Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)

WEBVTT
Kind: captions
Language: en

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Welcome back.

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We'll now try to see what
trigonometric identity we can

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come up with if we start off
with cosine of alpha plus beta.

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Let's see if we can rewrite
this as another combination

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of cosines and sines
of alpha and beta.

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So let's get started.

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And if you've already watched
the sine equivalent of this,

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this proof will be pretty
similar in how we operate.

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And we get a kind
of similar answer.

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And something interesting is to
kind of compare the difference

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between sine of alpha plus beta
and cosine of alpha plus

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beta after we're done.

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So just like that last proof,
let's say that this angle-- no,

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that color isn't bright enough.

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Let's do yellow.

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Let's say that this angle
right here is alpha, and this

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angle right here is beta.

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Right?

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We want to know cosine
of alpha plus beta.

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So alpha plus beta is this
large angle right here.

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Right?

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So what's the cosine of that?

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SOH, CAH, TOA.

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So cosine is adjacent
over hypotenuse.

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Right?

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SOH, CAH, TOA.

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CAH.

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So cosine is equal to
adjacent over hypotenuse.

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So for this large angle,
what's the adjacent?

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It's line AC.

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So that equals-- line AC,
that's the adjacent.

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The length of line AC over
the length of line--

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what's the hypotenuse?

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AB, right?

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All right.

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AC over AB.

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Now let's see what we
can do with this.

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AC-- adjacent over hypotenuse.

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Now can we write AC in any
other interesting way-- a

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combination of some of the
other lines on this very

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fortunately designed graph?

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Let's see.

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Well isn't AC the same thing
as AF, this big line,

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minus-- what is this?

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This is a D, right?

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OK.

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That's a D.

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Let me rewrite that as a D.

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D as in dog.

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There you go.

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So AF minus DE.

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Right?

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Oh, I forgot to
draw some things.

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We assume that this line is
perpendicular to that line.

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We assume this line is
perpendicular to that line.

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We assume that line is
perpendicular to that line.

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Right?

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And then by definition, that is
because we drew it that way.

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But anyway.

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So now you know that this line
is parallel to this line and

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this line is perpendicular.

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So we know that AF, this long
line, minus DE is equal to AC.

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Does that make sense?

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AF, this big line, minus
the shorter line is

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the same thing as AC.

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Right?

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So let me write that down.

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That equals AF minus DE,
all of that over AB.

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And then, of course, we can
rewrite that as-- and I'm going

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to switch to some different
colors-- as AF over AB.

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Let me switch to maybe green.

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Minus DE over AB.

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So we have now AF over
AB minus DE over AB.

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And those are kind of
nonsensical ratios to me.

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Wouldn't it be great if
we could express it

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somehow as AF over AE?

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Because then we could say well
that's cosine of alpha, and

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do something from there.

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Well let's try.

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So let's try to rewrite
this first expression.

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So I'll switch back to
the purple just so you

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know where this first
expression is coming from.

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Let's see if we can break
this down as AF over AE

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times something else.

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Well, we could just
algebraically do it.

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That's equivalent to AF over--
I might run out of space--

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over AE times AE over AB.

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And you're saying, Sal where
did you get that from?

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Well, you can kind of say my
motivation was to have AF

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as a ratio of over AE.

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And I just set it up so that
the multiplication would cancel

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out, because the AE's would
cancel out and you'd be

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left with AF over AB.

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Right?

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So this is a reasonable
thing to do.

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I hope you see.

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And let me switch to the green
and do something similar.

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DE over AB doesn't
make much sense.

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But if I could maybe do DE over
BE, then if this angle I can

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see is similar to alpha or
beta then maybe I can

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make some progress.

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So let's say DE over BE times--
and we'll do the same thing.

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You just have to multiply
times BE over AB.

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And just like in that sine
proof, and we'll do the same

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thing here, let's figure out
what this angle up here is.

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Right?

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Because if we know that then
these ratios become useful.

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So if this angle is an alpha,
then we know that this

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angle right here is alpha.

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Right?

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Because DE-- because this
line-- is parallel to AF.

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You learned that in geometry.

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And if this angle is alpha we
know that this angle right here

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is 90 minus alpha, because
it's complementary.

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Right?

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Because this whole angle
is 90 degrees, so this

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is 90 minus alpha.

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And since this angle 90 minus
alpha, this angle 90, and this

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angle add up to 180, we could
figure out that this is alpha.

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And if you don't believe me add
up alpha plus 90 plus 90 minus

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alpha, and you will
get 180 degrees.

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So this angle up here,
angle DBE, is alpha.

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So that's very interesting.

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So can we rewrite these ratios
as the sines or cosines

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of alpha's or beta's?

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Well, let's try.

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Let me switch back to purple.

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So that equals--
what is AF over AE?

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Well if we look at this right
triangle, that's the adjacent

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over the hypotenuse for alpha.

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Right?

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Adjacent over hypotenuse,
that's cosine.

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So it's cosine of alpha.

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And what's AE over AB?

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Well, they're similar.

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If we look at this big right
triangle right here that is the

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adjacent over the hypotenuse
for beta, so it's

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cosine of beta.

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Switch my colors.

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Minus DE over BE.

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Well this is alpha, right?

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Now there's a little smudge.

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You probably can't read it.

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But that was alpha.

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We showed that that was alpha.

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So DE is the opposite and
BE is the hypotenuse.

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Opposite over hypotenuse
is sine, right?

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So that's sine of alpha.

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And what is BE over AB?

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Look at this triangle again.

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Well, for beta that is
BE is the opposite and

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AB is the hypotenuse.

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So opposite over
hypotenuse for beta.

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So it's the sine of beta.

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Times the sine of beta.

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I'm running out of space.

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I have to go to another line.

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Pretty neat.

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I'll rewrite everything in
a new and exciting color.

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OK.

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Let me do it in light blue.

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So we now know that the cosine
of alpha plus beta is equal to

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the cosine of both of them
multiplied-- so cosine of

00:09:00.970 --> 00:09:10.070
alpha, cosine of beta-- minus
the sine of both of

00:09:10.070 --> 00:09:10.880
them multiplied.

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Minus sine of alpha
times sine of beta.

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I hope you found that
as satisfying as I do.

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See you in the next
presentation.


Proof: sin(a+b)  = (cos a)(sin b) + (sin a)(cos b)

WEBVTT
Kind: captions
Language: en

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Welcome back.

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I'm now going to do a proof
of a trig identity, which

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I think is pretty amazing.

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Although, I think, the
proof isn't that obvious.

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And I'll have to admit ahead of
time, this isn't something that

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would have occurred
to me naturally.

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I wouldn't have naturally
drawn this figure just

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to start off with.

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Let's just say we want to
figure out some other way to

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write the sine of alpha plus
beta, where alpha and beta are

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let's say, two separate angles.

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So if I had the sine of 40 and
50 degrees, I'd want to know--

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this would obviously be the
sine of 90, which is easy.

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But could I rewrite that as
some combination of the sine

00:00:47.040 --> 00:00:49.280
of 40 and the sine
of 50 or whatever?

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I think you'll see
where this is going.

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So let's go back to this
diagram and let's say

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that this-- let me
pick a better color.

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Let's say that this is angle
alpha and this is angle beta.

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Than this whole angle right
here is angle alpha plus beta.

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So we want to figure out the
sine of alpha plus beta.

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Well, the sine of alpha plus
beta, the sine of this

00:01:16.810 --> 00:01:19.120
whole angle, opposite
over hypotenuse.

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Opposite this whole angle is if
we use this right angle-- or

00:01:24.440 --> 00:01:26.830
this right triangle,
triangle BAC.

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Opposite is BC, so
that equals BC.

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I'll draw a little
line over it.

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BC over the hypotenuse, AB.

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BC over AB is the sine
of alpha plus beta.

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Well, can be write BC
over AB differently?

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Let's see if we can.

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And probably, the person who
first figured out this proof

00:01:56.920 --> 00:01:58.440
was just playing around.

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They drew this diagram,
they said, can I write

00:02:01.090 --> 00:02:02.480
BC any differently?

00:02:02.480 --> 00:02:08.730
Well BC-- this whole length--
is the sum of BD and EF.

00:02:08.730 --> 00:02:11.070
And we know that because this
is a horizontal line right now

00:02:11.070 --> 00:02:12.620
and you can figure that out
just by looking at all

00:02:12.620 --> 00:02:13.750
the right angles.

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But this is a horizontal line.

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So BC is the same
thing is BD plus EF.

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Let's write that one down.

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BC is the same thing
as BD plus EF.

00:02:34.050 --> 00:02:38.290
And then still, all
of that, over AB.

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All I did is I rewrote BC as a
sum of this segment and this

00:02:42.730 --> 00:02:44.700
segment, which should make
sense to you, hopefully.

00:02:44.700 --> 00:02:52.970
And then we can of course,
rewrite that as equal to BD

00:02:52.970 --> 00:03:05.460
over AB plus EF over AB.

00:03:05.460 --> 00:03:13.060
So BD over AB plus EF over AB.

00:03:13.060 --> 00:03:15.020
And these are kind of
nonsensical ratios, right?

00:03:15.020 --> 00:03:16.920
BD over AB, what can
I do with that?

00:03:16.920 --> 00:03:19.420
And EF over AB, what
can I do with that?

00:03:19.420 --> 00:03:23.190
Wouldn't it be more interesting
if I could do like BD over BE.

00:03:23.190 --> 00:03:25.520
That'd be an interesting ratio
because that would be a

00:03:25.520 --> 00:03:27.500
segment over its hypotenuse.

00:03:27.500 --> 00:03:30.400
So let's see if we can rewrite
it somehow like that.

