Atomic Energy Levels

WEBVTT
Kind: captions
Language: en

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- [Instructor] Here's a very
simplified model of an atom.

00:00:03.330 --> 00:00:05.080
The nucleus at the center of the atom

00:00:05.080 --> 00:00:07.910
is where the protons and neutrons live,

00:00:07.910 --> 00:00:09.870
but they're kind of boring,

00:00:09.870 --> 00:00:12.350
because for the most
part they just sit there.

00:00:12.350 --> 00:00:14.890
The real star of the show is the electron.

00:00:14.890 --> 00:00:17.360
The electron gets to do
all the interesting stuff,

00:00:17.360 --> 00:00:20.850
like move around, jump
around, bind with other atoms.

00:00:20.850 --> 00:00:24.090
These dashed lines represent
the different energy levels

00:00:24.090 --> 00:00:26.840
the electron can have while in the atom.

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We like representing these energy levels

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with an energy level diagram.

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The energy level diagram
gives us a way to show

00:00:33.970 --> 00:00:37.310
what energy the electron
has without having to draw

00:00:37.310 --> 00:00:40.380
an atom with a bunch of
circles all the time.

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Let's say our pretend atom
has electron energy levels

00:00:43.820 --> 00:00:48.820
of zero eV, four eV, six eV, and seven eV.

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Note that moving left or right
on an energy level diagram

00:00:52.720 --> 00:00:55.890
doesn't actually represent
anything meaningful,

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so technically there is no x-axis

00:00:58.560 --> 00:01:01.690
on an energy level diagram,
but we draw it there anyway

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because it makes it look nice.

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All that matters is what energy level

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or rung on the ladder the electron is at.

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Note that the electron for
our hypothetical atom here

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can only exist with zero
eV, four, six, or seven eV.

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The electron just cannot
exist between energy levels.

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It's always got to be right
on one of the energy levels.

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Okay, so let's say our electron

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starts off on the zero eV energy level.

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It's good to note that
the lowest energy level

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an electron can have in an atom
is called the ground state.

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So how could our electron
get from the ground state

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to any of the higher energy levels?

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Well, for the electron to
get to a higher energy level,

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we've got to give the
electron more energy,

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and we know how to give
an electron more energy.

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You just shoot light at it.

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If a photon of the right
energy can strike an electron,

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the electron will absorb
all the photon's energy

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and jump to a higher energy level.

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The electron in this
ground state needs four eV

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to jump to the next energy level.

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That means if a photon that
had an energy of four eV

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came in and struck the electron,

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the electron would absorb
the energy of the photon,

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causing the photon to disappear,

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and that electron would jump
up to the next energy level.

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We call the first energy
level after the ground state

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the first excited state.

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Once the electron's at
the higher energy level,

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it won't stay there long.

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Electrons, if given the
chance, will fall towards

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the lowest energy level they can.

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So our electron will fall
back down to the ground state

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and give up four eV of energy.

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The way an electron can give up energy

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is by emitting a photon.

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So after falling back
down to the ground state,

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this electron would emit a four eV photon.

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Electrons don't have to just jump

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one energy level at a time though.

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If the electron in our ground state

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were to absorb a six eV photon,

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the electron could jump all the way up

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to the six eV energy level.

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Now that the electron's
at a higher energy level,

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it's gonna try to fall back
down, but there's a couple ways

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it could fall back down in this case.

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The electron could fall
down to the ground state

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all in one shot, giving up a
six eV photon in the process,

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but since the electron started
at the six eV energy level,

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it could've also fallen first
to the four eV energy level,

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emitting a two eV photon in the process.

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It's a two eV photon
because the electron dropped

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two electron volts in energy,

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and now that the electron's
at the four eV energy level,

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it'll fall back down to the ground state,

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emitting a four eV photon in the process.

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So electrons will sometimes drop

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multiple energy levels at a time,

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and sometimes they'll choose
to take individual steps,

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but regardless, the energy
of the photon is always equal

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to the difference in
electron energy levels.

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What if our electron's in the ground state

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and we send a five eV photon at it?

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If the electron were to absorb

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all of the energy of the five eV photon,

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it would now have five electron volts,

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but that's not an allowed energy level,

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so the electron can't absorb this photon,

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and the photon will pass
straight through the atom.

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Keep in mind, the electron in the atom

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has to absorb all of the
photon's energy or none of it.

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It can't just absorb part of it.

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Alright, so now we could figure out

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every possible photon
this atom could absorb.

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If the electron's in the ground state,

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it could absorb a four eV photon,

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or a six eV photon, or a seven eV photon.

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If the electron's at
the second energy level,

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also called the first excited state,

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the electron could absorb a two eV photon

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or a three eV photon.

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And if the electron were
at the third energy level,

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or the second excited state,

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the electron could absorb a one eV photon.

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Those are the only photons that this atom

00:04:53.750 --> 00:04:56.130
will be seen to absorb.

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2.5eV photons will pass straight through,

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five eV photons will
pass straight through,

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6.3eV photons will pas straight through.

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What this means is that
if you were to shine light

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that consisted of all possible wavelengths

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through a gas that was
composed of our pretend atoms,

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all the wavelengths would
not make it through.

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Some of the wavelengths
would get absorbed,

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then scattered away in random directions.

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This would manifest itself as
dark lines in the spectrum,

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missing wavelengths or
missing energy levels

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that correspond to the energies of photons

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that our electron can absorb.

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This is like a fingerprint for an atom,

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and it's called that
atom's absorption spectrum.

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If you were to ever see this
progression of dark lines

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in these exact positions, you would know

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that the gas you were
looking at was composed

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at least partly of our hypothetical atom.

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This also allows astronomers to determine

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what stuff in our universe is made out of,

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even though we can't get close
enough to collect a sample.

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All we have to do is collect light

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from a distant star or quasar that shines

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through the stuff we're interested in,

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then just determine which wavelengths

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or energies got taken out.

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The details are a little
messier than that,

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but this provides astronomers

00:06:12.380 --> 00:06:15.870
with maybe the most important
tool at their disposal.

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Now the absorption spectrum
are all of the wavelengths

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or energies that an atom will absorb

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from light that passes through it.

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You could also ask about
the emission spectrum.

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The emission spectrum are
all of the wavelengths

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or energies that an atom will emit

00:06:31.380 --> 00:06:34.810
due to electrons falling
down in energy levels.

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You could go through all the possibilities

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of an electron falling down again,

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but you'd realize you're gonna get

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the exact same energies
for the emission spectrum

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that you got for the absorption spectrum.

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So instead of letting
light pass through a gas

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composed of your hypothetical atoms,

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let's say you made a container

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that had the gas of
your hypothetical atoms,

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and you ran an electric
current through it,

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exciting those electrons
to higher energy levels

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and letting them fall back
down to lower energy levels.

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This is what happens in neon lights,

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or if you're in science
class it's what happens

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in gas discharge tubes.

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So for the emission
spectrum, instead of seeing

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the whole electromagnetic
spectrum with a few lines missing,

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you're going to only
see a handful of lines

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that correspond to the
energies of those photons

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that that atom will emit.

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Okay I've gotta be honest about something.

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If any physicists are watching this video,

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they're cringing because the energies

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that electrons will have in
an atom are not positive.

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The energies an electron
can have in an atom

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are actually all negative values.

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This is because the
electron's bound to the atom.

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Anything that's bound to something else

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will have total energies
that are negative.

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This is analogous to a ball
stuck at the bottom of a ditch.

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If the ball's not moving
it has no kinetic energy,

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and if we assume that ground level

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is the H equals zero position,
then this ball's gonna have

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a negative gravitational potential energy.

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Since this ball has a
negative total energy,

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it's stuck and bound to the ditch.

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If someone could give
this ball enough energy

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so that it would have
positive total energy,

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the ball could leave the ditch.

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It would not be bound anymore.

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So to make our hypothetical
atom a little more realistic,

00:08:13.600 --> 00:08:17.290
let's subtract 10eV
from each energy level.

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This doesn't really change anything.

00:08:19.670 --> 00:08:21.210
In order for the electron to get

00:08:21.210 --> 00:08:26.210
from the -10eV ground state to
the -6eV first excited state,

00:08:26.680 --> 00:08:29.810
it's still gonna take a four eV photon.

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People do get confused with
the negative signs though,

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so be careful.

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In order to find the energy of the photon

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that was absorbed or emitted,

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you always take the higher energy level

00:08:40.470 --> 00:08:43.670
and subtract from it
the lower energy level.

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So in this case, we would take -6eV,

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and subtract from it -10eV, which tells us

00:08:50.450 --> 00:08:52.710
that it would take a four eV photon

00:08:52.710 --> 00:08:55.660
to bump an electron up
to that energy level,

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and the electron would
emit a four eV photon

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if it dropped back down from that level.

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Something else that's unrealistic

00:09:03.260 --> 00:09:06.610
about our hypothetical
atom is that real atoms

00:09:06.610 --> 00:09:11.400
wouldn't just stop at -3eV
for the highest energy level.

00:09:11.400 --> 00:09:14.240
Real atoms have energy
levels that get closer

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and closer together as
you approach a zero eV.

00:09:17.780 --> 00:09:22.270
What happens when an electron
gets more than zero eV energy?

00:09:22.270 --> 00:09:24.940
Well if an electron has
more than zero energy,

00:09:24.940 --> 00:09:27.290
that means it's got positive energy.

00:09:27.290 --> 00:09:29.010
And if it's got positive energy,

00:09:29.010 --> 00:09:31.870
it's not bound to the atom anymore.

00:09:31.870 --> 00:09:33.890
It'll be free to leave, it'll be gone,

00:09:33.890 --> 00:09:36.270
and we'll say that we've ionized the atom

00:09:36.270 --> 00:09:38.200
by removing an electron.

00:09:38.200 --> 00:09:40.510
So for example, say the electron started

00:09:40.510 --> 00:09:45.510
at the -4eV energy level and
it absorbed a seven eV photon,

00:09:46.240 --> 00:09:48.010
that electron would have a total energy

00:09:48.010 --> 00:09:52.003
of positive three eV, and so
it would be gone from the atom.

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(upbeat music)


Photon Momentum

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:01.480
- [Narrator] A while back,

00:00:01.480 --> 00:00:04.030
I was teaching physics in California.

00:00:04.030 --> 00:00:06.400
I got to class and I
was all like, hey class,

00:00:06.400 --> 00:00:08.160
you wanna hear a physics joke?

00:00:08.160 --> 00:00:10.400
Yeah, okay, totally.

00:00:10.400 --> 00:00:12.850
Does light hold mass?

00:00:12.850 --> 00:00:14.420
I don't know, does it?

00:00:14.420 --> 00:00:17.370
No, it's not even Catholic.

00:00:17.370 --> 00:00:19.000
Oh man.

00:00:19.000 --> 00:00:20.470
Shoulda went to the beach today.

00:00:20.470 --> 00:00:21.350
Dude, really?

00:00:21.350 --> 00:00:22.960
Totally lame.

00:00:22.960 --> 00:00:26.000
But wait, doesn't light really have mass?

00:00:26.000 --> 00:00:27.090
It should.

00:00:27.090 --> 00:00:29.120
It's got momentum so it's gotta have mass.

00:00:29.120 --> 00:00:32.440
It has momentum so it must have mass.

00:00:32.440 --> 00:00:33.730
Whoa, hey!

