24.850 --> 00:12:26.940
And we're just beginning to
understand how it actually

00:12:26.940 --> 00:12:30.900
works, but this will actually
replicate itself and give this

00:12:30.900 --> 00:12:33.290
guy a version of
the R plasmid.

00:12:33.290 --> 00:12:35.550
You could also have these
things, transposons, and I

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should make a whole video
on this because we have

00:12:37.240 --> 00:12:38.100
transposons, too.

00:12:38.100 --> 00:12:41.160
But there's parts of DNA that
can jump from one part of a

00:12:41.160 --> 00:12:43.870
fragment of DNA to another, and
these can also end up in

00:12:43.870 --> 00:12:44.750
the other one.

00:12:44.750 --> 00:12:47.230
So what you have is kind of--
it's not formal sexual

00:12:47.230 --> 00:12:48.890
reproduction, but what you
essentially have is a

00:12:48.890 --> 00:12:52.190
connection, and these bacteria
are just constantly swapping

00:12:52.190 --> 00:12:55.730
DNA with each other and DNA is
jumping back and forth, so you

00:12:55.730 --> 00:12:58.490
can imagine all sorts of
combinations of DNA happen

00:12:58.490 --> 00:13:02.840
even within what you used to
call one bacterial species and

00:13:02.840 --> 00:13:05.130
very quickly can turn to
multiple species and become

00:13:05.130 --> 00:13:06.300
resistant to different things.

00:13:06.300 --> 00:13:08.820
If this makes it resistant to
an antibiotic, then it can

00:13:08.820 --> 00:13:13.150
kind of spread the information
to produce those resistant

00:13:13.150 --> 00:13:15.850
proteins or whatever to
the other bacteria.

00:13:15.850 --> 00:13:19.460
So this is kind of a form of
introducing variation.

00:13:19.460 --> 00:13:23.940
And so when you transfer stuff
via this pilus, or the plural

00:13:23.940 --> 00:13:27.660
is pili, this is called
conjugation, bacterial

00:13:27.660 --> 00:13:28.910
conjugation.

00:13:32.520 --> 00:13:34.640
Now, the last thing I want to
talk about, because it's

00:13:34.640 --> 00:13:38.990
something that you've heard a
lot about, are antibiotics.

00:13:38.990 --> 00:13:39.930
A lot of people,
they get sick.

00:13:39.930 --> 00:13:41.550
The first thing they want
to get is an antibiotic.

00:13:44.680 --> 00:13:47.830
And an antibiotic is just a
whole class of chemicals and

00:13:47.830 --> 00:13:49.990
compounds, some of them
naturally derived, some of

00:13:49.990 --> 00:13:54.810
them not, that kill bacteria.

00:13:54.810 --> 00:13:58.660
So now if someone is undergoing
a surgery and they

00:13:58.660 --> 00:14:01.670
get a cut, instead of them
having to worry about getting

00:14:01.670 --> 00:14:04.240
an infection, they'll take some
antibiotics to prevent

00:14:04.240 --> 00:14:06.980
the bacteria from
growing on them.

00:14:06.980 --> 00:14:11.110
But the question is how was this
discovered or where does

00:14:11.110 --> 00:14:12.050
it come from?

00:14:12.050 --> 00:14:16.150
It actually came from
Alexander Fleming.

00:14:16.150 --> 00:14:17.000
Let me write him down.

00:14:17.000 --> 00:14:20.310
Very important, because the
discovery of antibiotics is,

00:14:20.310 --> 00:14:22.380
in my opinion, the most
important discovery in

00:14:22.380 --> 00:14:23.340
medicine so far.

00:14:23.340 --> 00:14:27.290
So, Alexander Fleming.

00:14:30.150 --> 00:14:33.130
He was studying-- I think
it was Staphylococcus.

00:14:33.130 --> 00:14:34.920
I forget which bacteria
it was, but it

00:14:34.920 --> 00:14:36.220
was in a Petri dish.

00:14:36.220 --> 00:14:37.765
He was using a Petri dish.

00:14:37.765 --> 00:14:38.985
Let me draw a Petri dish.

00:14:38.985 --> 00:14:40.460
There's a little circle.

00:14:40.460 --> 00:14:42.790
There's some nutrients that
the bacteria can grow on.

00:14:42.790 --> 00:14:46.180
So let's say the bacteria, you
know, it's growing on this

00:14:46.180 --> 00:14:47.170
Petri dish.

00:14:47.170 --> 00:14:49.510
And he went out, and he came
back into the room, and he saw

00:14:49.510 --> 00:14:56.880
that some mold, some fungus had
grown on this, kind of a

00:14:56.880 --> 00:15:01.310
bluish-greenish fungus had grown
on the center of his

00:15:01.310 --> 00:15:04.420
Petri dish.

00:15:04.420 --> 00:15:06.690
And the bacteria, there was kind
of this space around it,

00:15:06.690 --> 00:15:08.800
and the bacteria couldn't
get close to it.

00:15:08.800 --> 00:15:14.580
And this mold, this fungus was
called Penicillium, the

00:15:14.580 --> 00:15:15.730
Penicillium fungus.

00:15:15.730 --> 00:15:16.680
He was able to figure
that out.

00:15:16.680 --> 00:15:18.750
He took a sample of this and
then he cultured it, which

00:15:18.750 --> 00:15:21.300
means letting it grow and
then seeing what it is.

00:15:21.300 --> 00:15:22.810
This was Penicillium.

00:15:22.810 --> 00:15:26.650
And he figured out that, gee,
this fungus must have

00:15:26.650 --> 00:15:28.730
something, some chemical that
it's emitting that's

00:15:28.730 --> 00:15:31.070
essentially killing the bacteria
around it, that's not

00:15:31.070 --> 00:15:33.950
allowing the bacteria
to get near it.

00:15:33.950 --> 00:15:36.775
And so that led to the discovery
of penicillin.

00:15:47.660 --> 00:15:51.620
This was in the late
20s, 1920s.

00:15:51.620 --> 00:15:54.420
By the time World War II came
around, now people had gunshot

00:15:54.420 --> 00:15:56.890
wounds and they had to get
things amputated, whatnot, but

00:15:56.890 --> 00:15:58.850
for the first time, they could
actually give people

00:15:58.850 --> 00:16:02.970
antibiotics and not worry
about-- or they probably still

00:16:02.970 --> 00:16:05.990
worried about it, but didn't
have to worry about this thing

00:16:05.990 --> 00:16:07.670
as much as they did before.

00:16:07.670 --> 00:16:09.970
And now, you know, if you have
bacteria, if you have

00:16:09.970 --> 00:16:14.540
tuberculosis or Lyme disease or
anything, the treatment all

00:16:14.540 --> 00:16:16.390
involves taking antibiotics.

00:16:16.390 --> 00:16:18.600
And there are many, many more
types of antibiotics now

00:16:18.600 --> 00:16:21.140
coming from many, many more
different sources, but the

00:16:21.140 --> 00:16:22.570
general idea is the same.

00:16:22.570 --> 00:16:24.520
You want to kill bacteria.

00:16:24.520 --> 00:16:25.890
Although you don't want to kill
all bacteria, because

00:16:25.890 --> 00:16:26.550
some of it's good.

00:16:26.550 --> 00:16:30.190
In fact, we are made up a
whole lot of bacteria.

00:16:30.190 --> 00:16:31.140
I don't know if I even
mentioned this

00:16:31.140 --> 00:16:31.500
earlier in the video.

00:16:31.500 --> 00:16:34.540
There's bacteria in our skin
that helps take up oil and

00:16:34.540 --> 00:16:37.460
moisturizes and make our
skin nice and supple.

00:16:37.460 --> 00:16:38.930
So, you know, the way you think
about it, you could view

00:16:38.930 --> 00:16:41.000
them as negative or you could
view them as positive or you

00:16:41.000 --> 00:16:43.050
could view them as something
in between, but the really

00:16:43.050 --> 00:16:45.820
amazing thing, at least in my
mind, is that we're living in

00:16:45.820 --> 00:16:46.980
symbiosis with them.

00:16:46.980 --> 00:16:50.050
I remember I saw a Star Trek
episode once where you had

00:16:50.050 --> 00:16:52.700
these people.

00:16:52.700 --> 00:16:56.680
You had these people, and they
were some alien race.

00:16:56.680 --> 00:16:59.430
Jean-Luc Picard had--
they ran into them.

00:16:59.430 --> 00:17:02.470
They looked very humanoid
like that.

00:17:02.470 --> 00:17:05.109
But it turns out-- let me draw
this human-- that they had

00:17:05.109 --> 00:17:07.579
these little bugs in
their brain stem.

00:17:07.579 --> 00:17:10.000
So they had these big insects in
their brain stem and these

00:17:10.000 --> 00:17:14.310
insects started infecting the
crew of the Enterprise, and

00:17:14.310 --> 00:17:17.099
they were controlling their
brains and making them act

00:17:17.099 --> 00:17:19.450
weird and whatever not, and
this seemed like a very

00:17:19.450 --> 00:17:24.069
bizarre alien concept of some
creepy-crawlie living in us

00:17:24.069 --> 00:17:26.890
and affecting our brains and
affecting us in some ways.

00:17:26.890 --> 00:17:29.060
But if you really think about
it, we are doing this, and

00:17:29.060 --> 00:17:31.680
it's not just with one
little bug, it's on

00:17:31.680 --> 00:17:32.960
the order of trillions.

00:17:32.960 --> 00:17:36.540
Hundreds of trillions of bugs
are with us every day and they

00:17:36.540 --> 00:17:37.540
make us us.

00:17:37.540 --> 00:17:40.750
I mean, I'm here recording
videos along with-- or maybe I

00:17:40.750 --> 00:17:44.040
should even say the bacteria
is recording videos or it's

00:17:44.040 --> 00:17:46.250
maybe partially responsible
for controlling bacteria.

00:17:46.250 --> 00:17:48.870
And it's known that the bacteria
can even affect our

00:17:48.870 --> 00:17:49.910
mental state.

00:17:49.910 --> 00:17:52.220
There's a whole bunch of
research now that certain

00:17:52.220 --> 00:17:54.650
types of bacteria can
cause schizophrenia.

00:17:54.650 --> 00:17:56.780
Actually, syphilis does.

00:17:56.780 --> 00:17:58.920
Bacteria can cause depression.

00:17:58.920 --> 00:18:03.400
Lyme disease, it's known that
when you go into later phases

00:18:03.400 --> 00:18:07.660
of Lyme disease, it can affect
the mental condition of the

00:18:07.660 --> 00:18:11.320
person who has the infection,
so it affects every part of

00:18:11.320 --> 00:18:11.920
who we are.

00:18:11.920 --> 00:18:15.100
I mean, it would be hard to
even talk of being a human

00:18:15.100 --> 00:18:20.690
being without the 10% of our
mass or the 2,000 trillion

00:18:20.690 --> 00:18:23.670
cells or 2,000 trillion
bacteria that

00:18:23.670 --> 00:18:25.880
really make us us.


Derivative of x^(x^x)

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.660
A bit of a classic implicit
differentiation problem

00:00:03.660 --> 00:00:10.900
is the problem y is
equal to x to the x.

00:00:10.900 --> 00:00:13.810
And then to find out what
the derivative of y

00:00:13.810 --> 00:00:16.350
is with respect to x.

00:00:16.350 --> 00:00:19.520
And people look at that, oh you
know, I don't have just a

00:00:19.520 --> 00:00:21.800
constant exponent here, so I
can't just use the power

00:00:21.800 --> 00:00:23.300
rules, how do you do it.

00:00:23.300 --> 00:00:26.090
And the trick here is really
just to take the natural log of

00:00:26.090 --> 00:00:27.860
both sides of this equation.