00:03:30.400 --> 00:03:33.610
Well, we could just do
it mathematically.

00:03:33.610 --> 00:03:52.900
We could say this is equal to
BD over BE times BE over AB.

00:03:52.900 --> 00:03:54.560
So this might seem
non-intuitive to you, but

00:03:54.560 --> 00:03:55.820
it kind of makes sense.

00:03:55.820 --> 00:03:57.505
We didn't pick BE
completely arbitrarily.

00:03:57.505 --> 00:04:01.750
We said we know what BD is, so
let me pick another side that I

00:04:01.750 --> 00:04:05.130
can do something maybe
with real trig ratios.

00:04:05.130 --> 00:04:10.330
And so I said BD over
BE times BE over AB is

00:04:10.330 --> 00:04:12.380
equal to BD over AB.

00:04:12.380 --> 00:04:14.190
I hope I don't confuse you
with all these letters.

00:04:14.190 --> 00:04:15.210
But that makes sense, right?

00:04:15.210 --> 00:04:16.850
Because these two terms
would just cancel out.

00:04:16.850 --> 00:04:18.430
If we're just multiplying these
fractions then you would

00:04:18.430 --> 00:04:21.770
get back to this top term.

00:04:21.770 --> 00:04:24.090
Let me actually make sure
that you understand

00:04:24.090 --> 00:04:25.240
that this-- whoops.

00:04:25.240 --> 00:04:29.630
That this term and this
term are the same thing.

00:04:29.630 --> 00:04:31.810
And now let's do
that second term.

00:04:31.810 --> 00:04:34.960
We know EF, wouldn't it be good
if we could relate EF to

00:04:34.960 --> 00:04:37.570
something, like it's the
hypotenuse of this

00:04:37.570 --> 00:04:38.280
right triangle?

00:04:38.280 --> 00:04:39.220
Like AE.

00:04:39.220 --> 00:04:40.075
So let's do that.

00:04:42.810 --> 00:04:44.750
So let's put the
plus sign there.

00:04:44.750 --> 00:05:01.450
EF over AB is the same thing as
EF over AE times AE over AB.

00:05:01.450 --> 00:05:03.480
Once again, we're just
multiplying fractions.

00:05:03.480 --> 00:05:06.170
These would cancel out and
you would get this again.

00:05:06.170 --> 00:05:10.760
Let me make sure you understand
that this term is the

00:05:10.760 --> 00:05:11.660
same thing as this term.

00:05:11.660 --> 00:05:13.560
And you can just multiple out
the fractions and that's

00:05:13.560 --> 00:05:15.570
what you would get.

00:05:15.570 --> 00:05:19.920
Now before we progress
with this whole line of

00:05:19.920 --> 00:05:20.900
thought that we're doing.

00:05:20.900 --> 00:05:22.860
Let's see if we could figure
out something else interesting

00:05:22.860 --> 00:05:26.990
about this strange set of
triangles and shapes

00:05:26.990 --> 00:05:27.850
that I've drawn.

00:05:27.850 --> 00:05:30.060
It's actually pretty neat.

00:05:30.060 --> 00:05:36.390
IF this angle is alpha--
we have line AF.

00:05:36.390 --> 00:05:38.800
EF is perpendicular
to it, right?

00:05:38.800 --> 00:05:41.450
And DE is perpendicular
to EF, right?

00:05:41.450 --> 00:05:45.260
So DE, this line, and
AF are parallel.

00:05:45.260 --> 00:05:51.110
Since AF is parallel to DE and
then, AE intersects both,

00:05:51.110 --> 00:05:52.350
we know that, what is that?

00:05:52.350 --> 00:05:53.460
The inner angles?

00:05:53.460 --> 00:05:55.780
Yeah, I think that's
called inner angles

00:05:55.780 --> 00:05:56.730
with parallel lines.

00:05:56.730 --> 00:06:01.080
That this is also
equal to alpha.

00:06:01.080 --> 00:06:04.330
You can imagine long parallel
line here, long parallel here,

00:06:04.330 --> 00:06:05.890
and then this line
intersects both.

00:06:05.890 --> 00:06:08.500
So if this is a little
confusing maybe you want to

00:06:08.500 --> 00:06:11.880
review a little bit of the
parallel line geometry, but I

00:06:11.880 --> 00:06:13.150
think this might make sense.

00:06:13.150 --> 00:06:17.380
So if this angle is alpha,
then this angle right here

00:06:17.380 --> 00:06:19.300
is complementary to it.

00:06:19.300 --> 00:06:20.740
So it's 90 minus alpha.

00:06:23.470 --> 00:06:26.600
And if this angle is
90 minus alpha, this

00:06:26.600 --> 00:06:28.480
angle is obviously 90.

00:06:28.480 --> 00:06:31.210
Then we know that this angle
plus this angle plus this

00:06:31.210 --> 00:06:32.460
angle has to equal 180.

00:06:32.460 --> 00:06:35.620
So we know that this
is equal to alpha.

00:06:35.620 --> 00:06:38.980
If that doesn't make sense to
you, think about this: alpha

00:06:38.980 --> 00:06:44.070
plus 90 minus alpha plus
90-- that's a minus.

00:06:44.070 --> 00:06:45.370
Minus alpha.

00:06:45.370 --> 00:06:47.120
Plus 90 is what?

00:06:47.120 --> 00:06:48.760
Alpha plus 90 minus alpha.

00:06:48.760 --> 00:06:51.920
So this minus alpha and alpha
cancel out and you just have 90

00:06:51.920 --> 00:06:54.080
plus 90 and that equals 180.

00:06:54.080 --> 00:06:55.775
So we know that this angle
right here, I know it's

00:06:55.775 --> 00:06:57.980
getting really small and
probably hard to read.

00:06:57.980 --> 00:07:01.940
We know that this
angle here is alpha.

00:07:01.940 --> 00:07:03.730
So let's get back to what
we were progressing,

00:07:03.730 --> 00:07:05.060
what we were doing here.

00:07:05.060 --> 00:07:09.190
So what is BD over BE?

00:07:09.190 --> 00:07:12.850
BD over BE.

00:07:12.850 --> 00:07:15.990
Well, that's the adjacent
to this alpha, which is

00:07:15.990 --> 00:07:17.690
the same angle really.

00:07:17.690 --> 00:07:23.530
BD over BE, so it's
adjacent over hypotenuse.

00:07:23.530 --> 00:07:24.630
Cosine.

00:07:24.630 --> 00:07:28.780
So that is equal to
the cosine of alpha.

00:07:32.880 --> 00:07:35.360
And what's BE over AB?

00:07:39.900 --> 00:07:44.770
Well, if we look at this larger
right triangle, that is the

00:07:44.770 --> 00:07:50.230
opposite of beta times
its hypotenuse.

00:07:50.230 --> 00:07:52.620
So what's opposite
over hypotenuse?

00:07:52.620 --> 00:07:53.740
SOH.

00:07:53.740 --> 00:07:54.710
S O H.

00:07:54.710 --> 00:07:55.480
Sine.

00:07:55.480 --> 00:07:59.000
So sine of beta is BE over AB.

00:07:59.000 --> 00:08:00.310
So this is sine of beta.

00:08:06.940 --> 00:08:10.780
And now let me
switch to magenta.

00:08:10.780 --> 00:08:13.610
What's EF over AE?

00:08:17.210 --> 00:08:19.960
If we look at this right
triangle right here,

00:08:19.960 --> 00:08:24.220
is opposite over
hypotenuse for alpha.

00:08:24.220 --> 00:08:26.050
So it's sine of alpha.

00:08:26.050 --> 00:08:27.030
Opposite over hypotenuse.

00:08:30.000 --> 00:08:32.660
And what's AE over AB?

00:08:37.240 --> 00:08:40.330
So now we're looking at this
large right triangle here.

00:08:40.330 --> 00:08:42.780
AE over AB.

00:08:42.780 --> 00:08:46.710
Well, that's the adjacent of
beta over the hypotenuse.

00:08:46.710 --> 00:08:48.580
Well, what's adjacent
over hypotenuse?

00:08:48.580 --> 00:08:51.430
That's the cosine.

00:08:51.430 --> 00:08:53.230
CAH.

00:08:53.230 --> 00:08:57.140
Cosine of beta, of
this beta right here.

00:08:57.140 --> 00:08:58.650
I think we're done.

00:08:58.650 --> 00:09:02.140
This is to me, fairly
mind blowing.

00:09:02.140 --> 00:09:14.010
That the sine of alpha plus
beta is equal to the cosine of

00:09:14.010 --> 00:09:15.920
alpha times the sine of beta.

00:09:15.920 --> 00:09:20.310
Plus the sine of alpha
times the cosine of beta.

00:09:20.310 --> 00:09:22.590
What's neat about this is that
it kind of came out of this

00:09:22.590 --> 00:09:24.970
nice symmetric formula.

00:09:24.970 --> 00:09:27.370
It's not this big, hairy thing.

00:09:27.370 --> 00:09:28.530
You might have even guessed it.

00:09:28.530 --> 00:09:29.940
I don't know.

00:09:29.940 --> 00:09:30.950
I just find it very neat.

00:09:30.950 --> 00:09:33.430
We went through this big
convoluted proof with this big

00:09:33.430 --> 00:09:37.780
convoluted shape, but we got
this nice symmetric trig

00:09:37.780 --> 00:09:38.980
identity out of it.

00:09:38.980 --> 00:09:41.780
So hopefully you found that
amazing as well and in the next

00:09:41.780 --> 00:09:45.900
presentation I'll do a proof
for cosine of alpha plus beta.