00:00:33.730 --> 00:00:37.300
Everyone just calm the heck down.

00:00:37.300 --> 00:00:38.820
You see, the worst part of all this

00:00:38.820 --> 00:00:42.110
wasn't that my joke bombed,
it was that I actually managed

00:00:42.110 --> 00:00:44.420
to confuse people by telling it.

00:00:44.420 --> 00:00:47.450
My students had read
that light has momentum

00:00:47.450 --> 00:00:50.520
and they were right, light
really does have momentum.

00:00:50.520 --> 00:00:52.450
But then I come to class and I'm all like,

00:00:52.450 --> 00:00:55.570
hey, guess what, light has no mass.

00:00:55.570 --> 00:00:58.080
Now my students are
thinking to themselves,

00:00:58.080 --> 00:01:00.460
dude, but P equals MV.

00:01:00.460 --> 00:01:04.680
In other words, if momentum
equals mass times velocity,

00:01:04.680 --> 00:01:08.780
how could light, which has
no mass, ever have momentum?

00:01:08.780 --> 00:01:11.260
Well, I had to break it to my students

00:01:11.260 --> 00:01:13.568
that P equals MV

00:01:13.568 --> 00:01:16.830
isn't really accurate
for things that travel

00:01:16.830 --> 00:01:18.930
close to the speed of light.

00:01:18.930 --> 00:01:20.570
For things going that fast,

00:01:20.570 --> 00:01:23.050
you have to use special relativity.

00:01:23.050 --> 00:01:24.670
I don't wanna waste a lot of time

00:01:24.670 --> 00:01:27.560
talking about special
relativity in this video

00:01:27.560 --> 00:01:29.870
so you're just gonna have
to take my word for it

00:01:29.870 --> 00:01:33.810
that the rules of special
relativity allow for a loophole.

00:01:33.810 --> 00:01:37.730
This loophole provides a
way for massless objects

00:01:37.730 --> 00:01:39.640
to have momentum.

00:01:39.640 --> 00:01:42.940
Alright, so the bad news
is that we cannot use

00:01:42.940 --> 00:01:47.110
P equals MV to find the
momentum of a photon.

00:01:47.110 --> 00:01:49.540
The good news is that the
formula for the momentum

00:01:49.540 --> 00:01:52.610
of a photon is simple,
the momentum of a photon

00:01:52.610 --> 00:01:55.120
equals H over lambda.

00:01:55.120 --> 00:01:57.030
H is Planck's constant,

00:01:57.030 --> 00:02:02.030
6.626 times 10 to the
negative 34 joule-seconds.

00:02:02.410 --> 00:02:05.460
Lambda is the wavelength
of the light in meters.

00:02:05.460 --> 00:02:07.510
Be careful, don't use nanometers,

00:02:07.510 --> 00:02:09.490
you have to convert to meters.

00:02:09.490 --> 00:02:13.830
The momentum of a single photon
is gonna be extremely small,

00:02:13.830 --> 00:02:15.964
that's why it doesn't feel
like we're getting pushed on

00:02:15.964 --> 00:02:19.670
very much when light shines
on us, but theoretically,

00:02:19.670 --> 00:02:22.680
if you had a big enough
solar sail, the light

00:02:22.680 --> 00:02:26.260
bouncing off of that solar
sail could propel it forward

00:02:26.260 --> 00:02:30.320
due to the impulse imparted
by the momentum of the light.

00:02:30.320 --> 00:02:33.260
Okay, so very long story short,

00:02:33.260 --> 00:02:37.830
you should never ever use P equals MV

00:02:37.830 --> 00:02:40.310
to find the momentum of a photon.

00:02:40.310 --> 00:02:42.170
To find the momentum of a photon,

00:02:42.170 --> 00:02:45.991
you should always use
momentum equals H over lambda.

00:02:45.991 --> 00:02:49.241
(ambient techno music)


Quantum Wavefunction

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.870
- [Instructor] So when people first showed

00:00:01.870 --> 00:00:04.320
that matter particles like electrons

00:00:04.320 --> 00:00:07.060
can have wavelengths and
when DeBroglie showed

00:00:07.060 --> 00:00:09.620
that the wavelength is Planck's constant

00:00:09.620 --> 00:00:12.280
over the momentum, people were like cool,

00:00:12.280 --> 00:00:13.480
it's pretty sweet.

00:00:13.480 --> 00:00:15.920
But you know someone
was like wait a minute,

00:00:15.920 --> 00:00:19.235
if this particle has wavelike properties

00:00:19.235 --> 00:00:20.721
and it has a wavelength,

00:00:20.721 --> 00:00:23.690
what exactly is waving?

00:00:23.690 --> 00:00:26.350
What is this wave we're
even talking about?

00:00:26.350 --> 00:00:28.080
Conceptually it's a little strange.

00:00:28.080 --> 00:00:31.010
I mean a water wave, we know what that is.

00:00:31.010 --> 00:00:34.430
It's a bunch of water that's
oscillating up and down.

00:00:34.430 --> 00:00:37.190
A wave on a string, we know what that is.

00:00:37.190 --> 00:00:39.160
This string itself is moving up and down

00:00:39.160 --> 00:00:40.920
and it extends through space.

00:00:40.920 --> 00:00:42.240
But it's hard to imagine,

00:00:42.240 --> 00:00:45.720
how is this electron having a wavelength

00:00:45.720 --> 00:00:48.130
and what is the actual wave itself?

00:00:48.130 --> 00:00:50.860
So physicists were
grappling with this issue,

00:00:50.860 --> 00:00:52.556
trying to conceptually understand

00:00:52.556 --> 00:00:55.940
how to describe the wave of the electron.

00:00:55.940 --> 00:00:57.270
They wanted to do two things.

00:00:57.270 --> 00:00:59.510
They wanted a mathematical description

00:00:59.510 --> 00:01:01.240
for the shape of that wave,

00:01:01.240 --> 00:01:03.090
and that's called the wave function.

00:01:03.090 --> 00:01:06.740
So this wave function gives
you a mathematical description

00:01:06.740 --> 00:01:08.390
for what the shape of the wave is.

00:01:08.390 --> 00:01:11.210
So different electron
systems are gonna have

00:01:11.210 --> 00:01:12.880
different wave functions,

00:01:12.880 --> 00:01:14.730
and this is psi,

00:01:14.730 --> 00:01:17.170
it's the symbol for the wave function.

00:01:17.170 --> 00:01:20.010
So this is psi, the psi symbol.

00:01:20.010 --> 00:01:21.620
It's a function of x.

00:01:21.620 --> 00:01:24.010
So at different points in x,

00:01:24.010 --> 00:01:27.150
it may have a large value,
it may have a small value.

00:01:27.150 --> 00:01:29.690
This function would give
you the mathematical shape

00:01:29.690 --> 00:01:30.523
of this wave.

00:01:30.523 --> 00:01:32.900
So that was one of the things
they were trying to determine.

00:01:32.900 --> 00:01:34.580
But they also wanted to interpret it.

00:01:34.580 --> 00:01:37.580
Like what does this
wave function even mean?

00:01:37.580 --> 00:01:38.920
So we've got two problems.

00:01:38.920 --> 00:01:41.430
We want a mathematical
description of the wave

00:01:41.430 --> 00:01:44.490
and we wanna interpret what
does this wave even mean.

00:01:44.490 --> 00:01:48.250
Now the person that gave us
the mathematical description

00:01:48.250 --> 00:01:52.040
of this wave function
was Erwin Schrodinger.

00:01:52.040 --> 00:01:55.090
So Schrodinger is this guy right here.

00:01:55.090 --> 00:01:56.290
Schrodinger's right here.

00:01:56.290 --> 00:01:58.420
He wrote down Schrodinger's Equation,

00:01:58.420 --> 00:02:01.420
and his name now is basically synonymous

00:02:01.420 --> 00:02:04.110
with quantum mechanics
because this is arguably

00:02:04.110 --> 00:02:06.920
the most important equation
in all of quantum mechanics.

00:02:06.920 --> 00:02:09.340
There's a bunch of partial
derivatives in here

00:02:09.340 --> 00:02:11.150
and Planck's constants,

00:02:11.150 --> 00:02:12.930
but the important thing is that it's got

00:02:12.930 --> 00:02:14.610
the wave function in here.

00:02:14.610 --> 00:02:16.470
Now if you've never
seen partial derivatives

00:02:16.470 --> 00:02:17.810
or calculus, it's okay.

00:02:17.810 --> 00:02:21.290
All you need to know for our
purposes today in this video

00:02:21.290 --> 00:02:24.510
is that this equation
is a way to crank out

00:02:24.510 --> 00:02:26.460
the mathematical wave function.

00:02:26.460 --> 00:02:30.580
What is this function that
gives us the shape of the wave

00:02:30.580 --> 00:02:31.920
as a function of x?

00:02:31.920 --> 00:02:34.610
And you could imagine
plotting this on some graph.

00:02:34.610 --> 00:02:37.950
So once you solve for this
psi as a function of x,

00:02:37.950 --> 00:02:40.020
you could plot what this looks like.

00:02:40.020 --> 00:02:42.120
Maybe it looks something like this,

00:02:42.120 --> 00:02:44.010
and who knows, it could
do all kinds of stuff.

00:02:44.010 --> 00:02:45.070
Maybe it looks like that.

00:02:45.070 --> 00:02:47.260
But Schrodinger's Equation is the way

00:02:47.260 --> 00:02:49.300
you can get this wave function.

00:02:49.300 --> 00:02:51.380
So Schrodinger gave us a way to get

00:02:51.380 --> 00:02:53.110
the mathematical wave function,

00:02:53.110 --> 00:02:55.010
but we also wanted to interpret it.

00:02:55.010 --> 00:02:56.140
What does this even mean?

00:02:56.140 --> 00:02:59.750
To say that this wave function
represents the electrons

00:02:59.750 --> 00:03:00.850
is still strange.

00:03:00.850 --> 00:03:02.440
What does that mean?

00:03:02.440 --> 00:03:04.150
Schrodinger tried to
interpret it this way.

00:03:04.150 --> 00:03:08.600
He said, okay maybe this electron
really is like smudged out

00:03:08.600 --> 00:03:12.430
in space and its charge
is kinda distributed

00:03:12.430 --> 00:03:13.710
in different places.

00:03:13.710 --> 00:03:18.010
Schrodinger wanted to
interpret this wave function

00:03:18.010 --> 00:03:20.070
as charge density,

00:03:20.070 --> 00:03:22.540
and I mean it's kind of
a reasonable thing to do.

00:03:22.540 --> 00:03:24.900
The way you get a water
wave is by having water

00:03:24.900 --> 00:03:26.560
spread out through space.

00:03:26.560 --> 00:03:29.340
So maybe the way you get an electron wave

00:03:29.340 --> 00:03:31.640
is to have the charge of the electron

00:03:31.640 --> 00:03:33.320
spread out through space.

00:03:33.320 --> 00:03:35.360
But this description didn't work so well,

00:03:35.360 --> 00:03:36.820
which is kinda strange.

00:03:36.820 --> 00:03:39.010
Schrodinger invented this equation.

00:03:39.010 --> 00:03:40.370
He came up with this equation,

00:03:40.370 --> 00:03:42.140
but he couldn't even interpret

00:03:42.140 --> 00:03:44.140
what he was describing correctly.

00:03:44.140 --> 00:03:45.190
It took someone else.