00:00:27.860 --> 00:00:29.530
And this is going to build
up to what we're going to

00:00:29.530 --> 00:00:30.460
do later in this video.

00:00:30.460 --> 00:00:34.530
So If you take the natural log
on both sides of this equation,

00:00:34.530 --> 00:00:38.570
you get the natural log of y is
equal to the natural

00:00:38.570 --> 00:00:40.880
log of x to the x.

00:00:40.880 --> 00:00:44.750
Now our power rules, or I guess
our natural log rules, say

00:00:44.750 --> 00:00:46.550
look, if I'm taking the natural
log of something to the

00:00:46.550 --> 00:00:50.730
something, this is equivalent
to, I can rewrite the natural

00:00:50.730 --> 00:00:54.690
log of x to the x as being
equal to x times the

00:00:54.690 --> 00:00:56.550
natural log of x.

00:00:56.550 --> 00:00:59.380
So let me rewrite
everything again.

00:00:59.380 --> 00:01:01.910
If I take the natural log of
both sides of that equation, I

00:01:01.910 --> 00:01:06.190
get the natural log of y is
equal to x times the

00:01:06.190 --> 00:01:08.530
natural log of x.

00:01:08.530 --> 00:01:11.100
And now we can take the
derivative of both sides of

00:01:11.100 --> 00:01:12.350
this with respect to x.

00:01:12.350 --> 00:01:15.640
So the derivative with respect
to x of that, and then

00:01:15.640 --> 00:01:19.440
the derivative with
respect to x of that.

00:01:19.440 --> 00:01:24.710
Now we're going to apply a
little bit of the chain rule.

00:01:24.710 --> 00:01:25.520
So the chain rule.

00:01:25.520 --> 00:01:27.730
What's the derivative of
this with respect to x?

00:01:27.730 --> 00:01:30.840
What's the derivative of
our inner expression

00:01:30.840 --> 00:01:32.330
with respect to x?

00:01:32.330 --> 00:01:35.390
It's a little implicit
differentiation, so it's dy

00:01:35.390 --> 00:01:38.600
with respect to x times the
derivative of this whole

00:01:38.600 --> 00:01:40.710
thing with respect to
this inner function.

00:01:40.710 --> 00:01:44.150
So the derivative of
natural log of x is 1/x.

00:01:44.150 --> 00:01:45.690
So the derivative of
natural log of y with

00:01:45.690 --> 00:01:48.840
respect to y is 1/y.

00:01:48.840 --> 00:01:50.080
So times 1/y.

00:01:52.600 --> 00:01:55.450
And the derivative of this--
this is just the product rule,

00:01:55.450 --> 00:01:59.740
and I'll arbitrarily switch
colors here-- is the derivative

00:01:59.740 --> 00:02:03.610
of the first term, which is 1,
times the second term, so times

00:02:03.610 --> 00:02:08.760
the natural log of x plus the
derivative of the second term,

00:02:08.760 --> 00:02:11.550
which is 1/x times
the first term.

00:02:11.550 --> 00:02:13.380
So times x.

00:02:13.380 --> 00:02:23.280
And so we get dy/dx times 1/y
is equal to natural log of x

00:02:23.280 --> 00:02:28.140
plus-- this just turns out to
be 1-- x divided by x, and

00:02:28.140 --> 00:02:30.260
then you multiply both
sides of this by y.

00:02:30.260 --> 00:02:35.590
You get dy/dx is equal
to y times the natural

00:02:35.590 --> 00:02:37.610
log of x plus 1.

00:02:37.610 --> 00:02:40.360
And if you don't like this y
sitting here, you could just

00:02:40.360 --> 00:02:41.410
make the substitution.

00:02:41.410 --> 00:02:43.620
y is equal to x to the x.

00:02:43.620 --> 00:02:47.410
So you could say that the
derivative of y with respect to

00:02:47.410 --> 00:02:52.890
x is equal to x to the x times
the natural log of x plus 1.

00:02:52.890 --> 00:02:55.950
And that's a fun problem, and
this is often kind of given as

00:02:55.950 --> 00:02:59.810
a trick problem, or sometimes
even a bonus problem if people

00:02:59.810 --> 00:03:02.180
don't know to take the natural
log of both sides of that.

00:03:02.180 --> 00:03:05.290
But I was given an even more
difficult problem, and

00:03:05.290 --> 00:03:06.550
that's what we're going
to tackle in this.

00:03:06.550 --> 00:03:09.270
But it's good to see this
problem done first because it

00:03:09.270 --> 00:03:11.900
gives us the basic tools.

00:03:11.900 --> 00:03:14.350
So the more difficult
problem we're going to

00:03:14.350 --> 00:03:17.700
deal with is this one.

00:03:17.700 --> 00:03:19.610
Let me write it down.

00:03:19.610 --> 00:03:26.990
So the problem is y is equal
to x to the-- and here's the

00:03:26.990 --> 00:03:30.640
twist-- x to the x to the x.

00:03:30.640 --> 00:03:33.700
And we want to find out dy/dx.

00:03:33.700 --> 00:03:36.310
We want to find out
the derivative of y

00:03:36.310 --> 00:03:38.630
with respect to x.

00:03:38.630 --> 00:03:41.300
So to solve this problem we
essentially use the same tools.

00:03:41.300 --> 00:03:44.320
We use the natural log to
essentially breakdown this

00:03:44.320 --> 00:03:47.000
exponent and get it into
terms that we can deal with.

00:03:47.000 --> 00:03:48.780
So we can use the product rule.

00:03:48.780 --> 00:03:51.300
So let's take the natural log
of both sides of this equation

00:03:51.300 --> 00:03:52.810
like we did last time.

00:03:52.810 --> 00:03:58.570
You get the natural log of y
is equal to the natural log

00:03:58.570 --> 00:04:02.630
of x to the x to the x.

00:04:05.230 --> 00:04:06.910
And this is just the
exponent on this.

00:04:06.910 --> 00:04:12.820
So we can rewrite this as x to
the x times the natural log

00:04:12.820 --> 00:04:17.260
times the natural log of x.

00:04:17.260 --> 00:04:21.200
So now our expression our
equation is simplified to the

00:04:21.200 --> 00:04:25.770
natural log of y is equal to x
to the x times the

00:04:25.770 --> 00:04:26.940
natural log of x.

00:04:26.940 --> 00:04:29.890
But we still have this
nasty x to the x here.

00:04:29.890 --> 00:04:33.600
We know no easy way to take the
derivative there, although I've

00:04:33.600 --> 00:04:36.310
actually just shown you what
the derivative of this is, so

00:04:36.310 --> 00:04:38.720
we could actually just
apply it right now.

00:04:38.720 --> 00:04:40.890
I was going to take the natural
log again and it would turn

00:04:40.890 --> 00:04:45.250
into this big, messy, confusing
thing but I realized that

00:04:45.250 --> 00:04:47.340
earlier in this video I
just solved for what the

00:04:47.340 --> 00:04:50.230
derivative of x to the x is.

00:04:50.230 --> 00:04:51.660
It's this thing right here.

00:04:51.660 --> 00:04:53.390
It's this crazy
expression right here.

00:04:53.390 --> 00:04:57.890
So we just have to remember
that and then apply and

00:04:57.890 --> 00:04:59.620
then do our problem.

00:04:59.620 --> 00:05:01.350
So let's do our problem.

00:05:01.350 --> 00:05:05.480
And if we hadn't solved this
ahead of time, it was kind of

00:05:05.480 --> 00:05:08.470
an unexpected benefit of doing
the simpler version of the

00:05:08.470 --> 00:05:12.640
problem, you could just keep
taking the natural log of this,

00:05:12.640 --> 00:05:13.930
but it'll just get a
little bit messier.

00:05:13.930 --> 00:05:15.970
But since we already know what
the derivative of x to the

00:05:15.970 --> 00:05:18.150
x is, let's just apply it.

00:05:18.150 --> 00:05:20.540
So we're going to take
the derivative of both

00:05:20.540 --> 00:05:21.530
sides of the equation.

00:05:21.530 --> 00:05:26.060
Derivative of this is equal
to the derivative of this.

00:05:26.060 --> 00:05:28.160
We'll ignore this for now.

00:05:28.160 --> 00:05:31.780
Derivative of this with respect
to x is the derivative of

00:05:31.780 --> 00:05:34.700
the natural log of y
with respect to y.

00:05:34.700 --> 00:05:38.160
So that's 1/y times
the derivative of y

00:05:38.160 --> 00:05:38.965
with respect to x.

00:05:38.965 --> 00:05:40.630
That's just the chain rule.

00:05:40.630 --> 00:05:43.010
We learned that in
implicit differentiation.

00:05:43.010 --> 00:05:48.870
And so this is equal to the
derivative of the first term

00:05:48.870 --> 00:05:51.760
times the second term, and I'm
going to write it out here just

00:05:51.760 --> 00:05:54.280
because I don't want to skip
steps and confuse people.

00:05:54.280 --> 00:05:57.615
So this is equal to the
derivative with respect to x of

00:05:57.615 --> 00:06:03.370
x to the x times the natural
log of x plus the derivative

00:06:03.370 --> 00:06:05.690
with respect to x of
the natural log of

00:06:05.690 --> 00:06:11.040
x times x to the x.

00:06:11.040 --> 00:06:14.100
So let's focus on the right
hand side of this equation.

00:06:14.100 --> 00:06:17.980
What is the derivative of x
to the x with respect to x?

00:06:17.980 --> 00:06:19.920
Well we just solved that
problem right here.

00:06:19.920 --> 00:06:23.730
It's x to the x natural
log of x plus 1.

00:06:23.730 --> 00:06:30.040
So this piece right there--
I already forgot what it

00:06:30.040 --> 00:06:33.720
was-- it was x to the x
natural log of x plus 1.

00:06:33.720 --> 00:06:41.130
That is x to the x times the
natural log of x plus 1.

00:06:41.130 --> 00:06:42.810
And then we're going to
multiply that times

00:06:42.810 --> 00:06:44.070
the natural log of x.

00:06:48.050 --> 00:06:51.560
And then we're going to add
that to, plus the derivative

00:06:51.560 --> 00:06:54.910
of the natural log of x.

00:06:54.910 --> 00:06:59.310
That's fairly straightforward,
that's 1/x times x to the x.

00:07:03.010 --> 00:07:05.690
And of course the left
hand side of the equation

00:07:05.690 --> 00:07:10.432
was just 1/y dy/dx.

00:07:10.432 --> 00:07:14.740
And we can multiply both sides
of this now by y, and we get

00:07:14.740 --> 00:07:22.470
dy/dx is equal to y times all
of this crazy stuff-- x to the

00:07:22.470 --> 00:07:27.840
x times the natural log of x
plus 1 times the natural log of

00:07:27.840 --> 00:07:34.520
x plus 1/x times x to the x.

00:07:34.520 --> 00:07:36.000
That's x to the negative 1.

00:07:36.000 --> 00:07:38.890
We could rewrite this as x
to the minus 1, and then

00:07:38.890 --> 00:07:39.750
you add the exponents.

00:07:39.750 --> 00:07:44.650
You could write this as x
to the x minus 1 power.

00:07:44.650 --> 00:07:48.640
And if we don't like this
y here, we can just

00:07:48.640 --> 00:07:49.890
substitute it back.

00:07:49.890 --> 00:07:53.440
y was equal to this, this
crazy thing right there.

00:07:53.440 --> 00:07:59.110
So our final answer for this
seemingly-- well on one level

00:07:59.110 --> 00:08:01.120
looks like a very simple
problem, but on another level

00:08:01.120 --> 00:08:03.220
when you appreciate what it's
saying, it's like oh there's a

00:08:03.220 --> 00:08:07.480
very complicated problem-- you
get the derivative of y with

00:08:07.480 --> 00:08:11.370
respect to x is equal
to y, which is this.