00:09:45.900 --> 00:09:47.460
See you soon.


Trigonometric Identities

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:01.780
Welcome back.

00:00:01.780 --> 00:00:03.590
I'm now going to do a
series of videos on the

00:00:03.590 --> 00:00:05.430
trigonometric identities.

00:00:05.430 --> 00:00:08.250
So let's just review what
we already know about the

00:00:08.250 --> 00:00:09.920
trig function, so let me
just write SOHCAHTOA.

00:00:15.960 --> 00:00:18.320
That tells us, and we've
actually extended this with the

00:00:18.320 --> 00:00:20.670
unit circle definition, but if
you watch those videos, you'll

00:00:20.670 --> 00:00:23.040
realize that the unit circle
definition directly

00:00:23.040 --> 00:00:23.910
uses SOHCAHTOA.

00:00:23.910 --> 00:00:25.690
So we'll just stick with
SOHCAHTOA because I think it'll

00:00:25.690 --> 00:00:28.690
help make some of what we're
about to do seem a little bit

00:00:28.690 --> 00:00:31.380
more straightforward and will
kind of verge on the unit

00:00:31.380 --> 00:00:32.760
circle definition anyway.

00:00:32.760 --> 00:00:40.150
So we know that sine of
theta is equal to opposite

00:00:40.150 --> 00:00:41.390
over hypotenuse, right?

00:00:41.390 --> 00:00:49.290
So cosine of theta is equal to
adjacent over hypotenuse, and

00:00:49.290 --> 00:00:55.930
then the tangent of theta is
equal to opposite

00:00:55.930 --> 00:00:58.310
over adjacent.

00:00:58.310 --> 00:01:01.420
So let's draw that out
on a right triangle.

00:01:01.420 --> 00:01:03.230
We could do this with the
unit circle as well, and

00:01:03.230 --> 00:01:05.590
it would make sense.

00:01:05.590 --> 00:01:07.960
Let's see if we can find a
relationship between sine,

00:01:07.960 --> 00:01:10.760
cosine and tangent.

00:01:10.760 --> 00:01:11.915
There's my right triangle.

00:01:14.490 --> 00:01:16.690
Let's call this theta.

00:01:16.690 --> 00:01:18.910
This is the hypotenuse h.

00:01:18.910 --> 00:01:21.820
This is the opposite side,
right, opposite of theta.

00:01:21.820 --> 00:01:23.480
This is theta right here.

00:01:23.480 --> 00:01:27.090
This is the adjacent
side, right?

00:01:27.090 --> 00:01:28.940
Well, what do we know about
the relationship between

00:01:28.940 --> 00:01:31.860
the opposite adjacent side
and then the hypotenuse?

00:01:31.860 --> 00:01:34.030
What does the Pythagorean
theorem tell us?

00:01:34.030 --> 00:01:38.350
Oh, yeah, this side squared
plus this side squared is equal

00:01:38.350 --> 00:01:41.290
to the hypotenuse squared,
so we could write that down.

00:01:41.290 --> 00:01:48.100
a squared plus o squared is
equal to the hypotenuse

00:01:48.100 --> 00:01:49.900
squared, right?

00:01:49.900 --> 00:01:52.980
And then this is just an
equation, so if we want to, we

00:01:52.980 --> 00:01:56.620
could divide both sides of this
equation by h squared,

00:01:56.620 --> 00:01:57.790
and so what do we get?

00:01:57.790 --> 00:02:07.880
We get a squared over h squared
plus o squared over h squared

00:02:07.880 --> 00:02:10.590
is equal to 1, right?

00:02:10.590 --> 00:02:22.070
And then I could rewrite that
as a over h squared plus o

00:02:22.070 --> 00:02:26.710
over h squared is equal to 1.

00:02:26.710 --> 00:02:30.930
Now, do these look
at all familiar?

00:02:30.930 --> 00:02:32.080
Well, we have them here, right?

00:02:32.080 --> 00:02:35.570
This is a over h, this is
o over h, so we could

00:02:35.570 --> 00:02:36.750
just substitute.

00:02:36.750 --> 00:02:41.280
So this is just cosine
of theta squared.

00:02:41.280 --> 00:02:42.520
And this is how you
write cosine squared.

00:02:42.520 --> 00:02:44.280
You could put a parentheses
around the whole thing and then

00:02:44.280 --> 00:02:47.760
square it, but this is just
the notation people use.

00:02:47.760 --> 00:02:53.510
Plus opposite over adjacent
squared, so that's sine theta

00:02:53.510 --> 00:02:55.280
squared is equal to 1.

00:02:55.280 --> 00:02:56.530
So that's our first
trig identity.

00:02:56.530 --> 00:03:00.440
So if you know the sine of
theta, it's very easy to figure

00:03:00.440 --> 00:03:01.680
out the cosine of theta, right?

00:03:01.680 --> 00:03:04.550
You could just solve
this equation.

00:03:04.550 --> 00:03:06.430
If I know that the--
I don't know.

00:03:06.430 --> 00:03:14.050
Let's say I know that the
sine of theta is 1/2, right?

00:03:14.050 --> 00:03:16.351
Then what is the
cosine of theta?

00:03:16.351 --> 00:03:18.630
The cosine of theta is what?

00:03:18.630 --> 00:03:20.950
Well, I know the sine of
theta is 1/2, right?

00:03:20.950 --> 00:03:28.160
So I would say cosine squared
of theta plus sine of theta

00:03:28.160 --> 00:03:34.770
is 1/2, so 1/2 squared
is equal to 1, right?

00:03:34.770 --> 00:03:41.480
So cosine squared theta
plus 1/4 is equal to 1.

00:03:41.480 --> 00:03:47.870
So we have cosine squared theta
is equal to 3/4, or cosine of

00:03:47.870 --> 00:03:50.830
theta would be the square
root of this, right?

00:03:50.830 --> 00:03:52.010
We just take the square
root of both sides.

00:03:52.010 --> 00:03:54.510
It would be the
square root of 3/2.

00:03:54.510 --> 00:03:56.770
And you probably remember that
from our whole presentation

00:03:56.770 --> 00:03:57.900
on the 30-60-90 triangle.

00:03:57.900 --> 00:04:03.400
So I just wanted to show you
a use of this trig identity

00:04:03.400 --> 00:04:05.520
that's usually written
sine squared plus cosine

00:04:05.520 --> 00:04:07.860
squared is equal to 1.

00:04:07.860 --> 00:04:09.740
So let's extend that
one a little bit.

00:04:09.740 --> 00:04:12.750
Let's just play with the ratios
and see what else we can--

00:04:12.750 --> 00:04:15.950
other identities
we can discover.

00:04:15.950 --> 00:04:17.180
Whoops!

00:04:17.180 --> 00:04:20.370
Clear image, invert colors.

00:04:20.370 --> 00:04:28.310
So we know that sine squared
theta plus cosine squared

00:04:28.310 --> 00:04:31.390
theta is equal to 1.

00:04:31.390 --> 00:04:35.220
The one thing we could do is we
could divide both sides of this

00:04:35.220 --> 00:04:38.880
equation by cosine squared of
theta, and let's just see what

00:04:38.880 --> 00:04:40.260
happens when we do that.

00:04:40.260 --> 00:04:44.020
So if we say cosine
squared theta, right?

00:04:44.020 --> 00:04:46.530
You have to distribute
across both terms.

00:04:46.530 --> 00:04:57.710
Cosine squared of theta, and
then cosine squared of theta.

00:04:57.710 --> 00:05:00.360
Well, what's sine squared theta
over cosine squared theta?

00:05:00.360 --> 00:05:07.720
That's the same thing as sine
of theta over cosine of theta

00:05:07.720 --> 00:05:16.550
squared plus this is 1 over
cosine theta squared, right?

00:05:16.550 --> 00:05:19.240
I mean, 1 squared is 1,
so I just rewrote it.

00:05:19.240 --> 00:05:21.980
So sine over cosine theta, I
think we learned that already.

00:05:21.980 --> 00:05:23.410
That's just the
tangent of theta.

00:05:26.000 --> 00:05:28.280
And in case you actually
haven't learned that already,

00:05:28.280 --> 00:05:30.190
think about it this way.

00:05:30.190 --> 00:05:34.030
Sine is opposite over
the hypotenuse, right?

00:05:34.030 --> 00:05:36.350
So that's opposite
over hypotenuse.

00:05:36.350 --> 00:05:40.410
And then cosine is
adjacent over hypotenuse.

00:05:40.410 --> 00:05:42.290
So adjacent over hypotenuse.

00:05:42.290 --> 00:05:46.350
So then that equals opposite
over hypotenuse times

00:05:46.350 --> 00:05:49.030
hypotenuse over
adjacent, right?

00:05:49.030 --> 00:05:51.720
Just dividing by a fraction is
the same thing as multiplying

00:05:51.720 --> 00:05:53.160
by its reciprocal.

00:05:53.160 --> 00:05:54.000
That's all I did.

00:05:54.000 --> 00:05:56.780
And that equals opposite
over adjacent, right?

00:05:59.290 --> 00:06:01.930
So that just says sine of
theta over cosine of theta is

00:06:01.930 --> 00:06:03.720
equal to tangent of theta.

00:06:03.720 --> 00:06:08.270
So sine squared theta over
cosine squared theta is tan

00:06:08.270 --> 00:06:15.310
squared theta, then plus 1 is
1 over cosine theta squared.

00:06:15.310 --> 00:06:19.300
And now I'm going to introduce
a new trig ratio, it's really

00:06:19.300 --> 00:06:21.800
just 1 over cosine theta.