00:03:45.190 --> 00:03:48.000
It took a guy named Max Born to give us

00:03:48.000 --> 00:03:51.530
the interpretation we go with
now for this wave function.

00:03:51.530 --> 00:03:55.040
Max Born said no, don't interpret
it as the charge density.

00:03:55.040 --> 00:03:57.540
What you should do is interpret this psi

00:03:57.540 --> 00:04:00.150
is giving you a way to get the probability

00:04:00.150 --> 00:04:03.680
of finding the electron
at a given point in space.

00:04:03.680 --> 00:04:05.040
So Max Born said this,

00:04:05.040 --> 00:04:07.420
if you find your psi,
like he said go ahead

00:04:07.420 --> 00:04:10.500
and use Schrodinger's
equation, use it, get psi.

00:04:10.500 --> 00:04:14.980
Once you have psi, what you do
is you square this function.

00:04:14.980 --> 00:04:17.010
So take the absolute value, square it,

00:04:17.010 --> 00:04:19.820
and what that's gonna give
you is the probability

00:04:19.820 --> 00:04:22.930
of finding the electron at a given point.

00:04:22.930 --> 00:04:26.330
Now technically it's
the probability density,

00:04:26.330 --> 00:04:27.590
but for our purposes,

00:04:27.590 --> 00:04:29.400
you can pretty much just think about this

00:04:29.400 --> 00:04:31.530
as the probability of finding the electron

00:04:31.530 --> 00:04:32.400
at a given point.

00:04:32.400 --> 00:04:34.870
So if this was our wave
function in other words,

00:04:34.870 --> 00:04:38.070
Max Born would tell us that
points where it's zero,

00:04:38.070 --> 00:04:40.710
these points right here
where the value is zero,

00:04:40.710 --> 00:04:43.430
there is a zero percent
chance you're gonna find

00:04:43.430 --> 00:04:44.400
the electron there.

00:04:44.400 --> 00:04:47.820
Points where there's a large value of psi,

00:04:47.820 --> 00:04:50.160
be it positive or negative,

00:04:50.160 --> 00:04:52.500
there's gonna be a large probability

00:04:52.500 --> 00:04:54.470
of finding the electron at that point.

00:04:54.470 --> 00:04:57.210
And we could say the odds
of finding the electron

00:04:57.210 --> 00:04:59.400
at a given point here are gonna be largest

00:04:59.400 --> 00:05:01.630
for this value of x right here

00:05:01.630 --> 00:05:04.536
because that's the point
for which the wave function

00:05:04.536 --> 00:05:07.240
has the greatest magnitude.

00:05:07.240 --> 00:05:10.450
But you won't necessarily
find the electron there.

00:05:10.450 --> 00:05:13.400
If you repeat this
experiment over and over,

00:05:13.400 --> 00:05:15.950
you may find the electron here once,

00:05:15.950 --> 00:05:19.130
you may find it over here, you
may find it there next time.

00:05:19.130 --> 00:05:20.800
You have to keep taking measurements,

00:05:20.800 --> 00:05:22.140
and if you keep taking measurements,

00:05:22.140 --> 00:05:23.630
you'll get this
distribution where you find

00:05:23.630 --> 00:05:26.720
a lot of 'em here, a lot of 'em there,

00:05:26.720 --> 00:05:28.590
a lot of 'em here, and a lot of 'em here,

00:05:28.590 --> 00:05:30.867
always where there's these
peaks you get more of them

00:05:30.867 --> 00:05:32.910
than you would have at other points

00:05:32.910 --> 00:05:34.700
where the values are smaller.

00:05:34.700 --> 00:05:37.530
You build up a distribution
that's represented

00:05:37.530 --> 00:05:38.890
by this wave function.

00:05:38.890 --> 00:05:40.610
So the wave function does not tell you

00:05:40.610 --> 00:05:42.350
where the electron's gonna be.

00:05:42.350 --> 00:05:44.640
It just gives you the probability,

00:05:44.640 --> 00:05:46.800
and technically the square of it

00:05:46.800 --> 00:05:49.510
gives you the probability of
finding the electron somewhere.

00:05:49.510 --> 00:05:51.590
So even at points down here
where the wave function

00:05:51.590 --> 00:05:53.020
has a negative value,

00:05:53.020 --> 00:05:55.220
I mean you can't have
a negative probability.

00:05:55.220 --> 00:05:56.950
You square that value.

00:05:56.950 --> 00:05:59.770
That gives you the probability
of finding the electron

00:05:59.770 --> 00:06:00.760
in that region.

00:06:00.760 --> 00:06:01.593
So in other words,

00:06:01.593 --> 00:06:02.680
let's get rid of all this.

00:06:02.680 --> 00:06:06.160
Let's say we solved some
Schrodinger equation

00:06:06.160 --> 00:06:08.230
or we were just handed a wave function

00:06:08.230 --> 00:06:10.330
and we were told it looks
like this and we were asked,

00:06:10.330 --> 00:06:12.710
where are you most likely
to find the electron?

00:06:12.710 --> 00:06:15.450
Well the value of the
wave function is greatest

00:06:15.450 --> 00:06:16.530
at this point here,

00:06:16.530 --> 00:06:18.590
so you'd be most likely
to find the electron

00:06:18.590 --> 00:06:20.200
in this region right here.

00:06:20.200 --> 00:06:22.580
You'd have no shot of
finding it right there.

00:06:22.580 --> 00:06:24.810
You'd have pretty good odds
of finding it right here

00:06:24.810 --> 00:06:26.090
or right here,

00:06:26.090 --> 00:06:28.110
but you'd have the greatest
chance of finding it

00:06:28.110 --> 00:06:29.550
in this region right here.

00:06:29.550 --> 00:06:31.703
So you'd have to repeat
this measurement many times.

00:06:31.703 --> 00:06:33.440
In quantum mechanics,

00:06:33.440 --> 00:06:35.890
one measurement doesn't
verify that you've got

00:06:35.890 --> 00:06:37.220
the right wave function.

00:06:37.220 --> 00:06:38.630
Because if I do one experiment

00:06:38.630 --> 00:06:39.930
and measure one electron,

00:06:39.930 --> 00:06:41.690
boop I might find the
electron right there.

00:06:41.690 --> 00:06:43.230
That doesn't really tell me anything.

00:06:43.230 --> 00:06:45.960
I have to repeat this
over and over to make sure

00:06:45.960 --> 00:06:49.360
the relative frequency of
where I'm finding electrons

00:06:49.360 --> 00:06:52.207
matches the wave function
I'm using to model

00:06:52.207 --> 00:06:54.030
that electron system.

00:06:54.030 --> 00:06:55.710
So that's what the wave function is.

00:06:55.710 --> 00:06:57.360
That's what it can do for you,

00:06:57.360 --> 00:06:58.650
although if I were you,

00:06:58.650 --> 00:07:00.050
I'd still be unsatisfied.

00:07:00.050 --> 00:07:02.590
I'd be like, wait a minute,
okay, that's fine and good.

00:07:02.590 --> 00:07:05.117
Wave function can give us the probability

00:07:05.117 --> 00:07:07.960
or the probability density
of finding the electron

00:07:07.960 --> 00:07:09.050
in a given region,

00:07:09.050 --> 00:07:10.640
but we haven't answered the question,

00:07:10.640 --> 00:07:14.180
what is waving here and what exactly

00:07:14.180 --> 00:07:15.510
is this wave function?

00:07:15.510 --> 00:07:18.960
Is this a physical object
sort of like a water wave

00:07:18.960 --> 00:07:21.070
or even an electromagnetic wave?

00:07:21.070 --> 00:07:23.620
Or is this just some mathematical trickery

00:07:23.620 --> 00:07:26.760
that we're using that has
no physical interpretation

00:07:26.760 --> 00:07:29.776
other than giving us
information about where

00:07:29.776 --> 00:07:31.590
the electron's gonna be?

00:07:31.590 --> 00:07:34.310
And I've got good news and bad news.

00:07:34.310 --> 00:07:37.500
The bad news is that
people still don't agree

00:07:37.500 --> 00:07:40.160
on how to interpret this wave function.

00:07:40.160 --> 00:07:42.520
Yes they know that the
square of it gives you

00:07:42.520 --> 00:07:45.280
the probability of finding
the electron in some region,

00:07:45.280 --> 00:07:48.110
but people differ on how
they're supposed to interpret it

00:07:48.110 --> 00:07:49.200
past that point.

00:07:49.200 --> 00:07:51.810
For instance, is this wave function

00:07:51.810 --> 00:07:54.380
the wave function of a single electron

00:07:54.380 --> 00:07:57.150
or is this wave function
really the wave function

00:07:57.150 --> 00:08:00.750
of a system, an ensemble of electrons,

00:08:00.750 --> 00:08:03.050
all similarly prepared
that you're gonna do

00:08:03.050 --> 00:08:04.300
the experiment on?

00:08:04.300 --> 00:08:06.730
In other words, does it
describe one electron

00:08:06.730 --> 00:08:09.470
or only describe a system of electrons?

00:08:09.470 --> 00:08:11.550
Does it not describe the electron at all

00:08:11.550 --> 00:08:14.440
but only our measurement of the electron?

00:08:14.440 --> 00:08:16.560
And what happens to this wave function

00:08:16.560 --> 00:08:18.620
when you actually measure the electron?

00:08:18.620 --> 00:08:21.120
When you measure the electron
you find it somewhere,

00:08:21.120 --> 00:08:22.900
and at that moment there's no chance

00:08:22.900 --> 00:08:24.480
of finding it over here at all.

00:08:24.480 --> 00:08:26.689
So does the act of measuring the electron

00:08:26.689 --> 00:08:30.720
cause some catastrophic
collapse in this wave function

00:08:30.720 --> 00:08:33.280
that's not described by
Schrodinger's Equation?

00:08:33.280 --> 00:08:36.240
These and many more
questions are still debated

00:08:36.240 --> 00:08:37.980
and not completely understood.

00:08:37.980 --> 00:08:39.290
That's the bad news.

00:08:39.290 --> 00:08:41.940
The good news is that we don't really need

00:08:41.940 --> 00:08:44.290
to understand that to make progress.

00:08:44.290 --> 00:08:47.140
Everyone knows how to
use the wave function

00:08:47.140 --> 00:08:49.490
to get the probabilities of measurements.

00:08:49.490 --> 00:08:51.750
You can have your favorite interpretation,

00:08:51.750 --> 00:08:55.100
but luckily pretty much
regardless of how you interpret

00:08:55.100 --> 00:08:56.230
this wave function,

00:08:56.230 --> 00:08:57.870
as long as you're using it correctly

00:08:57.870 --> 00:09:00.070
to get the probabilities of measurements,

00:09:00.070 --> 00:09:01.727
you can continue making progress,

00:09:01.727 --> 00:09:03.540
testing different models,

00:09:03.540 --> 00:09:05.470
and correlating data to the measurements

00:09:05.470 --> 00:09:07.070
that people make in the lab.

00:09:07.070 --> 00:09:09.040
Now I'm not saying that interpretations

00:09:09.040 --> 00:09:11.500
of this wave function are not important.

00:09:11.500 --> 00:09:15.060
People have tried cracking
this nut for over 100 years,

00:09:15.060 --> 00:09:16.380
and it's resisted.