00:08:11.370 --> 00:08:21.480
So that's x to the x to the x
times all of this stuff-- times

00:08:21.480 --> 00:08:27.580
x to the x natural log of x
plus 1 times the natural log

00:08:27.580 --> 00:08:34.080
of x, and then all of that
plus x to the x minus 1.

00:08:34.080 --> 00:08:34.940
So who would have thought.

00:08:34.940 --> 00:08:36.460
Sometimes math is elegant.

00:08:36.460 --> 00:08:37.870
You take the derivative of
something like this and

00:08:37.870 --> 00:08:39.080
you get something neat.

00:08:39.080 --> 00:08:42.320
For example, when you take
the derivative of natural

00:08:42.320 --> 00:08:43.774
log of x you get 1/x.

00:08:43.774 --> 00:08:46.340
That's very simple and elegant,
and it's nice that math

00:08:46.340 --> 00:08:47.030
worked out that way.

00:08:47.030 --> 00:08:50.130
But sometimes you do something,
you take an operation on

00:08:50.130 --> 00:08:52.070
something that looks pretty
simple and elegant, and you get

00:08:52.070 --> 00:08:55.420
something that's hairy and not
that pleasant to look

00:08:55.420 --> 00:08:59.800
at, but is a pretty
interesting problem.

00:08:59.800 --> 00:09:00.080
And there you go.


Multiplying multiple digit numbers

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.100
We now have the general
tools to really tackle any

00:00:04.100 --> 00:00:05.390
multiplication problems.

00:00:05.390 --> 00:00:08.390
So in this video I'm just going
to do a ton of examples.

00:00:08.390 --> 00:00:12.670
So let's start off with--
and I'll start in yellow.

00:00:12.670 --> 00:00:17.075
Let's start off
with 32 times 18.

00:00:20.320 --> 00:00:24.260
Say 8 times 2 is 16.

00:00:24.260 --> 00:00:26.670
Well, I'll do it in our head
this time because you always

00:00:26.670 --> 00:00:28.510
don't have all this
space to work with.

00:00:28.510 --> 00:00:33.690
So 8 times 2 is 16.

00:00:33.690 --> 00:00:34.810
Put the 1 up there.

00:00:34.810 --> 00:00:37.500
8 times 3 is 24.

00:00:37.500 --> 00:00:39.750
24 plus 1 is 25.

00:00:42.540 --> 00:00:46.300
So 8 times 32 was 256.

00:00:46.300 --> 00:00:48.430
Now we're going to have to
multiply this 1, which is

00:00:48.430 --> 00:00:51.230
really a 10, times 32.

00:00:51.230 --> 00:00:53.220
I'll underline it
with the orange.

00:00:53.220 --> 00:00:56.430
1 times 2-- oh, we have
to be very careful here.

00:00:56.430 --> 00:00:58.770
1 times 2 is 2.

00:00:58.770 --> 00:01:01.080
So you might say hey, let
me stick a 2 down there.

00:01:01.080 --> 00:01:02.640
Remember, this isn't a 1.

00:01:02.640 --> 00:01:05.470
This is a 10, so we have
to stick a 0 there

00:01:05.470 --> 00:01:06.620
to remember that.

00:01:06.620 --> 00:01:10.860
So 10 times 2 is 20.

00:01:10.860 --> 00:01:13.520
Or you say 1 times 2 is 2, but
you're putting it in the 2's

00:01:13.520 --> 00:01:15.020
place, so you still get 20.

00:01:15.020 --> 00:01:16.320
So 10 times 2 is 20.

00:01:16.320 --> 00:01:17.660
It works out.

00:01:17.660 --> 00:01:19.510
Then 1 times 3.

00:01:19.510 --> 00:01:20.750
And we have to be very careful.

00:01:20.750 --> 00:01:22.780
Let's get rid of what
we had from before.

00:01:22.780 --> 00:01:25.600
1 times 3 is 3.

00:01:25.600 --> 00:01:29.390
There's nothing to add
here, so you just get a 3.

00:01:29.390 --> 00:01:34.530
And so you get 10
times 32 is 320.

00:01:34.530 --> 00:01:36.200
This 1 right here, that's a 10.

00:01:36.200 --> 00:01:38.830
10 plus 8 is 18.

00:01:38.830 --> 00:01:41.410
So now we just add
up the two numbers.

00:01:41.410 --> 00:01:42.510
You add them up.

00:01:42.510 --> 00:01:44.410
6 plus 0 is 6.

00:01:44.410 --> 00:01:46.180
5 plus 2 is 7.

00:01:46.180 --> 00:01:49.240
2 plus 3 is 5.

00:01:49.240 --> 00:01:50.850
Let's keep going.

00:01:50.850 --> 00:01:56.830
Let's do 99 times 88.

00:01:56.830 --> 00:01:58.450
So a big number.

00:01:58.450 --> 00:02:03.060
8 times 9 is 72.

00:02:03.060 --> 00:02:06.690
Stick the 7 up there.

00:02:06.690 --> 00:02:08.440
And then you have
8 times 9 again.

00:02:08.440 --> 00:02:11.350
8 times 9 is 72, but now
you have the 7 up here.

00:02:11.350 --> 00:02:16.280
So 72 plus 7 is 79.

00:02:16.280 --> 00:02:17.080
Fair enough.

00:02:17.080 --> 00:02:18.250
Now we're done with this.

00:02:18.250 --> 00:02:20.315
Let's just delete it just so
that we don't get confused

00:02:20.315 --> 00:02:22.160
in our next step.

00:02:22.160 --> 00:02:25.240
In our next we're going to
multiply this 8 now times 99.

00:02:25.240 --> 00:02:26.800
But this 8 is an 80.

00:02:26.800 --> 00:02:28.420
So let's stick a 0 down there.

00:02:28.420 --> 00:02:31.010
8 times 9 is 72.

00:02:31.010 --> 00:02:32.660
Stick a 7 up there.

00:02:32.660 --> 00:02:34.590
Then 8 times 9 is 72.

00:02:34.590 --> 00:02:35.630
Plus 7 is 79.

00:02:38.180 --> 00:02:40.330
2 plus 0 is 2.

00:02:40.330 --> 00:02:42.470
Let me switch colors.

00:02:42.470 --> 00:02:44.560
9 plus 2 is 11.

00:02:44.560 --> 00:02:45.820
Carry the 1.

00:02:45.820 --> 00:02:47.890
1 plus 7 is 8.

00:02:47.890 --> 00:02:50.150
8 plus 9 is 17.

00:02:50.150 --> 00:02:51.100
Carry the 1.

00:02:51.100 --> 00:02:53.530
1 plus 7 is 8.

00:02:53.530 --> 00:02:56.390
8,712.

00:02:56.390 --> 00:02:58.960
Let's keep going.

00:02:58.960 --> 00:03:01.120
Can't do enough of these.

00:03:01.120 --> 00:03:06.430
All right, 53 times 78.

00:03:09.380 --> 00:03:11.160
I think you're getting
the hang of it now.

00:03:11.160 --> 00:03:13.410
Let's multiply 8
times 53 first.

00:03:13.410 --> 00:03:17.330
So 8 times 3 is 24.

00:03:17.330 --> 00:03:18.890
Stick the 2 up there.

00:03:18.890 --> 00:03:21.980
8 times 5 is 40.

00:03:21.980 --> 00:03:23.610
40 plus 2 is 42.

00:03:26.920 --> 00:03:30.370
Now we're going to have to
deal with that 7 right

00:03:30.370 --> 00:03:31.980
there, which is really a 70.

00:03:31.980 --> 00:03:34.310
So we got to remember
to put the 0 there.

00:03:34.310 --> 00:03:37.200
7 times 3, and let's
get rid of this.

00:03:37.200 --> 00:03:38.450
Don't want to get confused.

00:03:38.450 --> 00:03:41.050
7 times 3 is 21.

00:03:41.050 --> 00:03:43.470
Put the 1 there and
put the 2 up here.

00:03:43.470 --> 00:03:46.840
7 times 5 is 35.

00:03:46.840 --> 00:03:50.280
Plus 2 is 37.

00:03:50.280 --> 00:03:52.450
Now we're ready to add.

00:03:52.450 --> 00:03:54.960
4 plus 0 is 4.

00:03:54.960 --> 00:03:56.940
2 plus 1 is 3.

00:03:56.940 --> 00:04:00.220
4 plus 7 is 11.

00:04:00.220 --> 00:04:00.960
Carry the 1.

00:04:00.960 --> 00:04:03.150
1 plus 3 is 4.

00:04:03.150 --> 00:04:06.980
4,134.

00:04:06.980 --> 00:04:08.510
Let's up the stakes
a little bit.

00:04:12.130 --> 00:04:27.240
So let's say I had
796 times 58.

00:04:27.240 --> 00:04:28.850
Let's mix it up well.

00:04:28.850 --> 00:04:33.090
All right, so first we're just
going to multiply 8 times 796.

00:04:33.090 --> 00:04:35.410
And notice, I've thrown in
an extra digit up here.

00:04:35.410 --> 00:04:39.876
So 8 times 6 is 48.

00:04:39.876 --> 00:04:41.950
Put the 4 up there.

00:04:41.950 --> 00:04:45.460
8 times 9 is 72.

00:04:45.460 --> 00:04:47.240
Plus 4 is 76.

00:04:50.760 --> 00:04:54.620
And then 8 times 7 is 56.

00:04:54.620 --> 00:04:58.310
56 plus 7 is 63.

00:05:01.450 --> 00:05:03.290
I'm sure I'll make a
careless mistake at some

00:05:03.290 --> 00:05:04.080
point in this video.

00:05:04.080 --> 00:05:07.570
And the goal for you is to
identify if and when I do.

00:05:07.570 --> 00:05:09.410
All right, now we're ready,
so we can get rid of

00:05:09.410 --> 00:05:10.480
these guys up here.

00:05:10.480 --> 00:05:14.440
Now we can multiply this 5,
which is in the 10's place.

00:05:14.440 --> 00:05:15.730
It's really a 50.

00:05:15.730 --> 00:05:16.780
Times this up here.

00:05:16.780 --> 00:05:20.170
Because it's a 50 we
stick a 0 down there.

00:05:20.170 --> 00:05:24.230
5 times 6 is 30.

00:05:24.230 --> 00:05:26.990
Put the 0 there, put
the 3 up there.

00:05:26.990 --> 00:05:29.860
5 times 9 is 45.

00:05:29.860 --> 00:05:31.430
Plus the 3 is 48.

00:05:34.310 --> 00:05:36.950
5 times 7 is 35.

00:05:36.950 --> 00:05:38.020
Plus 4 is 39.

00:05:40.800 --> 00:05:43.420
Now we're ready to add.

00:05:43.420 --> 00:05:45.630
8 plus 0 is 8.

00:05:45.630 --> 00:05:49.110
6 plus 0 is 6.

00:05:49.110 --> 00:05:52.650
3 plus 8 is 11.

00:05:52.650 --> 00:05:54.070
1 plus 6 is 7.

00:05:54.070 --> 00:05:57.120
7 plus 9 is 16.

00:05:57.120 --> 00:05:59.480
And then 1 plus 3 is 4.

00:05:59.480 --> 00:06:05.190
So 796 times 58 is 46,168.

00:06:05.190 --> 00:06:10.220
And that sounds about right
because 796-- it's almost 800.

00:06:10.220 --> 00:06:12.190
You know, which
is almost 1,000.

00:06:12.190 --> 00:06:16.480
So if we multiplied 1,000
times 58 we'd get 58,000.

00:06:16.480 --> 00:06:19.160
But we're multiplying something
a little bit smaller than 1,000

00:06:19.160 --> 00:06:20.740
times 58, so we're getting
something a little bit

00:06:20.740 --> 00:06:22.160
smaller than 58,000.