00:06:21.800 --> 00:06:24.945
So 1 over cosine theta-- and
I'm going to summarize this at

00:06:24.945 --> 00:06:28.050
the end, just so it's not too
confusing-- is actually

00:06:28.050 --> 00:06:30.340
the secant of theta.

00:06:30.340 --> 00:06:31.780
And this is just
another ratio, right?

00:06:31.780 --> 00:06:35.170
The secant of theta, instead of
being the adjacent over the

00:06:35.170 --> 00:06:38.580
hypotenuse, would be the
hypotenuse over the

00:06:38.580 --> 00:06:39.180
adjacent, right?

00:06:39.180 --> 00:06:40.370
It's just 1 over cosine theta.

00:06:40.370 --> 00:06:42.710
Nothing fancy here.

00:06:42.710 --> 00:06:44.000
So secant of theta.

00:06:44.000 --> 00:06:47.580
So that equals secant
squared of theta.

00:06:47.580 --> 00:06:49.280
I know it can be a little
overwhelming initially, just

00:06:49.280 --> 00:06:51.750
because I'm, you know, throwing
out all these new terms, secant

00:06:51.750 --> 00:06:54.670
is 1 over cosine theta, but
once you just play around with

00:06:54.670 --> 00:06:56.790
these enough and get familiar
with the terms, it'll make

00:06:56.790 --> 00:06:58.130
sense, and it'll be a little
more natural to you.

00:06:58.130 --> 00:06:59.760
So this could be-- you
could view this as

00:06:59.760 --> 00:07:01.970
another trig identity.

00:07:01.970 --> 00:07:04.160
And actually, I don't
even remember if I've

00:07:04.160 --> 00:07:05.690
taught it already.

00:07:05.690 --> 00:07:07.460
I mean, you could view this as
a trig identity, although

00:07:07.460 --> 00:07:09.110
that's almost definitional.

00:07:09.110 --> 00:07:12.730
And then, of course, you can--
in case I haven't done it

00:07:12.730 --> 00:07:18.340
already, you now know that sine
of theta over cosine of theta

00:07:18.340 --> 00:07:22.570
is equal to tangent of theta.

00:07:22.570 --> 00:07:25.210
And that's right here
with, I guess you could

00:07:25.210 --> 00:07:27.770
say, the proof of it.

00:07:27.770 --> 00:07:30.750
So let me keep introducing you
to more things, and if this is

00:07:30.750 --> 00:07:33.540
really daunting, maybe you
just can rewatch it, and

00:07:33.540 --> 00:07:34.890
hopefully, it'll make sense.

00:07:34.890 --> 00:07:37.810
Let me see, clear image.

00:07:37.810 --> 00:07:39.150
So what have we learned so far?

00:07:39.150 --> 00:07:45.540
We learned that sine squared
theta plus cosine squared

00:07:45.540 --> 00:07:47.540
theta is equal to 1.

00:07:47.540 --> 00:07:53.880
We learned that sine of theta
over cosine of theta is

00:07:53.880 --> 00:07:56.700
equal to tangent of theta.

00:07:56.700 --> 00:08:04.750
We learned that the tangent
squared of theta plus 1 is

00:08:04.750 --> 00:08:08.650
equal to the secant of theta.

00:08:08.650 --> 00:08:11.030
And here, let me actually
write this definition down.

00:08:11.030 --> 00:08:13.675
The secant of theta-- oops,
is equal to the secant

00:08:13.675 --> 00:08:15.790
squared of theta, sorry.

00:08:15.790 --> 00:08:19.800
And the secant of theta is
just 1 over cosine of theta.

00:08:19.800 --> 00:08:21.670
This is something you really
should just memorize, that

00:08:21.670 --> 00:08:22.950
secant is 1 over cosine.

00:08:22.950 --> 00:08:27.430
And if you're wondering what 1
over sine is, 1 over sine of

00:08:27.430 --> 00:08:33.350
theta, it's the cosecant-- the
abbreviation is csc-- of theta.

00:08:33.350 --> 00:08:37.690
And if you're wondering what
1 over the tangent is,

00:08:37.690 --> 00:08:39.730
it's the cotangent.

00:08:39.730 --> 00:08:41.660
And you just might want
to memorize these.

00:08:41.660 --> 00:08:46.600
And this often confuses me,
that 1 over the cosine is the

00:08:46.600 --> 00:08:50.340
secant, but 1 over the sine is
the cosecant, so it's kind of

00:08:50.340 --> 00:08:52.250
almost the opposite, right?

00:08:52.250 --> 00:08:55.230
1 over the sine has a co in
it, while 1 over the cosine

00:08:55.230 --> 00:08:56.060
doesn't have the co in it.

00:08:56.060 --> 00:09:00.150
So that might help
you remember things.

00:09:00.150 --> 00:09:02.760
So I think that's all
I have time for now.

00:09:02.760 --> 00:09:06.120
In the next presentation, I'm
going to introduce you to a

00:09:06.120 --> 00:09:07.575
couple more trig identities.

00:09:07.575 --> 00:09:09.380
See you soon.


Rates-of-change (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:01.790
Welcome back.

00:00:01.790 --> 00:00:04.320
I'm now going to actually do a
couple more rates of change

00:00:04.320 --> 00:00:06.480
problems because I think the
first one I did was probably a

00:00:06.480 --> 00:00:09.420
little bit more complicated
than I wanted, and the hard

00:00:09.420 --> 00:00:11.460
part was actually the geometry
and not the calculus.

00:00:11.460 --> 00:00:14.660
So let's just say I drop
a rock into a pond

00:00:14.660 --> 00:00:16.140
and it has a ripple.

00:00:16.140 --> 00:00:18.250
And the ripple spreads out--
it'll probably have multiple

00:00:18.250 --> 00:00:20.760
ripples, but let's just say
we focus on one ripple.

00:00:20.760 --> 00:00:24.450
And the ripple is
spreading out.

00:00:24.450 --> 00:00:25.550
So let me draw.

00:00:25.550 --> 00:00:28.560
This is where I dropped the
rock, and then this is

00:00:28.560 --> 00:00:31.000
a ripple in the pond.

00:00:31.000 --> 00:00:33.310
It would be an actual
circle, which isn't what

00:00:33.310 --> 00:00:34.780
I drew, but close enough.

00:00:38.620 --> 00:00:41.310
So let me say that--
let's call r.

00:00:41.310 --> 00:00:42.800
r is not for ripple.

00:00:42.800 --> 00:00:45.330
r is the distance that the
ripple is from the center

00:00:45.330 --> 00:00:48.440
from where it started
from the get go.

00:00:48.440 --> 00:00:52.320
And it's not by chance that
that's also for radius.

00:00:52.320 --> 00:00:55.700
So let's say that the
ripple is moving out at

00:00:55.700 --> 00:00:57.720
2 meters per second.

00:00:57.720 --> 00:01:03.400
So the rate at which r is
changing with respect to t

00:01:03.400 --> 00:01:07.660
is 2 meters per second.

00:01:07.660 --> 00:01:14.190
And what I want to know is what
is the rate at which the area

00:01:14.190 --> 00:01:20.250
that's kind of included by this
ripple-- this entire area-- how

00:01:20.250 --> 00:01:24.430
fast is that changing
with respect to time.

00:01:24.430 --> 00:01:31.840
When the ripple is let's say
3 meters from the center.

00:01:31.840 --> 00:01:35.430
When the ripple is
equal to 3 meters.

00:01:35.430 --> 00:01:42.530
Well do we know any relation
between the area of the circle

00:01:42.530 --> 00:01:46.690
essentially, and the distance
of the ripple, or the radius?

00:01:46.690 --> 00:01:49.250
Well yes we do.

00:01:49.250 --> 00:01:57.770
Area is equal to pi r squared.

00:01:57.770 --> 00:02:00.290
So we want to figure out the
rate at which area changes

00:02:00.290 --> 00:02:01.220
with respect to time.

00:02:01.220 --> 00:02:02.630
So what does a chain
rule tell us?

00:02:02.630 --> 00:02:07.270
The chain rule tells us that
the rate at which a changes

00:02:07.270 --> 00:02:11.890
with respect to t is equal to
the rate at which a changes

00:02:11.890 --> 00:02:17.490
with respect to r times the
rate at which r changes

00:02:17.490 --> 00:02:19.910
with respect to t.

00:02:19.910 --> 00:02:22.120
Well we already know
this piece, right?

00:02:22.120 --> 00:02:23.990
We already know the rate
at which r is changing

00:02:23.990 --> 00:02:25.270
with respect to time.

00:02:25.270 --> 00:02:28.010
That's right here.

00:02:28.010 --> 00:02:31.270
All we have to do to figure out
the rate at which a is changing

00:02:31.270 --> 00:02:34.320
with respect to time is we have
to figure out the rate at which

00:02:34.320 --> 00:02:37.320
a changes with respect to r.

00:02:37.320 --> 00:02:38.410
Well that's just
the derivative.

00:02:38.410 --> 00:02:42.230
The derivative with
respect to r.

00:02:42.230 --> 00:02:48.260
The derivative with respect to
r-- do it of both sides-- of a

00:02:48.260 --> 00:02:58.076
equals the derivative with
respect to r of pi r squared.

00:02:58.076 --> 00:03:04.160
So we get da/dr is equal
to-- what's the derivative

00:03:04.160 --> 00:03:05.250
with respect to r here?

00:03:05.250 --> 00:03:05.850
Well that's easy.

00:03:05.850 --> 00:03:13.390
Just 2 times pi r to the 1.

00:03:13.390 --> 00:03:18.660
That's pretty interesting by
itself that the rate at which

00:03:18.660 --> 00:03:23.650
the derivative of the area--
formula really-- is what?