00:09:16.380 --> 00:09:18.492
Maybe that's because it's a waste of time

00:09:18.492 --> 00:09:21.500
or maybe it's because the
difficulty of figuring this out

00:09:21.500 --> 00:09:24.270
is so great that whoever
does it will go down

00:09:24.270 --> 00:09:27.230
in history as one of the
great physicists of all time.

00:09:27.230 --> 00:09:29.130
It's hard to tell right now,

00:09:29.130 --> 00:09:32.320
but what's undebatable is
for about 100 years now,

00:09:32.320 --> 00:09:35.700
we've been able to make
progress with quantum mechanics

00:09:35.700 --> 00:09:38.740
even though we differ on
how exactly to interpret

00:09:38.740 --> 00:09:41.380
what this wave function really represents.

00:09:41.380 --> 00:09:43.260
So recapping the wave function gives you

00:09:43.260 --> 00:09:45.400
the probability of finding a particle

00:09:45.400 --> 00:09:47.130
in that region of space,

00:09:47.130 --> 00:09:50.090
specifically the square
of the wave function

00:09:50.090 --> 00:09:52.610
gives you the probability density

00:09:52.610 --> 00:09:55.510
of finding a particle
at that point in space.

00:09:55.510 --> 00:09:57.660
This almost everyone has agreed upon.

00:09:57.660 --> 00:10:00.600
Whether the wave function
has deeper implications

00:10:00.600 --> 00:10:02.960
besides this, people differ,

00:10:02.960 --> 00:10:05.190
but that hasn't yet
stopped us from applying

00:10:05.190 --> 00:10:07.840
quantum mechanics correctly in a variety

00:10:07.840 --> 00:10:09.403
of different scenarios.


Center of mass equation

WEBVTT
Kind: captions
Language: en

00:00:00.020 --> 00:00:01.191
- [Tutor] So let's say you wanted to know

00:00:01.191 --> 00:00:03.530
where the center of mass was between

00:00:03.530 --> 00:00:06.484
this two kilogram mass and
this six kilogram mass,

00:00:06.484 --> 00:00:08.690
now they're separated by 10 centimeters,

00:00:08.690 --> 00:00:10.670
so it's somewhere in between them

00:00:10.670 --> 00:00:14.030
and we know it's gonna be
closer to the larger mass,

00:00:14.030 --> 00:00:16.250
'cause the center of mass is always closer

00:00:16.250 --> 00:00:19.820
to the larger mass, but
exactly where is it gonna be?

00:00:19.820 --> 00:00:22.410
We need a formula to figure this out

00:00:22.410 --> 00:00:25.600
and the formula for the center
of mass looks like this,

00:00:25.600 --> 00:00:27.962
it says the location
of the center of mass,

00:00:27.962 --> 00:00:29.810
that's what this is,

00:00:29.810 --> 00:00:33.730
this Xcm is just the location
of the center of mass,

00:00:33.730 --> 00:00:37.790
it's the position of the
center of mass is gonna equal,

00:00:37.790 --> 00:00:40.270
you take all the masses
that you're trying to find

00:00:40.270 --> 00:00:42.050
the center of mass between,

00:00:42.050 --> 00:00:44.630
you take all those masses
times their positions

00:00:44.630 --> 00:00:47.690
and you add up all of these M times Xs,

00:00:47.690 --> 00:00:50.440
until you've accounted
for every single M times X

00:00:50.440 --> 00:00:51.273
there is in your system

00:00:51.273 --> 00:00:54.837
and then you just divide by all
of the masses added together

00:00:54.837 --> 00:00:56.590
and what you get out of this

00:00:56.590 --> 00:00:59.210
is the location of the center of mass.

00:00:59.210 --> 00:01:01.691
So let's use this, let's use
this for this example problem

00:01:01.691 --> 00:01:03.470
right here and let's see what we get,

00:01:03.470 --> 00:01:05.050
we'll have the center of mass,

00:01:05.050 --> 00:01:08.668
the position of the center
of mass is gonna be equal to,

00:01:08.668 --> 00:01:10.850
alright, so we'll take M1,

00:01:10.850 --> 00:01:13.090
which you could take either one as M1,

00:01:13.090 --> 00:01:14.273
but I already colored this one red,

00:01:14.273 --> 00:01:17.041
so we'll just say the
two kilogram mass is M1

00:01:17.041 --> 00:01:20.340
and we're gonna have to multiply by X1,

00:01:20.340 --> 00:01:22.970
the position of mass
one and at this point,

00:01:22.970 --> 00:01:25.733
you might be confused, you
might be like the position,

00:01:25.733 --> 00:01:27.370
I don't know what the position is,

00:01:27.370 --> 00:01:29.670
there's no coordinate system up here,

00:01:29.670 --> 00:01:31.976
well, you get to pick,
so you get to decide

00:01:31.976 --> 00:01:34.870
where you're measuring
these positions from

00:01:34.870 --> 00:01:37.150
and wherever you decide
to measure them from

00:01:37.150 --> 00:01:38.016
will also be the point,

00:01:38.016 --> 00:01:40.445
where the center of mass is measured from,

00:01:40.445 --> 00:01:44.050
in other words, you get to
choose where X equals zero.

00:01:44.050 --> 00:01:45.519
Let's just say for the sake of argument,

00:01:45.519 --> 00:01:48.720
the left-hand side over
here is X equals zero,

00:01:48.720 --> 00:01:52.014
let's say right here is X
equals zero on our number line

00:01:52.014 --> 00:01:54.530
and then it goes this way,
it's positive this way,

00:01:54.530 --> 00:01:58.152
so if this is X equals zero,
halfway would be X equals five

00:01:58.152 --> 00:02:00.940
and then over here, it
would be X equals 10,

00:02:00.940 --> 00:02:03.540
we're free to choose that,
in fact, it's kind of cool,

00:02:03.540 --> 00:02:05.650
because if this is X equals zero,

00:02:05.650 --> 00:02:08.710
the position of mass one is zero meters,

00:02:08.710 --> 00:02:11.440
so it's gonna be, this
term's just gonna go away,

00:02:11.440 --> 00:02:14.380
which is okay, we're gonna
have to add to that M2,

00:02:14.380 --> 00:02:18.220
which is six kilograms
times the position of M2,

00:02:18.220 --> 00:02:19.940
again we can choose
whatever point we want,

00:02:19.940 --> 00:02:22.710
but we have to be consistent,
we already chose this

00:02:22.710 --> 00:02:24.483
as X equals zero for mass one,

00:02:24.483 --> 00:02:27.465
so that still has to be X
equals zero for mass two,

00:02:27.465 --> 00:02:30.317
that means this has to
be 10 centimeters now

00:02:30.317 --> 00:02:32.586
and then those are our only
two masses, so we stop there

00:02:32.586 --> 00:02:35.445
and we just divide by all
the masses added together,

00:02:35.445 --> 00:02:38.500
which is gonna be two kilograms for M1

00:02:38.500 --> 00:02:42.970
plus six kilograms for M2
and what we get out of this

00:02:42.970 --> 00:02:46.560
is two times zero, zero plus six times 10

00:02:46.560 --> 00:02:50.946
is 60 kilogram centimeters
divided by two plus six

00:02:50.946 --> 00:02:53.380
is gonna be eight kilograms,

00:02:53.380 --> 00:02:56.742
which gives us 7.5 centimeters,

00:02:56.742 --> 00:03:00.570
so it's gonna be 7.5
centimeters from the point

00:03:00.570 --> 00:03:03.490
we called X equals zero,
which is right here,

00:03:03.490 --> 00:03:05.776
that's the location of the center of mass,

00:03:05.776 --> 00:03:08.940
so in other words, if you
connected these two spheres

00:03:08.940 --> 00:03:12.560
by a rod, a light rod and
you put a pivot right here,

00:03:12.560 --> 00:03:15.090
they would balance at
that point right there

00:03:15.090 --> 00:03:16.397
and just to show you, you might be like,

00:03:16.397 --> 00:03:19.347
"Wait, we can choose any
point as X equals zero,

00:03:19.347 --> 00:03:21.050
"won't we get a different number?"

00:03:21.050 --> 00:03:23.070
You will, so let's say you did this,

00:03:23.070 --> 00:03:26.000
instead of picking that as X equals zero,

00:03:26.000 --> 00:03:28.155
let's say we pick this
side as X equals zero,

00:03:28.155 --> 00:03:30.730
let's say we say X equals zero

00:03:30.730 --> 00:03:33.238
is this six kilogram mass's position,

00:03:33.238 --> 00:03:34.980
what are we gonna get then?

00:03:34.980 --> 00:03:37.710
We'll get that the location
of the center of mass

00:03:37.710 --> 00:03:38.949
for this calculation is gonna be,

00:03:38.949 --> 00:03:40.960
well, we'll have two kilograms,

00:03:40.960 --> 00:03:44.770
but now the location of the
two kilogram mass is not zero,

00:03:44.770 --> 00:03:45.920
it's gonna be if this is zero

00:03:45.920 --> 00:03:48.410
and we're considering
this way is positive,

00:03:48.410 --> 00:03:50.096
it's gonna be negative 10 centimeters,

00:03:50.096 --> 00:03:51.931
'cause it's 10 centimeters to the left,

00:03:51.931 --> 00:03:54.500
so this is gonna be
negative 10 centimeters

00:03:54.500 --> 00:03:57.380
plus six kilograms times,

00:03:57.380 --> 00:04:00.370
now the location of the
six kilogram mass is zero,

00:04:00.370 --> 00:04:02.130
using this convention and we divide

00:04:02.130 --> 00:04:03.690
by both of the masses added up,

00:04:03.690 --> 00:04:06.835
so that's still two
kilograms plus six kilograms

00:04:06.835 --> 00:04:07.726
and what are we gonna get?

00:04:07.726 --> 00:04:10.430
We're gonna get two times negative 10

00:04:10.430 --> 00:04:12.165
plus six times zero,
well, that's just zero,

00:04:12.165 --> 00:04:16.140
so it's gonna be negative
20 kilogram centimeters

00:04:16.140 --> 00:04:18.360
divided by eight kilograms

00:04:18.360 --> 00:04:21.059
gives us negative 2.5 centimeters,

00:04:21.059 --> 00:04:23.247
so you might be worried,
you might be like, "What?

00:04:23.247 --> 00:04:24.667
"We got a different answer.

00:04:24.667 --> 00:04:26.177
"The location can't change,

00:04:26.177 --> 00:04:28.400
"based on where we're measuring from,"

00:04:28.400 --> 00:04:31.160
and it didn't change, it's still
in the exact same position,

00:04:31.160 --> 00:04:34.200
because now this negative 2.5 centimeters

00:04:34.200 --> 00:04:37.129
is measured relative
to this X equals zero,

00:04:37.129 --> 00:04:40.050
so what's negative 2.5
centimeters from here?

00:04:40.050 --> 00:04:42.680
It's 2.5 centimeters to the left,

00:04:42.680 --> 00:04:46.240
which lo and behold is
exactly at the same point,

00:04:46.240 --> 00:04:49.580
since this was 7.5 and
this is negative 2.5

00:04:49.580 --> 00:04:51.960
and the whole thing is 10 centimeters,

00:04:51.960 --> 00:04:54.980
it gives you the exact same
location for the center of mass,

00:04:54.980 --> 00:04:57.570
it has to, it can't change based on

00:04:57.570 --> 00:05:00.586
whether you're calling this
point zero or this point zero,

00:05:00.586 --> 00:05:03.850
but you have to be careful and
consistent with your choice,

00:05:03.850 --> 00:05:06.438
any choice will work, but you
have to be consistent with it

00:05:06.438 --> 00:05:08.300
and you have to know at the end

00:05:08.300 --> 00:05:10.470
where is this answer measured from,

00:05:10.470 --> 00:05:12.520
otherwise you won't be able to interpret

00:05:12.520 --> 00:05:14.310
what this number means at the end.