00:06:22.160 --> 00:06:26.800
So the number is in
the correct ballpark.

00:06:26.800 --> 00:06:30.810
Now let's do one more here
where I'm really going

00:06:30.810 --> 00:06:33.310
to step up the stakes.

00:06:33.310 --> 00:06:41.520
Let's do 523 times-- I'm going
to do a three-digit number now.

00:06:41.520 --> 00:06:45.180
Times 798.

00:06:45.180 --> 00:06:47.940
That's a big
three-digit number.

00:06:47.940 --> 00:06:50.590
But it's the same
exact process.

00:06:50.590 --> 00:06:53.190
And once you kind of see the
pattern you say, hey, this'll

00:06:53.190 --> 00:06:55.920
apply to any number of digits
times any number of digits.

00:06:55.920 --> 00:06:59.040
It'll just start taking you a
long time and your chances of

00:06:59.040 --> 00:07:00.820
making a careless mistake are
going to go up, but

00:07:00.820 --> 00:07:02.570
it's the same idea.

00:07:02.570 --> 00:07:06.070
So we start with 8 times 523.

00:07:06.070 --> 00:07:09.740
8 times 3 is 24.

00:07:09.740 --> 00:07:13.060
Stick the 2 up there.

00:07:13.060 --> 00:07:16.340
Now 8 times 2 is 16.

00:07:16.340 --> 00:07:18.846
16 plus 2 is 18.

00:07:18.846 --> 00:07:20.580
Put the 1 up there.

00:07:20.580 --> 00:07:22.390
8 times 5 is 40.

00:07:22.390 --> 00:07:24.550
Plus 1 is 41.

00:07:24.550 --> 00:07:29.240
So 8 times 523 is 4,184.

00:07:29.240 --> 00:07:30.050
We're not done.

00:07:30.050 --> 00:07:34.260
We have to multiply times
the 90 and by the 700.

00:07:34.260 --> 00:07:37.190
So let's do the 90 right there.

00:07:37.190 --> 00:07:39.240
So it's a 90, so we'll
stick a 0 there.

00:07:39.240 --> 00:07:40.850
It's not a 9.

00:07:40.850 --> 00:07:43.080
And let's get rid of
these guys right there.

00:07:43.080 --> 00:07:45.000
9 times 3 is 27.

00:07:49.870 --> 00:07:54.000
9 times 2 is 18.

00:07:54.000 --> 00:07:56.285
18 plus 2 is 20.

00:07:58.960 --> 00:08:03.910
And then we have
9 times 5 is 45.

00:08:03.910 --> 00:08:09.625
45 plus 2 is 47.

00:08:09.625 --> 00:08:12.460
I don't want to
write that thick.

00:08:12.460 --> 00:08:13.580
47.

00:08:13.580 --> 00:08:15.500
Let me make sure I did that
one right, and let's just

00:08:15.500 --> 00:08:16.460
review it a little bit.

00:08:16.460 --> 00:08:19.540
9 times 3 was 27.

00:08:19.540 --> 00:08:22.180
We wrote the 7 down here
and put the 2 up there.

00:08:22.180 --> 00:08:24.490
9 times 2 is 18.

00:08:24.490 --> 00:08:27.110
We added 2 to that,
so we wrote 20.

00:08:27.110 --> 00:08:29.320
Wrote the 0 down there
and the 2 up there.

00:08:29.320 --> 00:08:31.620
The 9 times 5 was 45.

00:08:31.620 --> 00:08:33.630
Plus 2 is 47.

00:08:33.630 --> 00:08:35.400
You really have to make sure
you don't make careless

00:08:35.400 --> 00:08:37.360
mistakes with these.

00:08:37.360 --> 00:08:39.380
Then finally, we have to
multiply the 7, which is

00:08:39.380 --> 00:08:43.780
really a 700 times 523.

00:08:43.780 --> 00:08:46.090
When it was just an 8 we just
started multiplying here.

00:08:46.090 --> 00:08:48.190
When it was a 90, when we
were dealing with the 10's

00:08:48.190 --> 00:08:49.720
place, we put a 0 there.

00:08:49.720 --> 00:08:51.100
Now that we're dealing with
something that's in the

00:08:51.100 --> 00:08:53.630
100's, we're going to
put two 0's there.

00:08:56.770 --> 00:08:59.000
And so you have 7-- and let's
get rid of this stuff.

00:08:59.000 --> 00:09:00.610
That'll just mess us up.

00:09:00.610 --> 00:09:04.470
7 times 3 is 21.

00:09:04.470 --> 00:09:06.430
Put the 1 there.

00:09:06.430 --> 00:09:08.470
Stick the 2 up there.

00:09:08.470 --> 00:09:11.070
7 times 2 is 14.

00:09:11.070 --> 00:09:16.130
14 plus our 2 is 16.

00:09:16.130 --> 00:09:17.640
Put the 1 up there.

00:09:17.640 --> 00:09:20.410
7 times 5 is 35.

00:09:20.410 --> 00:09:21.690
Plus 1 is 36.

00:09:24.530 --> 00:09:26.820
And now we're ready to add.

00:09:26.820 --> 00:09:29.440
And hopefully we didn't make
any careless mistakes.

00:09:29.440 --> 00:09:30.940
So 4 plus 0 plus 0.

00:09:30.940 --> 00:09:31.550
That's easy.

00:09:31.550 --> 00:09:32.480
That's 4.

00:09:32.480 --> 00:09:34.100
8 plus 7 plus 0.

00:09:34.100 --> 00:09:36.030
That's 15.

00:09:36.030 --> 00:09:36.760
Carry the 1.

00:09:36.760 --> 00:09:39.530
1 plus 1 plus 1 is 3.

00:09:39.530 --> 00:09:41.560
4 plus 7 plus 6.

00:09:41.560 --> 00:09:42.080
That's what's?

00:09:42.080 --> 00:09:43.030
4 plus 6 is 10.

00:09:43.030 --> 00:09:43.950
It's 17.

00:09:46.580 --> 00:09:48.920
And then we have 1 plus 4 is 5.

00:09:48.920 --> 00:09:51.210
5 plus 6 is 11.

00:09:51.210 --> 00:09:52.230
Carry the 1.

00:09:52.230 --> 00:09:54.350
1 plus 3 is 4.

00:09:54.350 --> 00:10:03.570
So 523 times 798 is 417,354.

00:10:03.570 --> 00:10:05.730
Now we can even
check to make sure.

00:10:05.730 --> 00:10:07.240
And so this is the
moment of truth.

00:10:07.240 --> 00:10:08.620
Let's see if we
have-- let's see.

00:10:08.620 --> 00:10:15.560
523 times 798.

00:10:15.560 --> 00:10:16.430
There you go.

00:10:16.430 --> 00:10:17.150
Moment of truth.

00:10:17.150 --> 00:10:19.220
I don't have to
re-record this video.

00:10:19.220 --> 00:10:22.010
It's 417,354.

00:10:22.010 --> 00:10:24.510
But we did it without the
calculator, which is

00:10:24.510 --> 00:10:26.390
the important point.


2-digit times a 2-digit number

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.250
Let's start with a warm-up
problem to avoid getting

00:00:03.250 --> 00:00:06.050
any mental cramps as
we learn new things.

00:00:06.050 --> 00:00:08.490
So this is a problem that
hopefully, if you understood

00:00:08.490 --> 00:00:11.400
what we did in the last video,
you can kind of understand what

00:00:11.400 --> 00:00:12.800
we're about to do right now.

00:00:12.800 --> 00:00:14.320
And I'm going to
escalate it even more.

00:00:14.320 --> 00:00:17.590
In the last video I think
we finished what a

00:00:17.590 --> 00:00:19.420
four-digit number times
a one-digit number.

00:00:19.420 --> 00:00:22.710
Let's up the stakes to
a five-digit number.

00:00:28.260 --> 00:00:36.940
Let's do 64,329 times-- let
me think of a nice number.

00:00:36.940 --> 00:00:38.140
Times 4.

00:00:40.780 --> 00:00:42.770
I'm going to show you right now
that we're going to do the

00:00:42.770 --> 00:00:44.740
exact same process that we
did in the last video.

00:00:44.740 --> 00:00:48.260
We just have to do it a little
bit longer than we did before.

00:00:48.260 --> 00:00:51.100
So we just start off saying,
OK, what's 4 times 9?

00:00:51.100 --> 00:00:56.260
4 times 9 is equal to 36.

00:00:56.260 --> 00:00:56.560
Right?

00:00:56.560 --> 00:00:57.410
18 times 2.

00:00:57.410 --> 00:00:58.420
Yep, 36.

00:00:58.420 --> 00:01:02.280
So we write the 6 down here,
carry the 3 up there.

00:01:02.280 --> 00:01:05.850
Just put the 3 up there,
then you got 4 times 2.

00:01:09.280 --> 00:01:10.930
And they're going to
have to add the 3.

00:01:10.930 --> 00:01:12.110
So let me just
write that there.

00:01:12.110 --> 00:01:15.590
Plus 3 is equal to-- you do
the multiplication first.

00:01:15.590 --> 00:01:18.990
You can even think of it as
order of operations, but you

00:01:18.990 --> 00:01:22.050
just should know that you do
the multiplication first.

00:01:22.050 --> 00:01:23.310
So 4 times 2 is 8.

00:01:23.310 --> 00:01:27.030
Plus 3 is equal to 11.

00:01:27.030 --> 00:01:32.370
Put this 1 down here and put
the one 10 and 11 up there.

00:01:32.370 --> 00:01:34.080
Then you got 4 times 3.

00:01:38.660 --> 00:01:40.440
You got that one up there, so
you're going to have to add

00:01:40.440 --> 00:01:43.740
that plus 1 is equal to--
that's going to equal 12.

00:01:43.740 --> 00:01:46.750
Plus 1, which is equal to 13.

00:01:46.750 --> 00:01:50.120
So it's 13.

00:01:50.120 --> 00:01:52.190
Then you have 4 times 4.

00:01:56.460 --> 00:01:58.270
You have this little one
hanging out here from the

00:01:58.270 --> 00:01:59.650
previous multiplications,
so you're going to

00:01:59.650 --> 00:02:00.850
to have to add that.

00:02:00.850 --> 00:02:04.060
And that's equal to 16 plus 1.

00:02:04.060 --> 00:02:06.310
It's equal to 17.

00:02:06.310 --> 00:02:09.680
Stick the 7 down here,
put the 1 up there.

00:02:09.680 --> 00:02:13.040
We're almost done.

00:02:13.040 --> 00:02:15.650
And then we have 4 times 6.

00:02:18.290 --> 00:02:21.640
Plus 1.

00:02:21.640 --> 00:02:22.310
What is that?

00:02:22.310 --> 00:02:24.740
4 times 6 is 24.

00:02:24.740 --> 00:02:26.930
Plus 1 is 25.

00:02:26.930 --> 00:02:28.330
Put the 5 down here.

00:02:28.330 --> 00:02:29.630
There's no where to put the 2.

00:02:29.630 --> 00:02:32.200
There's no more multiplications
to do, so we just put

00:02:32.200 --> 00:02:33.680
the 2 down there.

00:02:33.680 --> 00:02:44.940
So 64,329 is 257,316.

00:02:44.940 --> 00:02:47.660
And in case you're wondering,
these commas don't mean much.

00:02:47.660 --> 00:02:49.460
They just help me
read the number.

00:02:49.460 --> 00:02:52.210
So I put it after every 3
digits, so I know that for

00:02:52.210 --> 00:02:55.435
example, that everything
after this-- so s is 7,000.

00:02:55.435 --> 00:02:57.920
If I had another comma
here, this is millions.

00:02:57.920 --> 00:03:01.050
So it just helps me read
the problem a bit.