00:03:23.650 --> 00:03:27.100
This formula by itself
looks interesting, right?

00:03:27.100 --> 00:03:29.350
That's a formula for the
circumference of a circle.

00:03:29.350 --> 00:03:32.590
So the derivative of an area of
a circle with respect to the

00:03:32.590 --> 00:03:34.690
radius is a circumference.

00:03:34.690 --> 00:03:37.510
That is mildly fascinating.

00:03:37.510 --> 00:03:44.070
Something even more interesting
is to figure out what the

00:03:44.070 --> 00:03:47.270
antiderivative is of the area
and compare that to the

00:03:47.270 --> 00:03:48.740
volume of a sphere.

00:03:48.740 --> 00:03:51.700
You can look it up on
Wikipedia or something.

00:03:51.700 --> 00:03:54.465
Or even compare that to the
surface area of the sphere, and

00:03:54.465 --> 00:03:57.570
just keep picking derivatives
and integrals and they'll be

00:03:57.570 --> 00:03:58.960
some pretty interesting
relationships.

00:03:58.960 --> 00:03:59.350
But anyway.

00:03:59.350 --> 00:04:00.030
Back to the problem.

00:04:00.030 --> 00:04:03.060
Not to go on too
far of a tangent.

00:04:03.060 --> 00:04:04.420
So we figured out the
rate at which 8 changes

00:04:04.420 --> 00:04:05.460
with respect to r.

00:04:05.460 --> 00:04:09.380
So going back to our original
thing, we now know that the

00:04:09.380 --> 00:04:15.230
rate at which a changes with
respect to t is equal to the

00:04:15.230 --> 00:04:17.790
rate at which a changes
with respect to r.

00:04:17.790 --> 00:04:20.116
Well that's just
right here, right?

00:04:20.116 --> 00:04:22.280
The rate at which a changes
with respect to r.

00:04:22.280 --> 00:04:28.760
Well that's just 2 pi r times
the rate at which r is

00:04:28.760 --> 00:04:31.700
changing with respect to t.

00:04:31.700 --> 00:04:32.720
Well the rate at which r
changes with respect to

00:04:32.720 --> 00:04:34.880
t is just right here.

00:04:34.880 --> 00:04:37.400
2 meters per second,
we figured that out.

00:04:37.400 --> 00:04:45.970
So 2 pi r times 2-- well I
won't include the units

00:04:45.970 --> 00:04:47.195
because that might
confuse you right now.

00:04:47.195 --> 00:04:49.135
So the rate at at which a
is changing with respect

00:04:49.135 --> 00:04:50.560
to t is 2 pi r times 2.

00:04:50.560 --> 00:04:51.840
Well are we done yet?

00:04:51.840 --> 00:04:53.880
Well no, because we haven't
substituted yet for r.

00:04:53.880 --> 00:04:54.930
So what is r?

00:04:54.930 --> 00:05:03.300
Well we know. r is three
meters, so da/dt is equal to

00:05:03.300 --> 00:05:06.230
2 times pi times 3 times 2.

00:05:06.230 --> 00:05:07.090
So what is that?

00:05:07.090 --> 00:05:09.610
2 times 3 is 6 times 2 is 12.

00:05:09.610 --> 00:05:14.850
12 pi and it's meters squared
per second, because it's the

00:05:14.850 --> 00:05:17.280
rate at which area-- square
meters-- is changing

00:05:17.280 --> 00:05:18.110
with respect to time.

00:05:18.110 --> 00:05:20.320
And if we multiply out the
units we would have gotten

00:05:20.320 --> 00:05:22.490
the same thing right here.

00:05:22.490 --> 00:05:24.990
So in the next video i'm going
to do a slightly harder rate of

00:05:24.990 --> 00:05:28.700
change problem just so that you
see that it doesn't only apply

00:05:28.700 --> 00:05:30.730
to kind of purely
geometric things.

00:05:30.730 --> 00:05:32.350
It applies to pretty much
anything that you can

00:05:32.350 --> 00:05:35.990
find a relationship
between two values.

00:05:35.990 --> 00:05:36.360
See you in the next module.


Introduction to rate-of-change problems

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.300
We've learned a lot about
derivatives, and now we will

00:00:03.300 --> 00:00:07.840
use them to solve something
that is hopefully may

00:00:07.840 --> 00:00:08.970
be kind of useful.

00:00:08.970 --> 00:00:11.980
So let's just start with a
review of the chain rule,

00:00:11.980 --> 00:00:13.760
and I'm going to write
in a different way.

00:00:13.760 --> 00:00:19.920
So let's say I had the
function f of g of x.

00:00:19.920 --> 00:00:22.530
So what I'm going to do is I'm
going actually write this in a

00:00:22.530 --> 00:00:25.430
way that might be a bit foreign
to you, but I think with a

00:00:25.430 --> 00:00:27.370
little bit of explanation
you'll realize that this is the

00:00:27.370 --> 00:00:32.500
same thing as the chain rule
that we all know and love.

00:00:32.500 --> 00:00:34.830
I changed colors
really for no reason.

00:00:34.830 --> 00:00:36.220
Just to change colors.

00:00:36.220 --> 00:00:37.230
Sometimes I do that.

00:00:37.230 --> 00:00:38.570
I'll change colors again.

00:00:38.570 --> 00:00:45.500
The derivative of f of g of x
is equal to the rate at which

00:00:45.500 --> 00:00:48.790
f changes with respect to g.

00:00:48.790 --> 00:01:01.870
So that's df to dg times the
rate at which g changes

00:01:01.870 --> 00:01:04.290
with respect to x.

00:01:04.290 --> 00:01:06.900
And you're saying Sal, this
looks completely foreign to me.

00:01:06.900 --> 00:01:09.550
And one, you could think
about what I just said and

00:01:09.550 --> 00:01:10.490
I think it'll make sense.

00:01:10.490 --> 00:01:12.760
Or I could just rewrite
this in the traditional

00:01:12.760 --> 00:01:14.540
chain rule format.

00:01:14.540 --> 00:01:16.130
And this isn't really
the traditional.

00:01:16.130 --> 00:01:17.440
What I wrote is the
traditional, but the

00:01:17.440 --> 00:01:19.100
way I've taught you.

00:01:19.100 --> 00:01:22.300
So the way I taught you,
what's the change of

00:01:22.300 --> 00:01:24.450
f with respect to g?

00:01:24.450 --> 00:01:34.130
Well that's just f prime of g
of x times-- and what's the

00:01:34.130 --> 00:01:37.025
rate at which g changes
with respect to x?

00:01:37.025 --> 00:01:40.870
Well that's just g prime of x.

00:01:40.870 --> 00:01:43.270
So hopefully this make sense
to you that these are just

00:01:43.270 --> 00:01:45.410
two different ways of
writing the chain rule.

00:01:45.410 --> 00:01:47.670
I think what was this, this is
either Lagrange or Leibniz's

00:01:47.670 --> 00:01:49.430
notation, and this
the other guy.

00:01:49.430 --> 00:01:50.640
But I forget.

00:01:50.640 --> 00:01:53.040
And this actually makes sense
just from a fractions point of

00:01:53.040 --> 00:01:55.500
view that this term and that
term cancels out and you get,

00:01:55.500 --> 00:01:59.730
you know that this equals that
rate at which f changes

00:01:59.730 --> 00:02:02.190
with respect x.

00:02:02.190 --> 00:02:04.960
So with that out of the way,
let's use this to solve

00:02:04.960 --> 00:02:08.020
something kind of hopefully
a little bit neat.

00:02:08.020 --> 00:02:11.635
Let's say I have a cone.

00:02:11.635 --> 00:02:13.830
Let's say it's one of
those cups you have

00:02:13.830 --> 00:02:15.280
at the water cooler.

00:02:15.280 --> 00:02:16.350
That's the top of the cup.

00:02:20.510 --> 00:02:27.500
And let's say at any point in
the clone the ratio of the

00:02:27.500 --> 00:02:31.090
radius of the cone to the
height-- if this is the

00:02:31.090 --> 00:02:35.110
height, then the radius
is 1/2 of height.

00:02:35.110 --> 00:02:36.340
At any point.

00:02:36.340 --> 00:02:40.750
As you know, it's a fixed cone,
these are lines on both sides.

00:02:40.750 --> 00:02:42.215
So that ratio stays fixed.

00:02:46.170 --> 00:02:49.850
So just to start off with
some-- you maybe never

00:02:49.850 --> 00:02:50.605
even learned this.

00:02:53.180 --> 00:02:54.900
This will help us with
the problem later on.

00:02:54.900 --> 00:02:59.020
What's the volume of-- let's
say I'm filling water

00:02:59.020 --> 00:03:00.500
up to height h.

00:03:00.500 --> 00:03:04.650
Let's say this is where
the water line is.

00:03:04.650 --> 00:03:09.130
But that's where the
water is in this cone.

00:03:09.130 --> 00:03:13.870
What's the volume assuming
that the height is h.

00:03:13.870 --> 00:03:17.680
Assuming that I've let's say
8 centimeters of water.

00:03:17.680 --> 00:03:20.210
What is the volume of water
that I put in the cone?

00:03:20.210 --> 00:03:23.450
Well, if you don't know it, and
I sometimes-- well actually

00:03:23.450 --> 00:03:28.180
later when we do rotations of
solid into integration modules

00:03:28.180 --> 00:03:33.030
I can actually prove this to
you-- but the volume of a cone

00:03:33.030 --> 00:03:38.300
is equal to 1/3 base times
height, where the base is

00:03:38.300 --> 00:03:40.120
actually, you can kind of
view it as the surface

00:03:40.120 --> 00:03:42.550
area of the water.