00:05:14.310 --> 00:05:17.000
So recapping, you can use
the center of mass formula

00:05:17.000 --> 00:05:19.640
to find the exact location
of the center of mass

00:05:19.640 --> 00:05:21.100
between a system of objects,

00:05:21.100 --> 00:05:24.020
you add all the masses
times their positions

00:05:24.020 --> 00:05:26.010
and divide by the total mass,

00:05:26.010 --> 00:05:27.440
the position can be measured

00:05:27.440 --> 00:05:29.748
relative to any point
you call X equals zero

00:05:29.748 --> 00:05:32.550
and the number you get
out of that calculation

00:05:32.550 --> 00:05:35.370
will be the distance from X equals zero

00:05:35.370 --> 00:05:38.163
to the center of mass of that system.


Classifying shapes of distributions

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:01.490
- [Instructor] What we have here

00:00:01.490 --> 00:00:03.560
are six different distributions.

00:00:03.560 --> 00:00:05.080
And what we're gonna do with this video

00:00:05.080 --> 00:00:07.280
is think about how to classify them,

00:00:07.280 --> 00:00:09.300
or use the words people typically use

00:00:09.300 --> 00:00:11.078
to classify distributions.

00:00:11.078 --> 00:00:13.533
So, let's first look at this
distribution right over here,

00:00:13.533 --> 00:00:16.030
it's the distribution of
the lengths of houseflies.

00:00:16.030 --> 00:00:17.410
So, someone went out there and measured

00:00:17.410 --> 00:00:18.520
a bunch of houseflies.

00:00:18.520 --> 00:00:19.860
And then said, "Hey look,

00:00:19.860 --> 00:00:21.662
there's many houseflies that are between

00:00:21.662 --> 00:00:23.460
six tenths of a centimeter

00:00:23.460 --> 00:00:25.720
and six and a half
tenths of a centimeter."

00:00:25.720 --> 00:00:28.220
Looks like there's about
40 houseflies there.

00:00:28.220 --> 00:00:31.880
And then if you say between six
and a half and seven tenths,

00:00:31.880 --> 00:00:33.710
there's about 30 houseflies.

00:00:33.710 --> 00:00:35.820
And if you were to say
between five and a half tenths

00:00:35.820 --> 00:00:38.390
and six tenths, it looks like
it's about the same amount.

00:00:38.390 --> 00:00:41.100
This type of distribution
is usually described

00:00:41.100 --> 00:00:43.890
as being symmetric.

00:00:43.890 --> 00:00:45.630
Why is it called that?

00:00:45.630 --> 00:00:47.990
Because if you were to draw a line

00:00:47.990 --> 00:00:50.061
down the middle of this distribution,

00:00:50.061 --> 00:00:54.070
both sides look like mirror
images of each other.

00:00:54.070 --> 00:00:57.660
This one looks pretty exactly symmetric.

00:00:57.660 --> 00:00:59.410
But more typically when
you're collecting data,

00:00:59.410 --> 00:01:01.744
you'll see roughly
symmetric distributions.

00:01:01.744 --> 00:01:04.150
Now, here we have a distribution

00:01:04.150 --> 00:01:06.560
that gives us the dates on pennies.

00:01:06.560 --> 00:01:07.527
So, someone went out there,

00:01:07.527 --> 00:01:09.190
observed a bunch of pennies,

00:01:09.190 --> 00:01:10.450
looked at the dates on them.

00:01:10.450 --> 00:01:11.580
They saw many pennies,

00:01:11.580 --> 00:01:13.970
looks like a little bit
more than 55 pennies,

00:01:13.970 --> 00:01:16.950
had a date between 2010 and 2020.

00:01:16.950 --> 00:01:18.452
While very few pennies had a date

00:01:18.452 --> 00:01:21.224
older than 1980 on them.

00:01:21.224 --> 00:01:23.290
And this type of distribution

00:01:23.290 --> 00:01:26.580
when you have a tail to the left,

00:01:26.580 --> 00:01:27.730
you can see it right over here,

00:01:27.730 --> 00:01:29.646
you have a long tail to the left,

00:01:29.646 --> 00:01:34.010
this is known as a
left-skewed distribution.

00:01:34.010 --> 00:01:36.550
Left-skewed.

00:01:36.550 --> 00:01:38.490
Now in future videos,
we'll come up with more

00:01:38.490 --> 00:01:41.990
technical definitions of
what makes it left-skewed,

00:01:41.990 --> 00:01:43.345
but the way that you can recognize it is,

00:01:43.345 --> 00:01:47.150
you have the high points of
your distribution on the right,

00:01:47.150 --> 00:01:48.711
but then you have this long tail

00:01:48.711 --> 00:01:51.980
that skews it to the left.

00:01:51.980 --> 00:01:53.570
Now, the other side of a left-skewed,

00:01:53.570 --> 00:01:54.403
you might say, well,

00:01:54.403 --> 00:01:56.220
that would be a right-skewed distribution,

00:01:56.220 --> 00:01:58.890
and that's exactly what
we see right over here.

00:01:58.890 --> 00:02:01.580
This is a distribution
of state representatives,

00:02:01.580 --> 00:02:04.070
and as you can see, most of
the states in the United States

00:02:04.070 --> 00:02:06.500
have between zero and ten representatives.

00:02:06.500 --> 00:02:09.040
It looks like it's a little over 35.

00:02:09.040 --> 00:02:10.400
None of them actually have zero,

00:02:10.400 --> 00:02:12.092
they all have at least one representative,

00:02:12.092 --> 00:02:14.380
but they would fall into this bucket,

00:02:14.380 --> 00:02:17.942
while very few have more
than 50 representatives.

00:02:17.942 --> 00:02:20.930
So, here where the bulk of our
distribution is to the left,

00:02:20.930 --> 00:02:24.320
where we have this tail
that skews us to the right,

00:02:24.320 --> 00:02:29.320
this is known as a
right-skewed distribution.

00:02:29.830 --> 00:02:31.630
Now, if we look at this next distribution,

00:02:31.630 --> 00:02:32.463
what would this be?

00:02:32.463 --> 00:02:34.460
Pause this video and think about it.

00:02:34.460 --> 00:02:36.300
Well, this could be a
distribution of maybe

00:02:36.300 --> 00:02:37.823
someone went around
the office and surveyed

00:02:37.823 --> 00:02:40.550
how many cups of coffee each person drank,

00:02:40.550 --> 00:02:41.440
and if they found someone

00:02:41.440 --> 00:02:43.440
who drank one cup of coffee per day,

00:02:43.440 --> 00:02:44.590
maybe this would be them.

00:02:44.590 --> 00:02:45.930
If they found another person who drinks

00:02:45.930 --> 00:02:47.268
one cup of coffee, that's them,

00:02:47.268 --> 00:02:49.150
then they found three people who drank

00:02:49.150 --> 00:02:50.300
two cups of coffee.

00:02:50.300 --> 00:02:51.980
Well, this is a very similar situation

00:02:51.980 --> 00:02:54.130
to what we saw on the dates on pennies.

00:02:54.130 --> 00:02:57.870
A large amount of our data
fell into this right bucket

00:02:57.870 --> 00:02:59.060
of three cups of coffee,

00:02:59.060 --> 00:03:01.328
but then we have this tail to the left.

00:03:01.328 --> 00:03:04.894
So, this would be left-skewed.

00:03:04.894 --> 00:03:07.980
Now, these right two
distributions are interesting.

00:03:07.980 --> 00:03:10.230
One could argue that
this distribution here,

00:03:10.230 --> 00:03:11.636
which is telling us the number of days

00:03:11.636 --> 00:03:14.540
that we had different high temperatures,

00:03:14.540 --> 00:03:16.710
that this looks roughly symmetric,

00:03:16.710 --> 00:03:18.720
or actually even looks exactly symmetric.

00:03:18.720 --> 00:03:20.186
'Cause if you did that little exercise

00:03:20.186 --> 00:03:22.620
of drawing a dotted line down the middle,

00:03:22.620 --> 00:03:25.372
it looks like the two sides are
mirror images of each other.

00:03:25.372 --> 00:03:28.088
Now, that would not be
technically incorrect.

00:03:28.088 --> 00:03:31.215
But typically when you
see these two peaks,

00:03:31.215 --> 00:03:34.670
this would typically be called
a bi-modal distribution.

00:03:34.670 --> 00:03:37.065
So, even though bi-modal distributions

00:03:37.065 --> 00:03:40.180
can sometimes be symmetric
or roughly symmetric,

00:03:40.180 --> 00:03:41.790
you wanna be more precise,

00:03:41.790 --> 00:03:44.170
and here when you have these two peaks,

00:03:44.170 --> 00:03:45.630
that's where the bi comes from.

00:03:45.630 --> 00:03:48.040
You'd call it bi-modal,
and this makes sense

00:03:48.040 --> 00:03:50.570
because you have a lot of days

00:03:50.570 --> 00:03:52.660
that are warm that might
happen during the summer

00:03:52.660 --> 00:03:54.129
and you might have a lot
of days that are cold

00:03:54.129 --> 00:03:56.870
that are happening during the winter.

00:03:56.870 --> 00:03:58.360
Now, this last distribution here,

00:03:58.360 --> 00:04:00.470
the results from die rolls,

00:04:00.470 --> 00:04:04.143
one could argue as well that
this is roughly symmetric.

00:04:04.143 --> 00:04:06.690
It's not exact, it's
not perfectly symmetric,

00:04:06.690 --> 00:04:08.610
but when you look at this dotted line here

00:04:08.610 --> 00:04:09.950
on the left and the right sides

00:04:09.950 --> 00:04:11.593
it looks roughly symmetric.

00:04:11.593 --> 00:04:14.730
But a more exact classification here

00:04:14.730 --> 00:04:17.400
would be that it looks
approximately uniform.

00:04:17.400 --> 00:04:19.750
So, rather than calling it
a symmetric distribution,

00:04:19.750 --> 00:04:21.146
or a roughly symmetric distribution,

00:04:21.146 --> 00:04:23.410
most people would classify this

00:04:23.410 --> 00:04:26.583
as an approximately uniform distribution.


Writing equations of perpendicular lines (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.736
- [Instructor] Find the
equation of a line perpendicular

00:00:03.736 --> 00:00:08.480
to this line that passes to
the point two comma eight.

00:00:08.480 --> 00:00:10.180
So this first piece of information that

00:00:10.180 --> 00:00:13.210
it's perpendicular to that
line right over there.

00:00:13.210 --> 00:00:14.750
What does that tell us?

00:00:14.750 --> 00:00:16.280
Well if it's perpendicular to this line,

00:00:16.280 --> 00:00:19.840
it's slope has to be the
negative inverse of two-fifths.

00:00:19.840 --> 00:00:22.250
So its slope, the negative
inverse of two-fifths,

00:00:22.250 --> 00:00:24.860
the inverse of two-fifths is five.

00:00:24.860 --> 00:00:25.990
Let me do it in a better color.