00:03:01.050 --> 00:03:05.330
So if you got that you're now
ready to escalate to a slightly

00:03:05.330 --> 00:03:06.840
more complicated situation.

00:03:06.840 --> 00:03:09.650
Although the first way that
we're going to do it it's

00:03:09.650 --> 00:03:11.270
actually not going to look
any more complicated.

00:03:11.270 --> 00:03:13.790
It's just going to
involve one more step.

00:03:13.790 --> 00:03:16.140
So everything we've done so
far are a bunch of digits

00:03:16.140 --> 00:03:18.090
times a one-digit number.

00:03:18.090 --> 00:03:21.910
Now let's do a bunch of digits
times a two-digit number.

00:03:21.910 --> 00:03:28.450
So let's say we want to
multiply 36 times-- instead of

00:03:28.450 --> 00:03:29.920
putting a one-digit number here
I'm going to put a

00:03:29.920 --> 00:03:31.140
two-digit number.

00:03:31.140 --> 00:03:32.305
So times 23.

00:03:35.500 --> 00:03:38.380
So you start off doing this
problem exactly the way you

00:03:38.380 --> 00:03:40.250
would have done it if there
was just a 3 down here.

00:03:40.250 --> 00:03:43.190
You can kind of ignore
the 2 for a little bit.

00:03:43.190 --> 00:03:48.270
So 3 times 6 is equal to 18.

00:03:48.270 --> 00:03:52.070
So you just put the 8 here, put
the 10 there, or the 1 there

00:03:52.070 --> 00:03:53.890
because it's 10 plus 8.

00:03:53.890 --> 00:03:59.160
3 times 3 is 9.

00:03:59.160 --> 00:04:04.640
Plus 1, so 3 times 3 plus
1 is equal to-- that's 9

00:04:04.640 --> 00:04:06.102
plus 1 is equal to 10.

00:04:06.102 --> 00:04:08.080
So you put the 10 there.

00:04:08.080 --> 00:04:09.200
There's nothing left.

00:04:09.200 --> 00:04:10.060
You put the 0 there.

00:04:10.060 --> 00:04:12.700
There's nothing left to put the
1, so you put the 10 there.

00:04:12.700 --> 00:04:17.300
So you essentially have solved
the problem that 36-- let me

00:04:17.300 --> 00:04:18.930
do this is another color.

00:04:18.930 --> 00:04:23.500
That 36 times 3
is equal to 108.

00:04:23.500 --> 00:04:25.360
That's what we've solved
so far, but we have this

00:04:25.360 --> 00:04:27.360
20 sitting out here.

00:04:27.360 --> 00:04:28.100
We have this 20.

00:04:28.100 --> 00:04:30.950
We have to figure out
what 20 times 360 is.

00:04:30.950 --> 00:04:34.880
Or sorry, what 20 times 36 is.

00:04:34.880 --> 00:04:36.660
This 2 is really a 20.

00:04:36.660 --> 00:04:39.730
And to make it all work out
like that, what we do is

00:04:39.730 --> 00:04:42.200
we throw a 0 down here.

00:04:42.200 --> 00:04:45.180
We throw a 0 right there.

00:04:45.180 --> 00:04:47.140
In a second I'm going
to explain why

00:04:47.140 --> 00:04:48.680
exactly we did that.

00:04:48.680 --> 00:04:50.620
So let's just do the
same process as we did

00:04:50.620 --> 00:04:51.280
before with the 3.

00:04:51.280 --> 00:04:53.460
Now we do it with a 2, but
we start filling up here

00:04:53.460 --> 00:04:55.110
and move to the left.

00:04:55.110 --> 00:04:56.510
So 2 times 6.

00:04:59.210 --> 00:04:59.730
That's easy.

00:04:59.730 --> 00:05:01.640
That's 12.

00:05:01.640 --> 00:05:04.460
So 2 times 6 is 12.

00:05:04.460 --> 00:05:07.370
We put the 1 up here and we
have to be very careful because

00:05:07.370 --> 00:05:10.570
we had this 1 from our previous
problem, which doesn't

00:05:10.570 --> 00:05:11.460
apply anymore.

00:05:11.460 --> 00:05:15.210
So we could erase it or that
1 we could get rid of.

00:05:15.210 --> 00:05:17.170
If you have an eraser get rid
of it, or you can just keep

00:05:17.170 --> 00:05:18.810
track in your head that the 1
you're about to write

00:05:18.810 --> 00:05:20.130
is a different 1.

00:05:20.130 --> 00:05:21.440
So what were we doing?

00:05:21.440 --> 00:05:23.970
We wrote 2 times 6 is 12.

00:05:23.970 --> 00:05:25.070
Put the 2 here.

00:05:25.070 --> 00:05:26.460
Put the 1 up here.

00:05:26.460 --> 00:05:28.210
And I got rid of the previous
1 because that would've

00:05:28.210 --> 00:05:29.630
just messed me up.

00:05:29.630 --> 00:05:32.160
Now I have 2 times 3.

00:05:32.160 --> 00:05:37.620
2 times 3 is equal to 6.

00:05:37.620 --> 00:05:41.030
But then I have this plus 1 up
here, so I have to add plus 1.

00:05:41.030 --> 00:05:42.550
So I get 7.

00:05:42.550 --> 00:05:44.560
So that is equal to 7.

00:05:44.560 --> 00:05:47.440
2 times 3 plus 1 is equal to 7.

00:05:47.440 --> 00:05:51.250
So this 720 we just solved,
that's literally-- let

00:05:51.250 --> 00:05:53.100
me write that down.

00:05:53.100 --> 00:05:53.710
What is that?

00:05:53.710 --> 00:05:57.910
That is 36 times 20.

00:05:57.910 --> 00:06:02.100
36 times 20 is equal to 720.

00:06:02.100 --> 00:06:03.740
And hopefully that should
explain why we had to

00:06:03.740 --> 00:06:05.470
throw this 0 here.

00:06:05.470 --> 00:06:08.390
If we didn't throw that 0 here
we would have just a 2--

00:06:08.390 --> 00:06:11.810
we would just have a 72
here, instead of 720.

00:06:11.810 --> 00:06:14.640
And 72 is 36 times 2.

00:06:14.640 --> 00:06:15.990
But this isn't a 2.

00:06:15.990 --> 00:06:17.920
This is a 2 in the 10's place.

00:06:17.920 --> 00:06:19.110
This is a 20.

00:06:19.110 --> 00:06:21.270
So we have to multiply
36 times 20, and that's

00:06:21.270 --> 00:06:23.150
why we got 720 there.

00:06:23.150 --> 00:06:26.810
So 36 times 23.

00:06:26.810 --> 00:06:28.790
Let's write it this way.

00:06:28.790 --> 00:06:31.560
Let me get some space up here.

00:06:31.560 --> 00:06:34.770
So we could write 30-- well,
actually, let me just finish

00:06:34.770 --> 00:06:38.400
the problem and then I'll
explain to you why it worked.

00:06:38.400 --> 00:06:42.000
So now to finish it up
we just add 108 to 720.

00:06:42.000 --> 00:06:44.300
So 8 plus 0 is 8.

00:06:44.300 --> 00:06:45.970
0 plus 2 is 2.

00:06:45.970 --> 00:06:48.650
1 plus 7 is 8.

00:06:48.650 --> 00:06:52.350
So 36 times 23 is 828.

00:06:52.350 --> 00:06:54.740
Now you're saying Sal,
why did that work?

00:06:54.740 --> 00:06:58.720
why were we able to figure out
separately 36 times 3 is equal

00:06:58.720 --> 00:07:04.190
to 108, and then 36 times 20 is
equal to 720, and then

00:07:04.190 --> 00:07:06.110
add them up like that?

00:07:06.110 --> 00:07:09.340
Because we could have rewritten
the problem like this.

00:07:09.340 --> 00:07:14.670
We could have rewritten
the problem as 36-- the

00:07:14.670 --> 00:07:16.320
original problem was this.

00:07:16.320 --> 00:07:22.770
We could have rewritten this
as 36 times 20 plus 3.

00:07:22.770 --> 00:07:25.560
And this, and I don't know if
you've learned the distributive

00:07:25.560 --> 00:07:28.150
property yet, but this is just
the distributive property.

00:07:28.150 --> 00:07:38.630
This is just the same thing as
36 times 20 plus 36 times 3.

00:07:38.630 --> 00:07:40.720
If that confuses you, then you
don't have to worry about it.

00:07:40.720 --> 00:07:42.740
But if it doesn't,
then this is good.

00:07:42.740 --> 00:07:44.330
It's actually teaching
you something.

00:07:44.330 --> 00:07:47.450
36 times 20 we saw was 720.

00:07:47.450 --> 00:07:50.520
We learned that 36
times 3 was 108.

00:07:50.520 --> 00:07:52.460
And when you added them
together we got what?

00:07:52.460 --> 00:07:55.150
828?

00:07:55.150 --> 00:07:55.720
Is that what we got?

00:07:55.720 --> 00:07:57.320
We got 828.

00:07:57.320 --> 00:07:59.060
And you could expand it
even more like we did

00:07:59.060 --> 00:08:00.020
in the previous video.

00:08:00.020 --> 00:08:07.010
You could write this out as
30 plus 6 times 20 plus 3.

00:08:07.010 --> 00:08:09.350
Actually, let me just do it
that way because I think

00:08:09.350 --> 00:08:11.100
that could help you
out a little bit.

00:08:11.100 --> 00:08:12.790
If it confuses you, ignore it.

00:08:12.790 --> 00:08:14.720
If it doesn't, that's good.

00:08:14.720 --> 00:08:17.160
So we could do it 3 times 6.

00:08:17.160 --> 00:08:19.570
3 times 6 is 18.

00:08:19.570 --> 00:08:21.450
18 is just 10 plus 8.

00:08:21.450 --> 00:08:24.670
So it's 8, then we
put a 10 up here.

00:08:24.670 --> 00:08:26.490
And ignore all this up here.

00:08:26.490 --> 00:08:28.450
3 times 30.

00:08:28.450 --> 00:08:31.580
3 times 30 is 90.

00:08:31.580 --> 00:08:34.890
90 plus 10 is 100.

00:08:34.890 --> 00:08:40.910
So 100 is zero 10's plus 100.

00:08:40.910 --> 00:08:42.510
I don't know if this
confuses you or not.

00:08:42.510 --> 00:08:43.750
If it does, ignore it.

00:08:43.750 --> 00:08:48.160
If it doesn't, well I don't
want to complicate the issue.

00:08:48.160 --> 00:08:49.430
And now we can multiply 20.

00:08:49.430 --> 00:08:52.110
We can ignore this thing
that we had before.

00:08:52.110 --> 00:08:55.680
20 times 6 is 120.

00:08:55.680 --> 00:09:01.950
So that's 20 plus 100.

00:09:01.950 --> 00:09:05.040
So I'll put that 100 up here.

00:09:05.040 --> 00:09:10.490
20 times 30 is 2 times 3 and
you have two 0's there.

00:09:10.490 --> 00:09:13.390
And I think I'm maybe jumping
the gun a little bit, assuming

00:09:13.390 --> 00:09:16.240
a little bit too much of what
you may or may not know.

00:09:16.240 --> 00:09:19.010
But 20 times 30 is
going to be 600.

00:09:19.010 --> 00:09:23.350
and you add another hundred
there, that's 700.

00:09:23.350 --> 00:09:24.490
And then you add them all up.

00:09:24.490 --> 00:09:25.960
You get 800.

00:09:25.960 --> 00:09:27.480
100 plus 700.

00:09:27.480 --> 00:09:32.520
Plus 20 plus 8, which
is equal to 828.

00:09:32.520 --> 00:09:35.860
My point here is to show you
why that system we did worked.