00:03:42.550 --> 00:03:44.370
And what's the base?

00:03:44.370 --> 00:03:47.595
Well that's just equal to
volume is equal to 1/3.

00:03:47.595 --> 00:03:52.230
The base is equal to pi r
squared, where r is the radius.

00:03:52.230 --> 00:03:54.770
This is just kind of
solid geometry review.

00:03:54.770 --> 00:03:57.000
And what's the radius
in this case?

00:03:57.000 --> 00:04:02.220
It's 1/3 pi r squared times h.

00:04:02.220 --> 00:04:04.370
And in this example I
said that the radius

00:04:04.370 --> 00:04:06.960
is 1/2 of the height.

00:04:06.960 --> 00:04:13.690
So this equals 1/3 pi 1/2 the
height-- I just replaced it

00:04:13.690 --> 00:04:15.450
for r squared-- times h.

00:04:15.450 --> 00:04:16.760
We haven't done
any calculus yet.

00:04:16.760 --> 00:04:21.100
This is just complicated-- not
complicated really-- just a

00:04:21.100 --> 00:04:23.770
little bit hairy
solid geometry.

00:04:23.770 --> 00:04:28.220
And if we simplify that we get
volume is equal to-- let's see.

00:04:28.220 --> 00:04:37.850
So we get 1/2 squared is 1/4
times 1/3 is 1/12 pi, and

00:04:37.850 --> 00:04:42.990
then h squared times
h, h to the third.

00:04:42.990 --> 00:04:48.200
Very interesting, Now let's
start doing some calculus.

00:04:48.200 --> 00:04:50.640
I think this might
blow your mind.

00:04:50.640 --> 00:05:03.030
Let's say I am pouring water
into this cup at a rate of 1

00:05:03.030 --> 00:05:06.410
cubic centimeter per second.

00:05:06.410 --> 00:05:10.800
And for you metric jocks,
you'll know sometimes doctors

00:05:10.800 --> 00:05:13.740
or nurses will say 1 cc, and
that's also equal

00:05:13.740 --> 00:05:15.090
to 1 millimeter.

00:05:15.090 --> 00:05:17.013
Just for other frames of
reference, but I like to stay

00:05:17.013 --> 00:05:19.070
in centimeters cubed per
second because we're going

00:05:19.070 --> 00:05:22.380
to work with centimeters
in multiple dimensions.

00:05:22.380 --> 00:05:25.960
But let's say we're pouring
1 centimeter cubed per

00:05:25.960 --> 00:05:28.270
second into this cup.

00:05:28.270 --> 00:05:34.460
I want to know-- this is an
interesting question-- how fast

00:05:34.460 --> 00:05:36.380
is the water level rising.

00:05:36.380 --> 00:05:40.750
How fast is h changing, the
height of the water, at the

00:05:40.750 --> 00:05:45.640
moment when h is equal
to 2 centimeters.

00:05:45.640 --> 00:05:46.600
So how do we do that?

00:05:46.600 --> 00:05:49.080
This just give us a static
picture of if we know the

00:05:49.080 --> 00:05:51.510
height we can figure
out the volume.

00:05:51.510 --> 00:05:54.980
But if we figure it out, if we
took the rate of change with

00:05:54.980 --> 00:05:57.700
respect to time off of both
sides of this equation,

00:05:57.700 --> 00:05:59.390
something interesting
might happen.

00:05:59.390 --> 00:06:01.310
So let's take the derivative
with respect time

00:06:01.310 --> 00:06:01.940
on both sides.

00:06:01.940 --> 00:06:06.970
So the derivative with respect
to time of the volume.

00:06:06.970 --> 00:06:08.860
Well that's just dv/dt.

00:06:11.930 --> 00:06:15.880
And what's the derivative
with respect to time of

00:06:15.880 --> 00:06:17.730
this side of the equation?

00:06:17.730 --> 00:06:21.530
Well it's the rate which v
changes with respect to h.

00:06:24.750 --> 00:06:27.970
This is v as a
function of h, right?

00:06:27.970 --> 00:06:36.040
So it's going to be dv as a
function-- how fast does v

00:06:36.040 --> 00:06:39.730
change with respect to h--
times how fast does h

00:06:39.730 --> 00:06:41.300
change with respect to t.

00:06:41.300 --> 00:06:43.175
This is just the
chain rule up here.

00:06:43.175 --> 00:06:44.740
I want you to think about
this a little bit.

00:06:44.740 --> 00:06:46.370
It might not seem obvious
but all we're doing

00:06:46.370 --> 00:06:47.280
is the chain rule.

00:06:47.280 --> 00:06:49.820
It's a little confusing because
I had no t's in this equation

00:06:49.820 --> 00:06:51.345
before, and all of a sudden I'm
picking a derivative

00:06:51.345 --> 00:06:52.690
with respect to t.

00:06:52.690 --> 00:06:56.670
But let's just say that h
actually is a function of t,

00:06:56.670 --> 00:06:59.020
which as you know it is.

00:06:59.020 --> 00:07:01.640
So let's then just solve that.

00:07:01.640 --> 00:07:06.020
So dv-- the derivative of v
with respect to t-- is equal

00:07:06.020 --> 00:07:08.840
to the derivative of
v with respect to h.

00:07:08.840 --> 00:07:09.740
Well that's easy.

00:07:09.740 --> 00:07:12.850
The derivative of v with
respect to h is just simple.

00:07:12.850 --> 00:07:17.230
So 3 times 1/12, that's 3/12.

00:07:17.230 --> 00:07:25.730
That's 1/4 pi, so we can just
write pi over 4 h squared.

00:07:25.730 --> 00:07:26.720
So that's this part.

00:07:29.550 --> 00:07:34.800
And in this part is just
still-- see, I can rewrite

00:07:34.800 --> 00:07:38.570
in this new color
dh/dt, times dh/dt.

00:07:42.200 --> 00:07:44.250
So you're saying Sal,
what have you now done.

00:07:44.250 --> 00:07:48.236
I said the rate of which the
volume is changing with respect

00:07:48.236 --> 00:07:52.810
to time is equal to pi over 4
times the height squared times

00:07:52.810 --> 00:07:56.560
the rate at which the height is
changing with respect to time.

00:07:56.560 --> 00:07:58.630
So can this solve
the problem for us?

00:08:04.590 --> 00:08:06.870
Well what do we know?

00:08:06.870 --> 00:08:08.860
We know the rate at which
the volume is changing

00:08:08.860 --> 00:08:09.980
with respect to time.

00:08:09.980 --> 00:08:12.330
1 centimeter cubed
per second squared.

00:08:12.330 --> 00:08:17.030
So we could say dv/dt
is equal to 1.

00:08:17.030 --> 00:08:18.210
I'm going to get
rid of the units.

00:08:18.210 --> 00:08:20.620
Most physics instructors
would cringe.

00:08:20.620 --> 00:08:28.080
But 1 centimeter cubed per
second is equal to pi over 4.

00:08:28.080 --> 00:08:32.390
This one is v the rate at
which the volume is changing

00:08:32.390 --> 00:08:34.450
with respect to time.

00:08:34.450 --> 00:08:35.520
pi over 4 h squared.

00:08:35.520 --> 00:08:38.210
We know what h is right
now. h is 2, so it's

00:08:38.210 --> 00:08:41.410
2 squared times 4.

00:08:41.410 --> 00:08:44.785
We said the height is 2
right then, so 2 squared

00:08:44.785 --> 00:08:46.330
is 4 times dh/dt.

00:08:49.720 --> 00:08:54.820
So this cancels out, and we get
1 is equal to pi times dh/dt.

00:08:54.820 --> 00:08:58.150
And we solve for dt/dt we
get-- let me make sure

00:08:58.150 --> 00:08:59.100
not to confuse you.

00:08:59.100 --> 00:09:02.423
The rate at which the height of
water is changing with respect

00:09:02.423 --> 00:09:04.860
to time is just 1 over pi.

00:09:04.860 --> 00:09:06.710
Fascinating.

00:09:06.710 --> 00:09:10.180
Or 1 over pi centimeters
per second.

00:09:10.180 --> 00:09:11.900
So we can figure out what this
number is, it's going to

00:09:11.900 --> 00:09:15.150
be like 0.3 something.

00:09:15.150 --> 00:09:18.660
So 0.3 something centimeters
per second is the rate at which

00:09:18.660 --> 00:09:22.900
the height of the water level
is going to change as I put 1

00:09:22.900 --> 00:09:28.300
centimeter cubed of volume
into this cup per second.

00:09:28.300 --> 00:09:29.620
I have probably confused you.

00:09:29.620 --> 00:09:31.090
You may want to
watch this again.

00:09:31.090 --> 00:09:33.110
And I'll do a couple more
videos with these rates of

00:09:33.110 --> 00:09:36.800
change problems because these
tend to confuse people, but

00:09:36.800 --> 00:09:39.400
once you get the hang of it, I
think you'll see that

00:09:39.400 --> 00:09:40.890
they're not so bad.

00:09:40.890 --> 00:09:41.650
I'll see you the
next presentation.


Definite Integrals (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.970
Welcome back.

00:00:01.970 --> 00:00:05.390
I'm now going to use definite
integrals to figure out the

00:00:05.390 --> 00:00:07.890
areas under a bunch of curves
and, if we have time, maybe

00:00:07.890 --> 00:00:09.290
even between some curves.

00:00:09.290 --> 00:00:11.660
So let me right down the
fundamental theorem

00:00:11.660 --> 00:00:12.350
of calculus.

00:00:12.350 --> 00:00:15.560
I know I covered it really fast
in the last presentation.