00:00:25.990 --> 00:00:27.450
A nicer green.

00:00:27.450 --> 00:00:30.110
If this lines slope is
negative two-fifths,

00:00:30.110 --> 00:00:32.260
the equation of the line
we have to figure out

00:00:32.260 --> 00:00:33.140
that's perpendicular,

00:00:33.140 --> 00:00:34.860
the slope is going to be the inverse.

00:00:34.860 --> 00:00:35.693
So instead of two-fifths,

00:00:35.693 --> 00:00:37.970
it's gonna be five halves.

00:00:37.970 --> 00:00:39.430
And instead of being a negative,

00:00:39.430 --> 00:00:40.430
it's going to be a positive.

00:00:40.430 --> 00:00:42.390
So this is the negative inverse

00:00:42.390 --> 00:00:44.450
of negative two-fifths, right.

00:00:44.450 --> 00:00:46.100
You take the negative
sign, it becomes positive.

00:00:46.100 --> 00:00:47.190
You swap the five and the two,

00:00:47.190 --> 00:00:48.580
you get five halves.

00:00:48.580 --> 00:00:52.260
So that is going to have to be our slope.

00:00:52.260 --> 00:00:55.300
And we can actually use the
point slope form right here.

00:00:55.300 --> 00:00:57.750
It goes through this point right there.

00:00:57.750 --> 00:00:59.700
So let's use point slope form.

00:00:59.700 --> 00:01:04.700
Y minus this Y value which
has to be on the line.

00:01:04.890 --> 00:01:07.050
Is equal to our slope,

00:01:07.050 --> 00:01:11.360
five halves times X minus this X value.

00:01:11.360 --> 00:01:13.990
The X value when Y is equal to eight.

00:01:13.990 --> 00:01:16.600
And this is the equation of
the line in point slope form

00:01:16.600 --> 00:01:18.500
if you wanna put it in

00:01:18.500 --> 00:01:19.590
slope intercept form.

00:01:19.590 --> 00:01:22.110
You can just do a little bit of algebra.

00:01:22.110 --> 00:01:23.350
Algebraic manipulation.

00:01:23.350 --> 00:01:26.580
Y minus eight is equal to let's
distribute the five halves.

00:01:26.580 --> 00:01:31.580
So five halves X minus five
halves times two is just five.

00:01:32.950 --> 00:01:34.420
And then add eight to both sides.

00:01:34.420 --> 00:01:38.770
You get Y is equal to five halves X.

00:01:38.770 --> 00:01:40.170
Add eight to negative five.

00:01:40.170 --> 00:01:42.160
So plus three.

00:01:42.160 --> 00:01:44.663
And we are done.


Parallel & perpendicular lines from graph

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.150
- [Instructor] In this video,
we're gonna do a couple

00:00:02.150 --> 00:00:06.320
of examples that deal with
parallel and perpendicular lines.

00:00:06.320 --> 00:00:08.850
So you have parallel,

00:00:08.850 --> 00:00:12.138
you have perpendicular,

00:00:12.138 --> 00:00:15.412
and of course you have lines
that are neither parallel

00:00:15.412 --> 00:00:17.100
nor perpendicular.

00:00:17.100 --> 00:00:18.340
And just as a bit of a review,

00:00:18.340 --> 00:00:19.990
if you've never seen this before,

00:00:19.990 --> 00:00:22.640
parallel lines they never intersect.

00:00:22.640 --> 00:00:25.300
So let me draw some axes.

00:00:25.300 --> 00:00:27.570
So if those are my
coordinate axes right there,

00:00:27.570 --> 00:00:31.530
it's my X-axis, that is my Y-axis,

00:00:31.530 --> 00:00:34.530
if this is a line that
I'm drawing in magenta,

00:00:34.530 --> 00:00:38.600
a parallel line might
look something like this.

00:00:38.600 --> 00:00:40.420
It's not the exact same line,

00:00:40.420 --> 00:00:42.720
but they have the exact same slope.

00:00:42.720 --> 00:00:44.458
If this moves a certain amount,

00:00:44.458 --> 00:00:47.770
if this change in Y over
change in X is a certain amount

00:00:47.770 --> 00:00:51.140
this change change in Y over
change in X is the same amount.

00:00:51.140 --> 00:00:53.750
And that's why they never intersect.

00:00:53.750 --> 00:00:58.750
So they have the same slope.

00:00:58.860 --> 00:01:01.490
Parallel lines have the same slope.

00:01:01.490 --> 00:01:04.130
Perpendicular lines, depending
on how you wanna view it,

00:01:04.130 --> 00:01:06.253
they're kind of the opposite.

00:01:06.253 --> 00:01:10.200
Let's say this is some line,
a line that is perpendicular

00:01:10.200 --> 00:01:13.240
to that, will not only intersect the line,

00:01:13.240 --> 00:01:14.810
it won't only intersect the line,

00:01:14.810 --> 00:01:17.470
it will intersect it at a right angle.

00:01:17.470 --> 00:01:19.450
At a 90 degree angle.

00:01:19.450 --> 00:01:22.730
At a 90 degree angle, and
I'm not going to prove it

00:01:22.730 --> 00:01:24.270
for you here, I actually prove it

00:01:24.270 --> 00:01:25.900
in the linear algebra playlist,

00:01:25.900 --> 00:01:28.470
but a perpendicular lined slope,

00:01:28.470 --> 00:01:31.750
let's say this one right here,
let's say that yellow line

00:01:31.750 --> 00:01:35.560
has a slope of M, then this orange line

00:01:35.560 --> 00:01:37.690
that's perpendicular to the yellow line

00:01:37.690 --> 00:01:42.120
is going to have a slope
of negative one over M.

00:01:42.120 --> 00:01:44.936
Their slopes are gonna
be the negative inverse

00:01:44.936 --> 00:01:46.810
of each other.

00:01:46.810 --> 00:01:48.790
Now, given this information,

00:01:48.790 --> 00:01:51.140
let's look at a bunch
of lines and figure out

00:01:51.140 --> 00:01:53.660
if they're parallel, if
they're perpendicular,

00:01:53.660 --> 00:01:55.140
or if they are neither.

00:01:55.140 --> 00:01:58.700
And to do that, we just have
to keep looking at the slopes.

00:01:58.700 --> 00:02:00.710
So let's see, they say
one line passes through

00:02:00.710 --> 00:02:03.350
the points four and negative three

00:02:03.350 --> 00:02:05.470
and negative eight, zero.

00:02:05.470 --> 00:02:10.150
Another line passes through
the points negative one,

00:02:10.150 --> 00:02:11.080
negative one,

00:02:11.080 --> 00:02:13.507
and negative two, six.

00:02:13.507 --> 00:02:14.910
So let's figure out the
slopes in each of these lines.

00:02:14.910 --> 00:02:16.810
I'll first do this one in pink.

00:02:16.810 --> 00:02:19.640
So this slope right here, so line one.

00:02:19.640 --> 00:02:21.920
So I'll call it slope one.

00:02:21.920 --> 00:02:25.130
Slope one is, let's just say it is,

00:02:25.130 --> 00:02:26.730
well I'll take this as
the finishing point,

00:02:26.730 --> 00:02:31.130
so negative three minus
zero, remember change in Y,

00:02:31.130 --> 00:02:35.263
negative three minus zero, over four,

00:02:37.040 --> 00:02:39.720
minus negative eight.

00:02:39.720 --> 00:02:41.240
Four minus negative eight,

00:02:41.240 --> 00:02:43.890
so this is equal to negative three,

00:02:43.890 --> 00:02:46.860
over, this is the same
thing as four plus eight,

00:02:46.860 --> 00:02:51.860
negative three over 12, which
is equal to negative 1/4.

00:02:52.340 --> 00:02:55.860
Divide the numerator by
the denominator by three,

00:02:55.860 --> 00:02:57.870
that's this line, the first line.

00:02:57.870 --> 00:02:59.800
Now what about the second line?

00:02:59.800 --> 00:03:03.630
The second line, the slope
for that second line is,

00:03:03.630 --> 00:03:06.740
well let's take, here negative one,

00:03:06.740 --> 00:03:10.197
negative one minus six, over negative one,

00:03:12.198 --> 00:03:16.615
minus negative two is equal
to negative one minus six

00:03:19.700 --> 00:03:24.586
is negative seven, over
negative one minus negative two,

00:03:24.586 --> 00:03:26.920
that's the same thing as
negative one plus two,

00:03:26.920 --> 00:03:28.430
well that's just one.

00:03:28.430 --> 00:03:32.460
So the slope here is negative seven.

00:03:32.460 --> 00:03:34.970
So here their slopes are neither equal,

00:03:34.970 --> 00:03:36.470
so they're not parallel.

00:03:36.470 --> 00:03:38.560
Nor are they the negative
inverse of each other,

00:03:38.560 --> 00:03:41.010
so this is neither, so
this is neither parallel

00:03:41.010 --> 00:03:43.410
or perpendicular.

00:03:43.410 --> 00:03:45.490
Neither parallel nor perpendicular,

00:03:45.490 --> 00:03:48.070
so these two lines they intersect

00:03:48.070 --> 00:03:51.913
but they're not going to
intersect at a 90 degree angle.

00:03:52.840 --> 00:03:54.234
Let's do a couple more of these.

00:03:54.234 --> 00:03:59.190
So I have here, once
again, one line passing

00:03:59.190 --> 00:04:03.650
through these points, and
then another line passing

00:04:03.650 --> 00:04:04.483
through these points.

00:04:04.483 --> 00:04:06.890
So let's just look at their slopes.

00:04:06.890 --> 00:04:11.120
So this one in green, what's the slope?

00:04:11.120 --> 00:04:14.500
The slope of the green one,
I'll call that the first line.

00:04:14.500 --> 00:04:16.680
We could say, let's see change in Y,

00:04:16.680 --> 00:04:20.913
so we could do negative two minus 14,

00:04:22.190 --> 00:04:24.960
over, I did negative two
first, so I'll do one first,

00:04:24.960 --> 00:04:29.230
over one minus negative three.

00:04:29.230 --> 00:04:33.270
So negative two minus 14 is negative 16.

00:04:33.270 --> 00:04:35.820
One minus negative three
is the same thing as one

00:04:35.820 --> 00:04:37.650
plus three, that's over four,

00:04:37.650 --> 00:04:40.120
so this is negative four.

00:04:40.120 --> 00:04:43.770
Now what's the slope of that
second line right there?

00:04:43.770 --> 00:04:47.550
So we have the slope of that second line,

00:04:47.550 --> 00:04:49.830
let's say five minus,

00:04:49.830 --> 00:04:54.660
so we'll say five minus negative three,

00:04:54.660 --> 00:04:56.440
that's our change in Y,

00:04:56.440 --> 00:05:00.883
over negative two minus zero.

00:05:01.720 --> 00:05:04.460
So this is equal to five
minus negative three

00:05:04.460 --> 00:05:06.650
that's the same thing as five plus three,

00:05:06.650 --> 00:05:08.320
that's eight.

00:05:08.320 --> 00:05:11.610
And then negative two
minus zero is negative two,

00:05:11.610 --> 00:05:14.830
so this is also equal to negative four.

00:05:14.830 --> 00:05:17.440
So these two lines are parallel.

00:05:17.440 --> 00:05:20.450
These two lines are parallel.

00:05:20.450 --> 00:05:23.770
They have the exact same slope.