00:09:35.860 --> 00:09:39.120
Why we added a 0
here to begin with.

00:09:39.120 --> 00:09:41.330
But if it confuses you, don't
worry about that right now.

00:09:41.330 --> 00:09:43.560
Learn how to do it and then
maybe re-watch this video.

00:09:43.560 --> 00:09:46.570
Let's just do a bunch of more
examples because I think the

00:09:46.570 --> 00:09:49.460
examples are what really,
hopefully, explain

00:09:49.460 --> 00:09:50.240
the situation.

00:09:50.240 --> 00:09:52.250
So let's do 77.

00:09:52.250 --> 00:09:52.910
Let's do a fun one.

00:09:52.910 --> 00:09:56.940
77 times 77.

00:09:56.940 --> 00:09:59.596
7 times 7 is 49.

00:09:59.596 --> 00:10:00.990
Put the 1 up here.

00:10:04.100 --> 00:10:06.300
7 times 7, well, that's 49.

00:10:06.300 --> 00:10:10.260
Plus 4 is 53.

00:10:10.260 --> 00:10:12.190
There's no where to put the
5, so we put it down here.

00:10:12.190 --> 00:10:13.980
7 times 7 is 49.

00:10:13.980 --> 00:10:15.600
Plus 4 is 53.

00:10:15.600 --> 00:10:17.680
Stick a 0 here.

00:10:17.680 --> 00:10:19.350
Now we're going to do this 7.

00:10:19.350 --> 00:10:21.910
So stick a 0 here.

00:10:21.910 --> 00:10:23.270
Let's get rid of this right
there because that'll

00:10:23.270 --> 00:10:24.340
just mess us up.

00:10:24.340 --> 00:10:26.460
7 times 7 is 49.

00:10:26.460 --> 00:10:28.130
Stick a 9 there.

00:10:28.130 --> 00:10:29.670
Put a 4 there.

00:10:29.670 --> 00:10:32.140
7 times 7 is 49.

00:10:32.140 --> 00:10:33.620
Plus 4, which is 53.

00:10:36.250 --> 00:10:40.880
So notice, when we multiplied
7 times 77 we got 539.

00:10:40.880 --> 00:10:45.500
When we multiplied 70
times 77 we got 5,390.

00:10:45.500 --> 00:10:46.870
And it makes sense.

00:10:46.870 --> 00:10:48.750
They just differ by a 0.

00:10:48.750 --> 00:10:50.630
By a factor of 10.

00:10:50.630 --> 00:10:54.190
And now we can just add them
up, and what do we get?

00:10:54.190 --> 00:10:58.060
9 plus 0 is 9.

00:10:58.060 --> 00:11:00.950
3 plus 9 is 12.

00:11:00.950 --> 00:11:02.360
Carry the 1.

00:11:02.360 --> 00:11:04.050
1 plus 5 is 6.

00:11:04.050 --> 00:11:06.150
6 plus 3 is 9.

00:11:09.000 --> 00:11:09.750
Then we have this 5.

00:11:09.750 --> 00:11:10.660
So it's 5,929.


Inverse trig functions: arccos

WEBVTT
Kind: captions
Language: en

00:00:00.030 --> 00:00:08.370
I've already made videos on the arc
sine and the arc tangent, so to kind

00:00:08.370 --> 00:00:13.590
of complete the trifecta I might as well
make a video on the arc cosine and just

00:00:13.590 --> 00:00:17.880
like the other inverse trigonometric
functions the arc cosine it's kind of

00:00:17.880 --> 00:00:23.130
the same thought process if I were to
tell you that the arc now I'm doing

00:00:23.130 --> 00:00:30.480
cosine if I were to tell you that the
arc cosine of X is equal to theta this

00:00:30.480 --> 00:00:36.390
is an equivalent statement to saying
that the inverse cosine of X is equal

00:00:36.390 --> 00:00:41.910
to theta these are just two different
ways of writing the exact same thing and

00:00:41.910 --> 00:00:46.800
as soon as I see either an arc anything
or an inverse trig function in general

00:00:46.800 --> 00:00:52.620
my brain immediately rearranges this my
brain regionally immediately says this

00:00:52.620 --> 00:00:59.010
is saying that if I take the cosine of
some angle theta that I'm going to get X

00:00:59.010 --> 00:01:04.080
or if I get in order the same statement
up here either of these should boil down

00:01:04.080 --> 00:01:08.760
to this if I say that the coast you know
what is the inverse cosine of X my brain

00:01:08.760 --> 00:01:14.220
says what angle can I take the cosine of
to get X so with that said let's try it

00:01:14.220 --> 00:01:22.650
out on an example let's say that I have
the arc I'm told no I put two CS there

00:01:22.650 --> 00:01:30.300
I'm told to evaluate the arc cosine of
minus 1/2 my brain as you know let's say

00:01:30.300 --> 00:01:33.720
that this is going to be equal to its
going to be equal to some angle and this

00:01:33.720 --> 00:01:40.980
is equivalent to saying that the cosine
of my mystery angle is equal to minus

00:01:40.980 --> 00:01:45.150
1/2 and as soon as you put it in this
way at least for my brain it becomes a

00:01:45.150 --> 00:01:50.340
lot easier to process so let's draw our
unit circle and see if we can make some

00:01:50.340 --> 00:01:55.710
headway here so that's my let me see I
could draw a little straighter actually

00:01:55.710 --> 00:01:59.940
maybe I could actually draw put rulers
here and if I put a ruler here maybe

00:01:59.940 --> 00:02:04.800
I can draw a straight line let me see
no that's too hard okay so that is my

00:02:04.800 --> 00:02:15.260
y-axis that is my x-axis not the neatest
most neatly drawn axes but it'll do and

00:02:15.260 --> 00:02:20.660
let me draw my unit circle looks more
like a unit ellipse but you get the

00:02:20.660 --> 00:02:25.730
idea and the cosine of an angle is a
defined on the unit circle definition

00:02:25.730 --> 00:02:31.580
is the x-value on the unit circle so if
we have some some angle the x-value is

00:02:31.580 --> 00:02:38.000
going to be equal to minus 1/2 so we
go to minus 1/2 right here and so the

00:02:38.000 --> 00:02:42.920
angle that we have to solve for R theta
is the angle that when we intersect the

00:02:42.920 --> 00:02:49.370
unit circle the x value is minus 1/2 so
let me see this is the angle that we're

00:02:49.370 --> 00:02:55.460
trying to figure out this is theta that
we needed to determine so how can we do

00:02:55.460 --> 00:03:00.830
that so if this is minus 1/2 right here
let's figure out these different angles

00:03:00.830 --> 00:03:05.030
and the way I like to think about it is
I like to figure out this angle right

00:03:05.030 --> 00:03:08.450
here and if I know that angle I can
just subtract that from 180 degrees

00:03:08.450 --> 00:03:15.170
to get this this light blue angle that's
kind of the solution to our problem so

00:03:15.170 --> 00:03:20.840
let me make this triangle a little bit
bigger so that triangle now let me do it

00:03:20.840 --> 00:03:26.300
like this that triangle looks something
like this where this distance right here

00:03:26.300 --> 00:03:36.230
is 1/2 that distance right there is 1/2
this distance right here is 1 hopefully

00:03:36.230 --> 00:03:39.860
you recognize that this is going to be
a 30-60-90 triangle you could actually

00:03:39.860 --> 00:03:44.240
solve for this other side you'll get to
square root of 3 over 2 and to solve for

00:03:44.240 --> 00:03:48.170
that other side you just need to do the
Pythagorean theorem actually let me just

00:03:48.170 --> 00:03:51.950
do that let me just call this I don't
know let me just call this a so you'd

00:03:51.950 --> 00:03:58.850
get a squared plus 1/2 squared which is
1/4 is equal to 1 squared which is 1 you

00:03:58.850 --> 00:04:07.070
get a squared is equal to 3/4 or a is
equal to the square root of 3 over 2 so

00:04:07.070 --> 00:04:11.360
you immediately notice 30-60-90 triangle
and you know that because the sides of

00:04:11.360 --> 00:04:16.760
a 30-60-90 triangle if the hypotenuse is
1 or 1/2 and square root of 3 over 2 and

00:04:16.760 --> 00:04:21.890
you'll also know that the side opposite
the square root of 3 over 2 side is the

00:04:21.890 --> 00:04:27.950
60 degrees that's 60 this is 90 this is
the right angle and this is 30 right up

00:04:27.950 --> 00:04:32.480
there but this is the one we care about
this angle right here we just figured

00:04:32.480 --> 00:04:38.090
out is 60 degrees so what's this what's
the bigger angle that we care about what

00:04:38.090 --> 00:04:45.920
is 60 degrees supplementary to it's
supplementary to 180 degrees so the

00:04:45.920 --> 00:04:53.210
arc cosine or the inverse cosine let me
write that down the arc cosine of minus

00:04:53.210 --> 00:05:05.570
1/2 is equal to 100 and 120 degrees I'll
write 180 there no it's 180 minus the 60

00:05:05.570 --> 00:05:11.090
this whole thing is 180 so this is right
here is 120 degrees right 120 plus 60

00:05:11.090 --> 00:05:19.070
is 180 or if we wanted to write that in
radians you just write 120 degrees times

00:05:19.070 --> 00:05:30.890
pi Radian per 180 degrees degrees cancel
out 12 over 18 is 2/3 so it equals 2 PI

00:05:30.890 --> 00:05:43.310
over 3 radians so this right here is
equal to 2 pi PI over 3 radians now

00:05:43.310 --> 00:05:50.000
just like we saw in the arc sine and
the arc tangent videos you probably say

00:05:50.000 --> 00:05:56.240
hey okay if I have 2 PI over 3 radians
that gives me a cosine of minus 1/2 and

00:05:56.240 --> 00:06:02.690
I could write that cosine of 2 pi over
3 is equal to minus 1/2 this gives you

00:06:02.690 --> 00:06:06.230
the same information as the statement
up here but I could just keep going

00:06:06.230 --> 00:06:09.800
around the unit circle for example I
could I will at this point over here

00:06:09.800 --> 00:06:13.940
cosine of this angle if I were to add
if I were to go this far would also

00:06:13.940 --> 00:06:17.900
be minus 1/2 and then I could go 2 pi
around and get back here so there's a

00:06:17.900 --> 00:06:22.670
lot of values that if I take the cosine
of those angles I'll get this minus 1/2

00:06:22.670 --> 00:06:28.100
so we have to restrict ourselves we
have to restrict the values that the

00:06:28.100 --> 00:06:33.710
arc cosine function can take on so we're
essentially restricting its range we're

00:06:33.710 --> 00:06:38.390
restricting its range what we do is
we restrict the range to this upper

00:06:38.480 --> 00:06:44.900
a hemisphere the first and second
quadrants so if we say if we make

00:06:44.900 --> 00:06:52.700
the statement that the arc cosine
of X is equal to theta we're going

00:06:52.700 --> 00:06:57.290
to restrict our range theta to that top
so theta is going to be greater than or

00:06:57.290 --> 00:07:10.130
equal to zero and less than or equal
to 102 PI less oh sorry not 2pi less

00:07:10.130 --> 00:07:16.790
than or equal to PI right or this is
also zero degrees or 180 degrees we're

00:07:16.790 --> 00:07:22.910
restricting ourselves to this part of
the hemisphere right there and so you

00:07:22.910 --> 00:07:26.420
can't do this this is the only point
where the cosine of the angle is equal

00:07:26.420 --> 00:07:30.830
to minus 1/2 we can't take this angle
because it's outside of our range and

00:07:30.830 --> 00:07:35.870
what are the valid values for X well
any angle if I take the cosine of it

00:07:35.870 --> 00:07:40.940
it can be between minus 1 and plus
1 so X the domain for the the domain