00:00:15.560 --> 00:00:17.310
Just to make sure you
understand this formula.

00:00:17.310 --> 00:00:19.020
The last couple presentations
were really to give you an

00:00:19.020 --> 00:00:20.800
intuition for this
exact formula.

00:00:20.800 --> 00:00:34.440
Let's say that big f prime of
x is equal to f of x, right?

00:00:34.440 --> 00:00:39.200
That's also like saying that
the -- that's equivalent to

00:00:39.200 --> 00:00:45.950
saying that f of x -- big f
of x -- is equal to the

00:00:45.950 --> 00:00:56.000
antiderivative of
f of x, right?

00:00:56.000 --> 00:01:01.880
Well, let's just that it's one
of the possible antiderivatives

00:01:01.880 --> 00:01:02.760
of f of x, right?

00:01:02.760 --> 00:01:04.420
Because there's always a
constant term here and you're

00:01:04.420 --> 00:01:06.090
not sure whether it is.

00:01:06.090 --> 00:01:09.760
And this is why people tend to
use this standard because we

00:01:09.760 --> 00:01:15.590
know that f of x is the
derivative of big f prime of x.

00:01:15.590 --> 00:01:18.850
Big f of x is just one of the
antiderivatives of f of x.

00:01:18.850 --> 00:01:21.100
So this is a little bit
not true, but I think

00:01:21.100 --> 00:01:21.800
you get the idea.

00:01:21.800 --> 00:01:26.430
But the fundamental theorem of
calculus tells us if this top

00:01:26.430 --> 00:01:38.480
line is true, then the definite
integral from a to b of f of x

00:01:38.480 --> 00:01:45.520
d x is equal to its
antiderivative evaluated at b

00:01:45.520 --> 00:01:50.010
minus its antiderivative
evaluated at a.

00:01:50.010 --> 00:01:53.300
And I know I said here that
big f isn't the only

00:01:53.300 --> 00:01:54.310
antiderivative, right?

00:01:54.310 --> 00:01:56.760
Because you could at any
constant to this and that would

00:01:56.760 --> 00:01:58.200
also be the antiderivative.

00:01:58.200 --> 00:02:01.110
But when you subtract here, the
constants will cancel out.

00:02:01.110 --> 00:02:02.575
So it really doesn't
matter which of the

00:02:02.575 --> 00:02:03.250
constants you pick.

00:02:03.250 --> 00:02:04.420
The constant actually
doesn't matter.

00:02:04.420 --> 00:02:06.765
So that's why I actually
said the antiderivative.

00:02:06.765 --> 00:02:08.150
But let's apply this.

00:02:08.150 --> 00:02:09.960
You might be
confused right now.

00:02:09.960 --> 00:02:13.795
So let me draw a graph.

00:02:21.740 --> 00:02:22.160
There you go.

00:02:22.160 --> 00:02:25.630
Look how straight that is.

00:02:25.630 --> 00:02:28.220
Draw the x-axis.

00:02:28.220 --> 00:02:29.510
Not perfect but it'll do.

00:02:37.640 --> 00:02:47.570
Let's say that my f of x is
equal to x squared plus 1.

00:02:47.570 --> 00:02:49.620
So f of x looks like this.

00:02:49.620 --> 00:02:50.630
This is 1.

00:02:53.810 --> 00:02:54.770
So it'll start at 1.

00:02:54.770 --> 00:02:56.880
It'll just be a parabola.

00:02:56.880 --> 00:03:00.600
Let me see how good
I can draw this.

00:03:00.600 --> 00:03:01.900
I've done worse.

00:03:01.900 --> 00:03:02.550
OK.

00:03:02.550 --> 00:03:03.550
So that's f of x.

00:03:03.550 --> 00:03:05.260
It's a parabola.
y-intercept at 1.

00:03:05.260 --> 00:03:08.660
And let's say I want to figure
out the area under the curve --

00:03:08.660 --> 00:03:11.570
between the curve and,
really, the x-axis.

00:03:11.570 --> 00:03:14.180
Let's say I want to figure out
the area between the curve and

00:03:14.180 --> 00:03:20.030
the x-axis from x equals
negative 1 to, I don't

00:03:20.030 --> 00:03:23.190
know, x equals 3.

00:03:23.190 --> 00:03:25.220
So this is the area I
want to figure out.

00:03:25.220 --> 00:03:26.220
I'm going to shade it in.

00:03:29.970 --> 00:03:33.470
So this is the area.

00:03:33.470 --> 00:03:34.980
All of this stuff.

00:03:34.980 --> 00:03:36.660
I want to figure out this area.

00:03:36.660 --> 00:03:39.620
And you could imagine, before
you knew calculus, figuring out

00:03:39.620 --> 00:03:41.870
an area of something with a
curve -- it's kind

00:03:41.870 --> 00:03:42.630
of top boundary.

00:03:42.630 --> 00:03:44.100
It would have been
very difficult.

00:03:44.100 --> 00:03:46.000
But we will now use the
fundamental theorem of calculus

00:03:46.000 --> 00:03:48.970
and hopefully you have an
intuition of why this works and

00:03:48.970 --> 00:03:52.470
how the integral is really just
a sum of a bunch of little,

00:03:52.470 --> 00:03:55.400
little, small squares with
infinitely small bases.

00:03:55.400 --> 00:03:57.030
But if you watched the last
videos, hopefully that

00:03:57.030 --> 00:03:57.540
hit the point home.

00:03:57.540 --> 00:04:00.310
But now we'll just mechanically
compute because, actually,

00:04:00.310 --> 00:04:02.925
understanding it is a bit
harder than just doing it.

00:04:02.925 --> 00:04:04.390
But let's just
mechanically compute it.

00:04:04.390 --> 00:04:07.470
So we are essentially just
going to figure out the

00:04:07.470 --> 00:04:16.780
integral from minus 1 to 3
of f of x, which is x

00:04:16.780 --> 00:04:23.970
squared plus 1 d x.

00:04:23.970 --> 00:04:27.850
What's the antiderivative
of x squared plus 1?

00:04:27.850 --> 00:04:30.010
This just equals the
antiderivative.

00:04:30.010 --> 00:04:34.630
So it's just x to the third --
we could say 1/3 x to the third

00:04:34.630 --> 00:04:39.860
or x to the third over
3 -- plus x, right?

00:04:39.860 --> 00:04:41.130
The derivative of x is 1.

00:04:41.130 --> 00:04:42.770
And then we don't have to worry
about plus c because we're

00:04:42.770 --> 00:04:44.600
going to subtract out the c's.

00:04:44.600 --> 00:04:45.030
You'll see.

00:04:45.030 --> 00:04:45.850
I think you'll get the point.

00:04:45.850 --> 00:04:46.340
It doesn't matter.

00:04:46.340 --> 00:04:47.800
You could pick an arbitrary
c right here and it'll

00:04:47.800 --> 00:04:48.900
just cancel out.

00:04:48.900 --> 00:04:53.960
And we're going to evaluate
that at 3 and negative 1 and

00:04:53.960 --> 00:04:57.590
we're going to subtract
out big f of negative

00:04:57.590 --> 00:04:59.650
1 from big f of 3.

00:04:59.650 --> 00:05:00.940
This is just the
notation they use.

00:05:00.940 --> 00:05:02.070
You figure out the
antiderivative and

00:05:02.070 --> 00:05:03.710
you say where you're
going to evaluate it.

00:05:03.710 --> 00:05:08.430
And then this is equal to
-- So if I evaluate 3.

00:05:08.430 --> 00:05:10.820
3 to the third power is what?

00:05:10.820 --> 00:05:12.370
That's 27.

00:05:12.370 --> 00:05:18.010
27 divided by 3 is 9.

00:05:18.010 --> 00:05:21.420
And then 9 plus 3 is 12.

00:05:21.420 --> 00:05:22.360
Right?

00:05:22.360 --> 00:05:27.150
This is just big f of 3, right?

00:05:27.150 --> 00:05:30.520
Because I figured out the
end -- This is big f of x.

00:05:30.520 --> 00:05:33.150
You can kind of view
this as big f of x.

00:05:33.150 --> 00:05:36.340
But not to be confused with
small, cursive f of x.

00:05:36.340 --> 00:05:37.860
This is big f of x.

00:05:37.860 --> 00:05:39.430
So this big f of 3.

00:05:39.430 --> 00:05:41.480
And then, from that
we'll subtract big f

00:05:41.480 --> 00:05:43.280
of negative 1, right?

00:05:43.280 --> 00:05:45.950
Minus big f of negative 1.

00:05:45.950 --> 00:05:47.640
And if we put minus 1 here.

00:05:47.640 --> 00:05:50.020
Let's see, minus 1 to the
third power is minus 1.

00:05:50.020 --> 00:05:54.570
So it's minus 1/3 and then
plus minus 1, right?

00:05:54.570 --> 00:05:57.610
So minus 1/3 plus minus 1.

00:05:57.610 --> 00:06:02.690
I think that equals
minus 4/3, correct?

00:06:02.690 --> 00:06:03.470
I think so.

00:06:03.470 --> 00:06:06.420
Maybe I'm making a mistake
with negative signs.

00:06:06.420 --> 00:06:08.790
Minus 1/3 minus 4/3.

00:06:08.790 --> 00:06:10.840
And I'm going to
subtract that, right?

00:06:10.840 --> 00:06:13.010
So if I'm subtracting minus
4/3, it's the same thing

00:06:13.010 --> 00:06:16.850
as adding minus 4/3.

00:06:16.850 --> 00:06:20.040
And then we have our answer.