00:05:23.770 --> 00:05:25.400
And I encourage you to find the equations

00:05:25.400 --> 00:05:27.750
of both of these lines and
graph both of these lines

00:05:27.750 --> 00:05:31.003
and verify for yourself that
they are indeed parallel.

00:05:32.190 --> 00:05:33.733
Let's do this one.

00:05:34.597 --> 00:05:37.380
Once again, this is just an
exercise in finding slopes.

00:05:37.380 --> 00:05:39.330
So this first line has those points.

00:05:39.330 --> 00:05:41.210
Let's figure out it's slope.

00:05:41.210 --> 00:05:42.830
The slope of this first line.

00:05:42.830 --> 00:05:44.820
One line passes through these points,

00:05:44.820 --> 00:05:49.820
so let's say three minus negative three,

00:05:49.871 --> 00:05:51.610
that's our change in Y,

00:05:51.610 --> 00:05:55.483
over three minus negative six.

00:05:56.370 --> 00:05:59.440
So this is the same thing as
three plus three which is six

00:05:59.440 --> 00:06:02.300
over three plus six which is nine.

00:06:02.300 --> 00:06:04.703
So the first line has a slope of 2/3.

00:06:05.560 --> 00:06:07.750
What is the second line's slope?

00:06:07.750 --> 00:06:09.180
This is the second line there.

00:06:09.180 --> 00:06:12.010
That's the other line
passing through these points.

00:06:12.010 --> 00:06:14.820
So the other line's slope, let's see,

00:06:14.820 --> 00:06:19.003
we could say negative eight minus four,

00:06:20.490 --> 00:06:23.150
over two minus negative six.

00:06:26.342 --> 00:06:28.840
So what is this equal to?

00:06:28.840 --> 00:06:32.280
Negative eight minus four is negative 12.

00:06:32.280 --> 00:06:34.330
Two minus negative six,
that's the same thing

00:06:34.330 --> 00:06:36.260
as two plus six, right?

00:06:36.260 --> 00:06:37.750
The negatives cancel out.

00:06:37.750 --> 00:06:41.220
So it's negative 12 over
eight, which is the same thing

00:06:41.220 --> 00:06:43.850
if we divide the numerator
and the denominator by four,

00:06:43.850 --> 00:06:45.753
that's negative 3/2.

00:06:46.820 --> 00:06:51.820
Notice these guys are the
negative inverse of each other.

00:06:51.930 --> 00:06:55.870
If I take negative one over 2/3,

00:06:55.870 --> 00:07:00.588
that is equal to negative one times 3/2,

00:07:00.588 --> 00:07:03.010
which is equal to negative 3/2.

00:07:03.010 --> 00:07:05.470
These guys are the negative
inverses of each other.

00:07:05.470 --> 00:07:07.940
You swap the numerator and
denominator make 'em negative,

00:07:07.940 --> 00:07:09.560
and they become equal to each other,

00:07:09.560 --> 00:07:13.513
so these two lines are perpendicular.

00:07:15.170 --> 00:07:17.880
I encourage you to find the equations,

00:07:17.880 --> 00:07:19.080
I've already got the slopes for you,

00:07:19.080 --> 00:07:21.150
but find the equations
of both of these lines

00:07:21.150 --> 00:07:23.750
plot them and verify for yourself

00:07:23.750 --> 00:07:25.163
that they are perpendicular.


Worked example: Approximation with local linearity

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:01.410
- [Instructor] We're told the function f

00:00:01.410 --> 00:00:04.300
is twice differentiable
with f of two equals one,

00:00:04.300 --> 00:00:05.960
f prime of two is equal to four,

00:00:05.960 --> 00:00:08.790
and f prime prime of
two is equal to three.

00:00:08.790 --> 00:00:11.020
What is the value of the approximation

00:00:11.020 --> 00:00:15.630
of f of 1.9 using the line
tangent to the graph of f

00:00:15.630 --> 00:00:17.280
at x equals two?

00:00:17.280 --> 00:00:19.300
So pause this video and see
if you can figure this out.

00:00:19.300 --> 00:00:22.483
This is an actual question
from a past AP Calculus exam.

00:00:23.330 --> 00:00:24.720
All right now let's do this together.

00:00:24.720 --> 00:00:26.330
And if I was actually
doing this on an exam,

00:00:26.330 --> 00:00:28.160
I would just cut to the
chase and I would figure out

00:00:28.160 --> 00:00:31.440
the equation of the
tangent line at x equals

00:00:31.440 --> 00:00:33.560
to go through the point two comma one,

00:00:33.560 --> 00:00:34.650
and then I would figure out okay,

00:00:34.650 --> 00:00:36.140
when x is equal to 1.9,

00:00:36.140 --> 00:00:37.690
what is the value of y?

00:00:37.690 --> 00:00:39.270
And that would be my approximation.

00:00:39.270 --> 00:00:42.640
But for the sake of learning
and getting the intuition here,

00:00:42.640 --> 00:00:45.220
let's just make sure we
understand what's happening.

00:00:45.220 --> 00:00:46.900
So let me graph this.

00:00:46.900 --> 00:00:50.190
So let's say that's my y-axis,

00:00:50.190 --> 00:00:54.370
and then this is my x-axis.

00:00:54.370 --> 00:00:56.820
And this is x equals one.

00:00:56.820 --> 00:00:58.820
This is x equals two.

00:00:58.820 --> 00:01:01.230
This is y equals one.

00:01:01.230 --> 00:01:03.400
We know that the point two comma one

00:01:03.400 --> 00:01:06.060
is on the graph of y is equal to f of x.

00:01:06.060 --> 00:01:09.040
So we know that point
right over there is there.

00:01:09.040 --> 00:01:11.760
And we also know the
slope of the tangent line.

00:01:11.760 --> 00:01:13.920
The slope of the tangent line is four.

00:01:13.920 --> 00:01:16.630
So it's gonna look something like this.

00:01:16.630 --> 00:01:18.780
It's gonna probably even be
a little steeper than that.

00:01:18.780 --> 00:01:20.850
The slope of the tangent
line is gonna look

00:01:20.850 --> 00:01:22.390
something like that.

00:01:22.390 --> 00:01:24.480
And we don't know much more about it.

00:01:24.480 --> 00:01:26.230
We know the second derivative here.

00:01:26.230 --> 00:01:28.260
But what they're asking us
to do is without knowing

00:01:28.260 --> 00:01:29.650
what the function actually looks like,

00:01:29.650 --> 00:01:31.850
the function might look
something like this.

00:01:31.850 --> 00:01:33.670
Let me just draw something.

00:01:33.670 --> 00:01:36.660
So the function might look,

00:01:36.660 --> 00:01:39.210
might look something like this.

00:01:39.210 --> 00:01:41.890
We're trying to figure
out what f of 1.9 is.

00:01:41.890 --> 00:01:42.883
So if x is 1.9,

00:01:43.840 --> 00:01:47.080
f of 1.9, if that's the way
the function actually looked,

00:01:47.080 --> 00:01:48.600
might be this value right over here.

00:01:48.600 --> 00:01:50.020
But we don't know for
sure because we don't know

00:01:50.020 --> 00:01:51.580
much more about the function.

00:01:51.580 --> 00:01:53.650
But what they're suggesting for us to do

00:01:53.650 --> 00:01:55.470
is use this tangent line.

00:01:55.470 --> 00:01:58.120
If we know the equation
of this tangent line here,

00:01:58.120 --> 00:02:00.400
we could say well what does
that tangent line equal

00:02:00.400 --> 00:02:02.190
when x equals 1.9?

00:02:02.190 --> 00:02:03.260
When x equals 1.9,

00:02:03.260 --> 00:02:05.770
it equals that point right over there.

00:02:05.770 --> 00:02:08.180
And then we could use
that as our approximation

00:02:08.180 --> 00:02:10.000
for f of 1.9.

00:02:10.000 --> 00:02:12.350
Well to do that we need
to know the equation

00:02:12.350 --> 00:02:14.000
of the tangent line.

00:02:14.000 --> 00:02:16.520
And we could do that in point-slope form.

00:02:16.520 --> 00:02:19.130
We would just have to say y minus

00:02:19.130 --> 00:02:21.520
the y value that we know is on that line,

00:02:21.520 --> 00:02:24.150
the point two comma one
we know is on that line,

00:02:24.150 --> 00:02:25.820
so y minus one

00:02:25.820 --> 00:02:29.030
is going to be equal to the
slope of our tangent line,

00:02:29.030 --> 00:02:31.150
which we know is going
to be equal to four,

00:02:31.150 --> 00:02:34.470
times x minus the x value that corresponds

00:02:34.470 --> 00:02:37.200
to that y value, so x minus two.

00:02:37.200 --> 00:02:39.970
So now we just have to
substitute x equals 1.9

00:02:39.970 --> 00:02:42.660
to get our approximation for f of 1.9.

00:02:42.660 --> 00:02:45.410
So we would say y minus
one is equal to four times

00:02:45.410 --> 00:02:48.100
1.9 minus two.

00:02:48.100 --> 00:02:52.520
1.9 minus two is negative 0.1.

00:02:52.520 --> 00:02:55.600
And let's see four times negative 0.1,

00:02:55.600 --> 00:02:59.100
this all simplifies to negative 0.4.

00:02:59.100 --> 00:03:00.710
Now you add one to both sides,

00:03:00.710 --> 00:03:02.720
you get y is equal to,

00:03:02.720 --> 00:03:05.100
if you add one here you're gonna get 0.6.

00:03:05.100 --> 00:03:07.010
So this, I didn't draw it quite to scale,

00:03:07.010 --> 00:03:10.080
0.6 might be something
closer to right around there,

00:03:10.080 --> 00:03:10.913
but there you go.

00:03:10.913 --> 00:03:14.780
That is our approximation for f of 1.9,

00:03:14.780 --> 00:03:16.060
which is choice B.

00:03:16.060 --> 00:03:17.040
And we're done.

00:03:17.040 --> 00:03:18.830
And one interesting thing to notice,

00:03:18.830 --> 00:03:21.260
we didn't have to use all
the information they gave us.

00:03:21.260 --> 00:03:23.270
We did not have to use this information

00:03:23.270 --> 00:03:26.020
about the second derivative
in order to solve the problem.

00:03:26.020 --> 00:03:28.030
So if you ever find
yourself in that situation,

00:03:28.030 --> 00:03:30.030
don't doubt yourself too much.

00:03:30.030 --> 00:03:33.823
Because they will sometimes
give you unneeded information.


Exchange rate primer

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:01.960
- [Instructor] You are
already likely familiar

00:00:01.960 --> 00:00:05.170
with the notion that a dollar in the U.S.,

00:00:05.170 --> 00:00:08.660
a U.S. dollar is not
necessarily equivalent

00:00:08.660 --> 00:00:13.070
to one currency unit in
another country, say one Euro.

00:00:13.070 --> 00:00:15.260
And so if you were
traveling to that country,

00:00:15.260 --> 00:00:16.650
and you are holding dollars,

00:00:16.650 --> 00:00:19.280
and you needed to exchange
those dollars into Euros,

00:00:19.280 --> 00:00:20.300
you'd say, well how many Euros

00:00:20.300 --> 00:00:22.080
am I going to get for my dollars?