00:07:40.940 --> 00:07:45.860
for the our cosine function is going
to be X has to be less than or equal

00:07:45.860 --> 00:07:52.340
to 1 and greater than or equal to minus
1 and once again let's just check our

00:07:52.340 --> 00:07:56.750
work let's see if if the value I got
here that the arc cosine of minus 1/2

00:07:56.750 --> 00:08:04.730
really is 2 PI over 3 as calculated by
the ti-85 let me turn it on so I need

00:08:04.730 --> 00:08:08.420
to figure out the inverse cosine which
is the same thing as the arc cosine of

00:08:08.420 --> 00:08:19.730
minus 1/2 of minus 0.5 it gives me that
decimal that strange number let's see

00:08:19.730 --> 00:08:27.380
if that's the same thing is 2 PI over
3 2 times pi divided by 3 is equal to

00:08:27.380 --> 00:08:31.910
that exact same number so the calculator
gave me the same value I got but this is

00:08:31.910 --> 00:08:35.150
kind of a useless what's not a useless
number it's it's a valid that's that is

00:08:35.150 --> 00:08:39.200
the answer but it's it doesn't it's not
a nice clean answer I didn't know that

00:08:39.200 --> 00:08:44.270
this is 2 PI over 3 radians and so when
we did it using the unit circle we were

00:08:44.270 --> 00:08:48.920
able to get that answer so hopefully
and actually let me ask you let me

00:08:48.920 --> 00:08:52.340
just finish this up with an interesting
question and this applies let's do all

00:08:52.340 --> 00:08:59.630
of them if I were to ask you you know
say I were to take the arc arc cosine

00:08:59.630 --> 00:09:08.120
of X and then I were to take the cosine
of that what is this what is this going

00:09:08.120 --> 00:09:14.330
to be equal to well this statement right
here could be said well let's say that

00:09:14.330 --> 00:09:23.000
the arc cosine of X is equal to theta
that means that the cosine of theta is

00:09:23.000 --> 00:09:30.680
equal to X right so if the arc cosine
of X is equal to theta we can replace

00:09:30.680 --> 00:09:35.570
this with theta and then the cosine
of theta well the cosine of theta is

00:09:35.570 --> 00:09:39.410
X so this whole thing is going to be
X hopefully I didn't confuse you there

00:09:39.410 --> 00:09:42.860
right I'm just saying look R cosine
of X let's just call that theta now

00:09:42.860 --> 00:09:47.780
it by definition this means that the
cosine of theta is equal to X these

00:09:47.780 --> 00:09:53.420
are equivalent statements these are
completely equivalent statements right

00:09:53.420 --> 00:09:57.140
here so if we put a theta right there
we take the cosine of theta has to be

00:09:57.140 --> 00:10:03.800
equal to X now let me ask you a bonus
slightly trickier question what if I

00:10:03.800 --> 00:10:08.570
were to ask you and this is true for any
X that you put in here this is true for

00:10:08.570 --> 00:10:13.760
any X any value between negative 1 and
1 including those two endpoints this is

00:10:13.760 --> 00:10:20.930
going to be true now what if I were to
ask you what the arc arc cosine of the

00:10:20.930 --> 00:10:29.210
cosine of theta is what is this going
to be equal to my answer is it depends

00:10:29.210 --> 00:10:35.720
it depends on the theta so if theta is
in the if theta is in the range if theta

00:10:35.720 --> 00:10:47.540
is between if theta is between 0 and pi
so it's in our valid range for for kind

00:10:47.540 --> 00:10:52.880
of our range for the product of the arc
cosine then this will be equal to theta

00:10:52.880 --> 00:10:58.160
if this is true for theta but what if we
take some data out of that range let's

00:10:58.160 --> 00:11:03.920
try it out let's sake so let me do it
1 with theta in that range let's take

00:11:03.920 --> 00:11:10.710
the arc cosine of the cosine of let's
just do some one of them that we know

00:11:10.710 --> 00:11:17.970
let's take the cosine of let's take
the cosine of 2 pi over 3 cosine of 2

00:11:17.970 --> 00:11:25.290
pi over 3 radians that's the same thing
as the arc cosine of minus 1/2 cosine

00:11:25.290 --> 00:11:29.310
of 2 pi over 3 is minus 1/2 we just saw
that in the earlier part of this video

00:11:29.310 --> 00:11:34.770
and then we solve this we said oh this
is equal to 2 PI over 3 so if we're in

00:11:34.770 --> 00:11:39.120
the range if theta is between 0 and
pi it worked and that's because the

00:11:39.120 --> 00:11:43.800
arc cosine function can only produce
values between 0 and PI but what if

00:11:43.800 --> 00:11:54.300
I were to ask you what is the arc arc
cosine of the cosine of I don't know

00:11:54.300 --> 00:12:00.810
of 3 PI of 3 PI so if I were to draw
the unit circle here let me draw the

00:12:00.810 --> 00:12:07.200
unit circle real quick one and that's my
axes what's 3 pi 2 pi is if I go around

00:12:07.200 --> 00:12:12.210
once and then I go around another pi so
I end up right here so I've gone around

00:12:12.210 --> 00:12:16.920
one and a half times the unit circle so
this is at 3 pi what's the x-coordinate

00:12:16.920 --> 00:12:24.570
here it's minus one so cosine of 3 pi is
minus one all right so what's what's arc

00:12:24.570 --> 00:12:32.040
cosine of minus one arc cosine of minus
one well remember the the range or the

00:12:32.040 --> 00:12:37.800
set of values that are cosine can be can
evaluate to is in this upper hemisphere

00:12:37.800 --> 00:12:46.890
it's between this can only be between PI
and 0 so arc cosine of negative one is

00:12:46.890 --> 00:12:52.230
just going to be PI so this is going
to be PI our cosine of negative this

00:12:52.230 --> 00:12:56.730
is this is negative one our cosine
of negative one is PI and that's

00:12:56.730 --> 00:13:00.780
a reasonable statement because the
difference between 3 PI and PI is just

00:13:00.780 --> 00:13:04.020
going around the unit circle a couple
of times and so you get an equivalent

00:13:04.020 --> 00:13:08.010
it's kind of your the equivalent point
on the unit circle so I just thought I

00:13:08.010 --> 00:13:11.940
would throw those two at you this one I
mean this is a useful one if I actually

00:13:11.940 --> 00:13:16.800
let me write it up here this one is a
useful one the cosine of the arc cosine

00:13:16.800 --> 00:13:23.620
of X is always going to be X I can so do
that with sign the sign of the arc sine

00:13:23.620 --> 00:13:30.820
of X is also going to be X and these are
just useful things to you shouldn't just

00:13:30.820 --> 00:13:33.280
memorize them because obviously you
might memorize it the wrong way but

00:13:33.280 --> 00:13:36.700
you just think a little bit about
it and it you'll never forget it


Inverse trig functions: arctan

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:03.490
In the last video, I showed you
that if someone were to walk

00:00:03.490 --> 00:00:10.010
up to you and ask you what
is the arcsine-- Whoops.

00:00:10.010 --> 00:00:13.190
--arcsine of x?

00:00:13.190 --> 00:00:16.360
And so this is going to be
equal to who knows what.

00:00:16.360 --> 00:00:20.100
This is just the same thing
as saying that the sine of

00:00:20.100 --> 00:00:22.470
some angle is equal to x.

00:00:22.470 --> 00:00:25.520
And we solved it in a couple
of cases in the last example.

00:00:25.520 --> 00:00:28.230
So using the same pattern--
Let me show you this.

00:00:28.230 --> 00:00:31.810
I could have also rewritten
this as the inverse sine

00:00:31.810 --> 00:00:33.540
of x is equal to what.

00:00:33.540 --> 00:00:35.140
These are equivalent
statements.

00:00:35.140 --> 00:00:37.480
Two ways of writing the
inverse sine function.

00:00:37.480 --> 00:00:39.790
This is more-- This is the
inverse sine function.

00:00:39.790 --> 00:00:41.460
You're not taking this to
the negative 1 power.

00:00:41.460 --> 00:00:45.340
You're just saying the sine of
what-- So what question mark--

00:00:45.340 --> 00:00:46.920
What angle is equal to x?

00:00:46.920 --> 00:00:48.470
And we did this in
the last video.

00:00:48.470 --> 00:00:52.020
So by the same pattern, if I
were to walk up to you on the

00:00:52.020 --> 00:00:58.770
street and I were to say the
tangent of-- the inverse

00:00:58.770 --> 00:01:02.010
tangent of x is equal to what?

00:01:02.010 --> 00:01:05.140
You should immediately in your
head say, oh he's just asking

00:01:05.140 --> 00:01:09.080
me-- He's just saying the
tangent of some angle

00:01:09.080 --> 00:01:10.440
is equal to x.

00:01:10.440 --> 00:01:13.380
And I just need to figure
out what that angle is.

00:01:13.380 --> 00:01:14.840
So let's do an example.

00:01:14.840 --> 00:01:17.060
So let's say I were walk
up to you on the street.

00:01:17.060 --> 00:01:20.150
There's a lot of a walking
up on a lot of streets.

00:01:20.150 --> 00:01:23.790
I would write -- And I were
to say you what is the

00:01:23.790 --> 00:01:27.970
arctangent of minus 1?

00:01:27.970 --> 00:01:30.190
Or I could have equivalently
asked you, what is the

00:01:30.190 --> 00:01:32.510
inverse tangent of minus 1?

00:01:32.510 --> 00:01:35.190
These are equivalent questions.

00:01:35.190 --> 00:01:37.430
And what you should do is you
should, in your head-- If you

00:01:37.430 --> 00:01:40.200
don't have this memorized, you
should draw the unit circle.

00:01:40.200 --> 00:01:42.640
Actually let me just do a
refresher of what tangent

00:01:42.640 --> 00:01:44.330
is even asking us.

00:01:44.330 --> 00:01:48.890
The tangent of theta-- this is
just the straight-up, vanilla,

00:01:48.890 --> 00:01:52.630
non-inverse function tangent
--that's equal to the sine of

00:01:52.630 --> 00:01:56.690
theta over the cosine of theta.

00:01:56.690 --> 00:02:00.670
And the sine of theta is the
y-value on the unit function--

00:02:00.670 --> 00:02:03.010
on the unit circle.

00:02:03.010 --> 00:02:06.730
And the cosine of
theta is the x-value.

00:02:06.730 --> 00:02:08.600
And so if you draw a line--
Let me draw a little

00:02:08.600 --> 00:02:11.110
unit circle here.

00:02:11.110 --> 00:02:14.770
So if I have a unit
circle like that.

00:02:14.770 --> 00:02:17.980
And let's say I'm
at some angle.

00:02:17.980 --> 00:02:20.940
Let's say that's
my angle theta.

00:02:20.940 --> 00:02:25.640
And this is my y-- my
coordinates x, y.

00:02:25.640 --> 00:02:29.380
We know already that
the y-value, this is

00:02:29.380 --> 00:02:30.880
the sine of theta.

00:02:30.880 --> 00:02:32.780
Let me scroll over here.

00:02:32.780 --> 00:02:34.210
Sine of theta.

00:02:34.210 --> 00:02:38.730
And we already know that this
x-value is the cosine of theta.

00:02:38.730 --> 00:02:40.200
So what's the tangent
going to be?

00:02:40.200 --> 00:02:46.670
It's going to be this distance
divided by this distance.

00:02:46.670 --> 00:02:49.970
Or from your algebra I, this
might ring a bell, because

00:02:49.970 --> 00:02:52.520
we're starting at the origin
from the point 0, 0.

00:02:52.520 --> 00:02:56.250
This is our change in y
over our change in x.

00:02:56.250 --> 00:02:58.700
Or it's our rise over run.

00:02:58.700 --> 00:03:01.950
Or you can kind of view the
tangent of theta, or it really

00:03:01.950 --> 00:03:04.570
is, as the slope of this line.

00:03:04.570 --> 00:03:05.730
The slope.

00:03:05.730 --> 00:03:11.660
So you could write slope is
equal to the tangent of theta.

00:03:11.660 --> 00:03:14.350
So let's just bear that in mind
when we go to our example.

00:03:14.350 --> 00:03:19.550
If I'm asking you-- and I'll
rewrite it here --what is the

00:03:19.550 --> 00:03:22.600
inverse tangent of minus 1?

00:03:22.600 --> 00:03:23.880
And I'll keep rewriting it.

00:03:23.880 --> 00:03:26.440
Or the arctangent of minus 1?

00:03:26.440 --> 00:03:29.830
I'm saying what angle gives
me a slope of minus 1

00:03:29.830 --> 00:03:31.320
on the unit circle?

00:03:31.320 --> 00:03:34.830
So let's draw the unit circle.

00:03:34.830 --> 00:03:37.960
Let's draw the unit
circle like that.

00:03:37.960 --> 00:03:42.880
Then I have my axes like that.

00:03:42.880 --> 00:03:44.440
And I want a slope of minus 1.

00:03:44.440 --> 00:03:46.450
A slope of minus 1
looks like this.

00:03:49.995 --> 00:03:52.430
If it was like that, it
would be slope of plus 1.

00:03:52.430 --> 00:03:55.580
So what angle is this?

00:03:55.580 --> 00:03:58.710
So in order to have a slope
of minus 1, this distance is

00:03:58.710 --> 00:04:00.580
the same as this distance.

00:04:00.580 --> 00:04:03.940
And you might already recognize
that this is a right angle.

00:04:03.940 --> 00:04:06.410
So these angles have
to be the same.

00:04:06.410 --> 00:04:09.250
So this has to be a
45 45 90 triangle.

00:04:09.250 --> 00:04:10.630
This is an isosceles triangle.

00:04:10.630 --> 00:04:12.880
These two have to add up to 90
and they have to be the same.

00:04:12.880 --> 00:04:15.120
So this is 45 45 90.

00:04:15.120 --> 00:04:18.680
And if you know your 45 45 90--
Actually, you don't even have

00:04:18.680 --> 00:04:20.250
to know the sides of it.

00:04:20.250 --> 00:04:22.440
In the previous video, we
saw that this is going

00:04:22.440 --> 00:04:23.810
to be-- Right here.

00:04:23.810 --> 00:04:28.040
This distance is going to be
square root of 2 over 2.

00:04:28.040 --> 00:04:31.610
So this coordinate in the
y-direction is minus

00:04:31.610 --> 00:04:33.380
square root of 2 over 2.

00:04:33.380 --> 00:04:36.210
And then this coordinate right
here on the x-direction is

00:04:36.210 --> 00:04:39.500
square root of 2 over 2 because
this length right

00:04:39.500 --> 00:04:40.960
there is that.

00:04:40.960 --> 00:04:43.430
So the square root of 2 over 2
squared plus the square root of

00:04:43.430 --> 00:04:46.170
2 over 2 squared is
equal to 1 squared.

00:04:46.170 --> 00:04:47.806
But the important thing
to realize is this is

00:04:47.806 --> 00:04:50.690
a 45 45 90 triangle.

00:04:50.690 --> 00:04:54.700
So this angle right here is--
Well if you're just looking at

00:04:54.700 --> 00:04:57.670
the triangle by itself, you
would say that this is

00:04:57.670 --> 00:04:59.360
a 45 degree angle.

00:04:59.360 --> 00:05:04.060
But since we're going clockwise
below the x-axis, we'll call

00:05:04.060 --> 00:05:05.980
this a minus 45 degree angle.

00:05:09.220 --> 00:05:13.710
So the tangent of minus 40--
Let me write that down.

00:05:13.710 --> 00:05:15.250
So if I'm in degrees.

00:05:15.250 --> 00:05:16.910
And that tends to
be how I think.

00:05:16.910 --> 00:05:25.160
So I could write the tangent of
minus 45 degrees it equals this

00:05:25.160 --> 00:05:28.167
negative value-- minus square
root of 2 over 2 over square

00:05:28.167 --> 00:05:31.200
root of 2 over 2, which
is equal to minus 1.

00:05:31.200 --> 00:05:36.670
Or I could write the arctangent
of minus 1 is equal

00:05:36.670 --> 00:05:39.110
to minus 45 degrees.

00:05:39.110 --> 00:05:40.920
Now if we're dealing with
radians, we just have to

00:05:40.920 --> 00:05:42.350
convert this to radians.

00:05:42.350 --> 00:05:47.530
So we multiply that times--
We get pi radians for

00:05:47.530 --> 00:05:49.880
every 180 degrees.

00:05:49.880 --> 00:05:51.890
The degrees cancel out.

00:05:51.890 --> 00:05:53.960
So you have a 45 over 180.

00:05:53.960 --> 00:05:55.160
This goes four times.

00:05:55.160 --> 00:05:57.570
So this is equal to-- you
have the minus sign--

00:05:57.570 --> 00:06:01.710
minus pi over 4 radians.

00:06:01.710 --> 00:06:06.450
So the arctangent of minus 1 is
equal to minus pi over 4 or the

00:06:06.450 --> 00:06:13.850
inverse tangent of minus 1 is
also equal to minus pi over 4.

00:06:13.850 --> 00:06:15.350
Now you could say, look.

00:06:15.350 --> 00:06:17.930
If I'm at minus pi
over 4, that's there.

00:06:17.930 --> 00:06:18.540
That's fine.

00:06:18.540 --> 00:06:22.360
This gives me a value of
minus 1 because the slope

00:06:22.360 --> 00:06:23.320
of this line is minus 1.

00:06:23.320 --> 00:06:25.120
But I can keep going
around the unit circle.

00:06:25.120 --> 00:06:26.880
I could add 2 pi to this.

00:06:26.880 --> 00:06:30.890
Maybe I could add 2 pi to this
and that would also give me--

00:06:30.890 --> 00:06:33.090
If I took the tangent of that
angle, it would also

00:06:33.090 --> 00:06:34.640
give me minus 1.

00:06:34.640 --> 00:06:39.170
Or I could add 2 pi again and
it'll, again, give me minus 1.

00:06:39.170 --> 00:06:42.100
In fact I could go to
this point right here.

00:06:42.100 --> 00:06:44.420
And the tangent would also
give me minus 1 because

00:06:44.420 --> 00:06:45.790
the slope is right there.

00:06:45.790 --> 00:06:49.460
And like I said in the sine--
in the inverse sine video, you

00:06:49.460 --> 00:06:51.960
can't have a function that
has a 1 to many mapping.

00:06:51.960 --> 00:06:58.190
You can't-- Tangent inverse
of x can't map to a bunch

00:06:58.190 --> 00:06:59.830
of different values.

00:06:59.830 --> 00:07:03.280
It can't map to
minus pi over 4.

00:07:03.280 --> 00:07:09.270
It can't map to 3-- what it
would be? --3 pi over 4.

00:07:09.270 --> 00:07:09.730
I don't know.

00:07:09.730 --> 00:07:14.310
It would be-- I'll just
say 2 pi minus pi over 4.

00:07:14.310 --> 00:07:16.200
Or 4 pi minus pi.

00:07:16.200 --> 00:07:18.550
It can't map to all of
these different things.

00:07:18.550 --> 00:07:20.740
So I have to constrict
the range on the

00:07:20.740 --> 00:07:22.270
inverse tan function.

00:07:22.270 --> 00:07:25.570
And we'll restrict it very
similarly to the way we

00:07:25.570 --> 00:07:29.100
restricted the sine--
the inverse sine range.

00:07:29.100 --> 00:07:32.510
We're going to restrict it to
the first and fourth quadrants.

00:07:32.510 --> 00:07:36.010
So the answer to your inverse
tangent is always going to be

00:07:36.010 --> 00:07:37.470
something in these quadrants.

00:07:37.470 --> 00:07:39.950
But it can't be this
point and that point.

00:07:39.950 --> 00:07:44.810
Because a tangent function
becomes undefined at pi over

00:07:44.810 --> 00:07:46.350
2 and at minus pi ever 2.

00:07:46.350 --> 00:07:48.110
Because your slope
goes vertical.

00:07:48.110 --> 00:07:50.260
You start dividing--
Your change in x is 0.

00:07:50.260 --> 00:07:52.890
You're dividing-- Your
cosine of theta goes to 0.

00:07:52.890 --> 00:07:55.560
So if you divide by
that, it's undefined.

00:07:55.560 --> 00:07:59.730
So your range-- So if I--
Let me write this down.

00:07:59.730 --> 00:08:03.430
So if I have an inverse tangent
of x, I'm going to-- Well,

00:08:03.430 --> 00:08:06.330
what are all the values that
the tangent can take on?

00:08:06.330 --> 00:08:11.800
So if I have the tangent of
theta is equal to x, what are

00:08:11.800 --> 00:08:13.810
all the different values
that x could take on?

00:08:13.810 --> 00:08:16.900
These are all the possible
values for the slope.

00:08:16.900 --> 00:08:18.810
And that slope can
take on anything.

00:08:18.810 --> 00:08:22.700
So x could be anywhere
between minus infinity

00:08:22.700 --> 00:08:24.870
and positive infinity.

00:08:24.870 --> 00:08:27.250
x could pretty much
take on any value.

00:08:27.250 --> 00:08:29.050
But what about theta?

00:08:29.050 --> 00:08:29.950
Well I just said it.

00:08:29.950 --> 00:08:33.910
Theta, you can only go
from minus pi over 2 all

00:08:33.910 --> 00:08:35.260
the way to pi over 2.

00:08:35.260 --> 00:08:37.680
And you can't even include pi
over 2 or minus pi over 2

00:08:37.680 --> 00:08:39.790
because then you'd be vertical.

00:08:39.790 --> 00:08:41.910
So then you say your--
So if I'm just dealing

00:08:41.910 --> 00:08:43.310
with vanilla tangent.

00:08:43.310 --> 00:08:44.470
Not the inverse.

00:08:44.470 --> 00:08:50.940
The domain-- Well the domain of
tangent can go multiple times

00:08:50.940 --> 00:08:53.410
around, so let me not
make that statement.

00:08:53.410 --> 00:08:55.580
But if I want to do inverse
tangent so I don't have

00:08:55.580 --> 00:08:56.610
a 1 to many mapping.

00:08:56.610 --> 00:08:58.870
I want to cross
out all of these.

00:08:58.870 --> 00:09:04.180
I'm going to restrict theta, or
my range, to be greater than

00:09:04.180 --> 00:09:10.160
the minus pi over 2 and less
than positive pi over 2.

00:09:10.160 --> 00:09:13.710
And so if I restrict my range
to this right here and I

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exclude that point
and that point.

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Then I can only get one answer.

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When I say tangent of what
gives me a slope of minus 1?

00:09:21.670 --> 00:09:24.360
And that's the question
I'm asking right there.

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There's only one answer.

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Because if I keep-- This
one falls out of it.

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And obviously as I go around
and around, those fall out of

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that valid range for theta
that I was giving you.

00:09:34.680 --> 00:09:38.270
And then just to kind of
make sure we did it right.

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Our answer was pi over 4.

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Let's see if we get that
when we use our calculator.

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So the inverse tangent of
minus 1 is equal to that.

00:09:50.340 --> 00:09:53.460
Let's see if that's the same
thing as minus pi over 4.

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Minus pi over 4 is
equal to that.

00:09:57.760 --> 00:09:59.100
So it is minus pi over 4.

00:09:59.100 --> 00:10:02.280
But it was good that we solved
it without a calculator because

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it's hard to recognize
this as minus pi over 4.