00:00:22.080 --> 00:00:23.490
Or if you were European,

00:00:23.490 --> 00:00:24.677
and you had a certain number of Euro,

00:00:24.677 --> 00:00:26.550
and you said how many
dollars am I going to get

00:00:26.550 --> 00:00:31.310
for my Euros, you would be
thinking about the exchange rate.

00:00:31.310 --> 00:00:33.410
And it's not just about
if you're a tourist

00:00:33.410 --> 00:00:35.540
or you're traveling,
it's also very important

00:00:35.540 --> 00:00:37.300
if you're an importer and exporter.

00:00:37.300 --> 00:00:39.760
If you make, say cars,
in the United States,

00:00:39.760 --> 00:00:42.400
where all of your costs are in dollars,

00:00:42.400 --> 00:00:44.300
but you're selling them in Europe

00:00:44.300 --> 00:00:46.720
where you're getting your
money in terms of Euros,

00:00:46.720 --> 00:00:47.880
and then you have to convert them

00:00:47.880 --> 00:00:49.880
back to dollars to pay your costs,

00:00:49.880 --> 00:00:52.800
well the idea of an
exchange rate matters a lot.

00:00:52.800 --> 00:00:54.790
And so, first let's make tangible

00:00:54.790 --> 00:00:56.580
what an exchange rate even looks like.

00:00:56.580 --> 00:00:59.300
And there's many different
ways of representing it.

00:00:59.300 --> 00:01:00.580
The first thing to appreciate,

00:01:00.580 --> 00:01:02.300
if someone says what's the exchange rate

00:01:02.300 --> 00:01:05.400
for the U.S. dollar, your
next reaction should be,

00:01:05.400 --> 00:01:08.090
with regards to what other currency?

00:01:08.090 --> 00:01:09.730
So if I said, what's the exchange rate

00:01:09.730 --> 00:01:12.960
of the U.S. dollar with
respect to the Euro,

00:01:12.960 --> 00:01:17.080
you could express that as one U.S. dollar,

00:01:17.080 --> 00:01:22.080
U.S. dollar, is equal to how many Euros?

00:01:22.890 --> 00:01:24.340
And at the time of this video,

00:01:24.340 --> 00:01:28.860
it's actually 0.87 Euros.

00:01:28.860 --> 00:01:31.350
Now another way to express this is

00:01:31.350 --> 00:01:36.350
one Euro, one Euro is equal
to how many U.S. dollars?

00:01:39.490 --> 00:01:41.400
And if you know this first equation,

00:01:41.400 --> 00:01:42.900
you can figure out the second one.

00:01:42.900 --> 00:01:46.220
You would essentially just
divide both sides by 0.87.

00:01:46.220 --> 00:01:49.080
If you divide this by 0.87 you get one.

00:01:49.080 --> 00:01:51.450
If you divide one by 0.87,

00:01:51.450 --> 00:01:54.273
you are going to get approximately 1.15.

00:01:55.280 --> 00:01:57.230
And so at the time of this video,

00:01:57.230 --> 00:02:00.900
1.15 U.S. dollars is equal to one Euro.

00:02:00.900 --> 00:02:02.400
Or another way to think about it is,

00:02:02.400 --> 00:02:04.930
if you took a dollar
and 15 cents in the U.S.

00:02:04.930 --> 00:02:08.550
and convert it to Euros,
you would get one Euro.

00:02:08.550 --> 00:02:10.570
And there's other ways of expressing it.

00:02:10.570 --> 00:02:14.370
You could say what is the
price of a U.S. dollar,

00:02:14.370 --> 00:02:19.370
so price of the U.S.
dollar in terms of Euro,

00:02:21.440 --> 00:02:25.760
in terms of Euro.

00:02:25.760 --> 00:02:27.600
So this is, how much would it cost

00:02:27.600 --> 00:02:30.760
to buy one U.S. dollar in terms of Euros?

00:02:30.760 --> 00:02:32.550
Well you could just
look at this over here,

00:02:32.550 --> 00:02:34.080
and you could say, well,

00:02:34.080 --> 00:02:37.880
one U.S. dollar is going
to cost 0.87 Euros.

00:02:37.880 --> 00:02:42.320
Or you could say 0.87 Euros,

00:02:42.320 --> 00:02:47.320
Euros per U.S., per U.S. dollar.

00:02:47.910 --> 00:02:49.340
And if you were to go
the other way around,

00:02:49.340 --> 00:02:52.760
if you wanted the price of the Euro,

00:02:52.760 --> 00:02:57.760
price of the Euro in terms,

00:02:59.000 --> 00:03:04.000
terms of the U.S. dollar, U.S. dollar,

00:03:05.220 --> 00:03:07.010
well that would be the
other right way around.

00:03:07.010 --> 00:03:09.490
One Euro would cost $1.15.

00:03:09.490 --> 00:03:14.450
So this would be 1.15 U.S. dollars,

00:03:14.450 --> 00:03:17.350
dollars per Euro.

00:03:17.350 --> 00:03:19.110
Now with all of this out of the way,

00:03:19.110 --> 00:03:21.290
let's actually do some calculations.

00:03:21.290 --> 00:03:23.670
Let's imagine that in the United States

00:03:23.670 --> 00:03:26.220
a yellow box goes for $20.

00:03:26.220 --> 00:03:28.800
These yellow boxes are in high demand.

00:03:28.800 --> 00:03:33.160
Someone from Europe wants to
buy one of these yellow boxes,

00:03:33.160 --> 00:03:35.600
and they need to buy
it for 20 U.S. dollars,

00:03:35.600 --> 00:03:37.200
but they only have Euro.

00:03:37.200 --> 00:03:39.830
How many Euro would they need in order

00:03:39.830 --> 00:03:44.120
to buy that box for 20 U.S. dollars.

00:03:44.120 --> 00:03:48.070
Well if one U.S. dollar
is equal 0.87 Euros,

00:03:48.070 --> 00:03:51.690
then 20 U.S. dollars is going
to be 20 times as many Euros.

00:03:51.690 --> 00:03:53.650
So we could say $20

00:03:54.860 --> 00:03:59.860
times 0.87 Euro per dollar.

00:03:59.960 --> 00:04:03.420
This symbol right over
here is shorthand for Euro.

00:04:03.420 --> 00:04:06.130
And so you are going to get, let's see,

00:04:06.130 --> 00:04:09.300
that would be 17, approximately,

00:04:09.300 --> 00:04:13.530
approximately 17.40 Euro.

00:04:17.810 --> 00:04:20.000
And if it went the other way around.

00:04:20.000 --> 00:04:23.400
Let's say that red boxes
are available for purchase

00:04:23.400 --> 00:04:26.030
only in Europe for 20 Euro.

00:04:26.030 --> 00:04:27.700
An American tourist goes there,

00:04:27.700 --> 00:04:29.870
and wants to figure out
how many U.S. dollars

00:04:29.870 --> 00:04:31.850
would they have to convert
in order to be able

00:04:31.850 --> 00:04:35.253
to pay for one of these
red boxes that are 20 Euro?

00:04:36.090 --> 00:04:40.260
Well here we see one Euro is
equal to 1.15 U.S. dollars.

00:04:40.260 --> 00:04:45.260
So here you would say 20
Euro times 1.15 U.S. dollars,

00:04:48.760 --> 00:04:52.990
dollar per Euro, and so
what is this going to be?

00:04:52.990 --> 00:04:56.210
This is going to be 20 times 1.15,

00:04:56.210 --> 00:04:59.450
which is going to be 23 U.S. dollars.

00:04:59.450 --> 00:05:02.850
23 U.S. dollars.

00:05:02.850 --> 00:05:05.990
Now another idea, a related idea to this,

00:05:05.990 --> 00:05:07.780
is this notion that currencies can

00:05:07.780 --> 00:05:10.570
appreciate and depreciate.

00:05:10.570 --> 00:05:15.060
So let's say that the Euro
appreciates, appreciates,

00:05:15.060 --> 00:05:17.720
and it always matters
relative to something.

00:05:17.720 --> 00:05:22.720
So you say relative to the dollar.

00:05:24.370 --> 00:05:26.930
So the Euro appreciates
relative to the dollar.

00:05:26.930 --> 00:05:28.700
Come up with an exchange rate

00:05:28.700 --> 00:05:31.790
that would show the Euro
appreciating relative to the dollar

00:05:31.790 --> 00:05:33.323
with this as a starting point.

00:05:34.210 --> 00:05:36.370
So one scenario where this is happening

00:05:36.370 --> 00:05:38.810
is if this was where you are starting,

00:05:38.810 --> 00:05:41.160
one Euro would now be more U.S. dollars,

00:05:41.160 --> 00:05:43.730
more than 1.15 U.S. dollars.

00:05:43.730 --> 00:05:45.960
So you have one Euro,

00:05:45.960 --> 00:05:50.210
would now be equal to, maybe
it's equal to $1.20 now.

00:05:50.210 --> 00:05:54.073
So it's now equal to $1.20.

00:05:56.440 --> 00:05:59.480
Now if someone said the
dollar appreciates relative

00:05:59.480 --> 00:06:02.070
to the Euro, then it would
go the other way around.

00:06:02.070 --> 00:06:05.330
That might be one U.S.
dollars is now equivalent

00:06:05.330 --> 00:06:10.330
to .9 Euros, or maybe
one Euro, or 1.05 Euros.

00:06:10.530 --> 00:06:14.190
And similarly, if someone
says that the Euro

00:06:14.190 --> 00:06:19.190
depreciates, depreciates relative,

00:06:19.630 --> 00:06:22.380
relative to the dollar.

00:06:22.380 --> 00:06:23.250
That's another way of saying

00:06:23.250 --> 00:06:26.230
that the dollar has appreciated
relative to the Euro.

00:06:26.230 --> 00:06:28.980
So this could be a situation where

00:06:28.980 --> 00:06:31.180
if the Euro's depreciating
relative to the dollar

00:06:31.180 --> 00:06:33.560
from this place right over here,

00:06:33.560 --> 00:06:37.681
one Euro, one Euro is now
equal to fewer dollars.

00:06:37.681 --> 00:06:39.381
And maybe it's now equal to $1.05.

00:06:41.670 --> 00:06:43.900
And with this as a starting point,

00:06:43.900 --> 00:06:46.450
this would also be a
situation where the dollar

00:06:46.450 --> 00:06:50.420
has appreciated relative to the Euro.

00:06:50.420 --> 00:06:51.430
So I'll leave you there.

00:06:51.430 --> 00:06:53.210
This is a primer on exchange rates.

00:06:53.210 --> 00:06:55.010
It can get a little bit confusing

00:06:55.010 --> 00:06:57.150
because you're talking for
the price of a currency

00:06:57.150 --> 00:06:59.180
in terms of another currency.

00:06:59.180 --> 00:07:01.340
But I encourage you to
keep thinking about things.

00:07:01.340 --> 00:07:03.840
Keep doing these little thought
experiments in your head,

00:07:03.840 --> 00:07:05.130
and thinking about, okay,

00:07:05.130 --> 00:07:07.910
if a Euro is equal to more than a dollar,

00:07:07.910 --> 00:07:10.450
if something's worth a
certain number of Euros,

00:07:10.450 --> 00:07:14.540
it should be a larger
number in terms of dollars.

00:07:14.540 --> 00:07:17.690
Similarly, if a dollar
is less than one Euro,

00:07:17.690 --> 00:07:20.410
if something costs a
certain number of dollars,

00:07:20.410 --> 00:07:24.363
it should be a smaller
number in terms of Euros.