00:06:20.040 --> 00:06:25.460
Actually, it's 12 and 4/3
-- whatever -- units.

00:06:25.460 --> 00:06:26.290
Squared units.

00:06:26.290 --> 00:06:27.880
12 and 4/3 squared units.

00:06:27.880 --> 00:06:30.480
We could write this as a
mixed number as well.

00:06:30.480 --> 00:06:31.520
Let's do another one.

00:06:31.520 --> 00:06:33.285
I'll do a slight variation.

00:06:40.020 --> 00:06:42.850
OK.

00:06:42.850 --> 00:06:45.110
Let me draw again.

00:06:45.110 --> 00:06:47.596
Some coordinates.

00:06:47.596 --> 00:06:49.210
I don't know if I'm going
to have time to do it in

00:06:49.210 --> 00:06:51.040
this video but I'll try.

00:06:51.040 --> 00:06:52.000
I always try.

00:06:55.060 --> 00:07:07.440
Let's say I have f of x is
equal to the square root of x.

00:07:07.440 --> 00:07:08.590
So it looks something
like this.

00:07:14.750 --> 00:07:17.910
That's actually a pretty nice
looking, kind of sideways

00:07:17.910 --> 00:07:19.790
parabola, I think.

00:07:19.790 --> 00:07:22.960
This is f of x.

00:07:22.960 --> 00:07:25.660
And let's say I have
another function.

00:07:25.660 --> 00:07:32.250
g of x which equals x squared.

00:07:32.250 --> 00:07:34.790
So g of x is actually going
to look something like this.

00:07:41.980 --> 00:07:42.840
Whoops!

00:07:42.840 --> 00:07:45.650
I was doing well and then
something happened.

00:07:45.650 --> 00:07:47.750
And, of course, it'll continue
on this side as well.

00:07:47.750 --> 00:07:50.810
Because it is defined
for negative numbers.

00:07:50.810 --> 00:07:53.990
But anyway, my question to you,
or my question to myself,

00:07:53.990 --> 00:07:58.170
really, is what is the
area between the curves

00:07:58.170 --> 00:07:59.730
where they intersect?

00:07:59.730 --> 00:08:01.340
What is this?

00:08:01.340 --> 00:08:02.420
What is this area?

00:08:08.980 --> 00:08:10.830
Well, the first thing you have
to figure out is just what

00:08:10.830 --> 00:08:11.690
are the boundary points?

00:08:11.690 --> 00:08:12.922
What is this point?

00:08:12.922 --> 00:08:15.350
And what is this point?

00:08:15.350 --> 00:08:16.800
Well this point, I
think, is pretty clear.

00:08:16.800 --> 00:08:19.220
It's 0, 0, right?

00:08:19.220 --> 00:08:21.570
They both intersect at 0, 0.

00:08:21.570 --> 00:08:24.640
And even this point, you could
probably do it from intuition.

00:08:24.640 --> 00:08:30.310
But if you don't, I guess, want
to do it through intuition, you

00:08:30.310 --> 00:08:32.990
could just set these 2
equations equal to

00:08:32.990 --> 00:08:33.580
each other, right?

00:08:33.580 --> 00:08:38.690
You could say x squared
is equal to the square

00:08:38.690 --> 00:08:41.790
root of x, right?

00:08:41.790 --> 00:08:44.330
And then you could do
a bunch of things.

00:08:44.330 --> 00:08:50.450
You could square both sides or
-- well, actually, this is the

00:08:50.450 --> 00:08:51.910
same thing as doing
it by intuition.

00:08:51.910 --> 00:08:54.170
But I think it's pretty obvious
that the only places where x

00:08:54.170 --> 00:08:57.390
squared is equal to the square
root of x are the points x

00:08:57.390 --> 00:09:02.510
equals 0, which you already
know, and x equals 1.

00:09:02.510 --> 00:09:06.060
So this is the point 1, 1.

00:09:06.060 --> 00:09:07.310
Which is true for both of them.

00:09:07.310 --> 00:09:09.005
And this is more algebra,
so I won't go into that

00:09:09.005 --> 00:09:09.890
in too much detail.

00:09:09.890 --> 00:09:12.220
I'm kind of running
out of time.

00:09:12.220 --> 00:09:15.400
So we want to figure out the
area between these 2 curves.

00:09:15.400 --> 00:09:18.280
So what we can do is -- maybe
you want to pause it and think

00:09:18.280 --> 00:09:22.320
about it yourself -- we
can figure out the area

00:09:22.320 --> 00:09:26.840
under the grey curve.

00:09:26.840 --> 00:09:28.180
We could figure out this area.

00:09:30.940 --> 00:09:34.840
So we want to figure out --
this is a boundary, right?

00:09:34.840 --> 00:09:35.760
Between 0 and 1.

00:09:35.760 --> 00:09:40.970
We could figure out this area
and then we could figure out

00:09:40.970 --> 00:09:44.640
the entire area under the green
curve separately and then we

00:09:44.640 --> 00:09:45.700
could subtract the difference.

00:09:45.700 --> 00:09:47.930
Which is exactly how we're
going to do it in the next

00:09:47.930 --> 00:09:50.630
video because I have
run out of time.


Definite Integrals (part 5)

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:01.940
Welcome back.

00:00:01.940 --> 00:00:05.730
Where I had just left off we
were trying to figure out this

00:00:05.730 --> 00:00:08.460
area between these two curves,
and we figured out that it's

00:00:08.460 --> 00:00:12.440
really the area between the
curves between the point 0,

00:00:12.440 --> 00:00:14.520
x equals 0 and x equals 1.

00:00:14.520 --> 00:00:16.840
And I was proposing
of a way to do it.

00:00:16.840 --> 00:00:19.620
Let's figure out the entire
area under the square root of

00:00:19.620 --> 00:00:26.570
x from 0 to 1, and we can
subtract from that, this

00:00:26.570 --> 00:00:29.580
purple area, which is the
area under x squared.

00:00:29.580 --> 00:00:32.220
So just based on the last
example we did, we could just

00:00:32.220 --> 00:00:34.490
write the indefinite integral,
and I'm not going to rewrite

00:00:34.490 --> 00:00:36.970
the fundamental theorem from
calculus, because I think

00:00:36.970 --> 00:00:38.770
you know that by now.

00:00:38.770 --> 00:00:41.080
Let me do it in a loud color.

00:00:41.080 --> 00:00:42.120
Magenta.

00:00:42.120 --> 00:00:45.770
So I want to know the
larger area, right.

00:00:45.770 --> 00:00:47.355
The area under just
the square root of x.

00:00:47.355 --> 00:00:49.660
That's Kind of like
the combined area.

00:00:49.660 --> 00:00:58.710
Well that's just from 0 to 1,
the integral of square root of

00:00:58.710 --> 00:01:04.050
x, because square root of x
is the green function, dx.

00:01:04.050 --> 00:01:13.020
And I want to subtract from
that the area from 0 to 1

00:01:13.020 --> 00:01:15.450
what's under x squared.

00:01:15.450 --> 00:01:18.110
x squared dx.

00:01:18.110 --> 00:01:19.240
And I just want
to make a point.

00:01:19.240 --> 00:01:21.420
This could have just been
rewritten this way.

00:01:21.420 --> 00:01:23.330
You could just rewrite a new
function, which is the

00:01:23.330 --> 00:01:25.410
difference of these two
functions, and it would

00:01:25.410 --> 00:01:26.120
have been equivalent.

00:01:26.120 --> 00:01:27.070
You could have said this.

00:01:27.070 --> 00:01:28.820
This isn't kind of a step of
the problem, but you could

00:01:28.820 --> 00:01:29.520
have done it this way.

00:01:29.520 --> 00:01:32.080
In fact, some people
start this way.

00:01:32.080 --> 00:01:35.960
See those are the same thing
as the integral from 0 to 1

00:01:35.960 --> 00:01:42.120
of square root of x
minus x squared dx.

00:01:42.120 --> 00:01:44.720
So you could do this two
separate problems, two separate

00:01:44.720 --> 00:01:47.450
indefinite integrals, or
you could do it as one

00:01:47.450 --> 00:01:48.080
indefinite integral.

00:01:48.080 --> 00:01:49.800
Actually that might be even
simpler because when you

00:01:49.800 --> 00:01:52.620
evaluate it from 1 to 0 it
simplify things a little bit.

00:01:52.620 --> 00:01:54.150
So let's stick with
the second one.

00:01:54.150 --> 00:01:55.710
So first of all we just have
to figure out what the

00:01:55.710 --> 00:01:58.930
antiderivative of this
inner expression is.

00:01:58.930 --> 00:02:02.720
So you haven't seen
square root of x yet.

00:02:02.720 --> 00:02:04.100
Do you think you
know how to do it?

00:02:04.100 --> 00:02:06.480
Well, I think you do.

00:02:06.480 --> 00:02:10.060
Let's say that equals square
root of x is just x to

00:02:10.060 --> 00:02:11.940
the 1/2 power, right?

00:02:11.940 --> 00:02:14.020
So we just use the same
antiderivative rules

00:02:14.020 --> 00:02:15.160
we've always used.

00:02:15.160 --> 00:02:21.110
We raise it one more power
so it becomes x to the 3/2.

00:02:21.110 --> 00:02:23.230
Right, it was 1/2,
we added 1 to it.

00:02:23.230 --> 00:02:25.860
And then we divide by
this new exponent.

00:02:25.860 --> 00:02:28.370
So dividing by a fraction
is like multiplying

00:02:28.370 --> 00:02:30.920
by its reciprocal.

00:02:30.920 --> 00:02:36.840
So it's 2/3 x to the 3/2 and
then minus-- I think the second

00:02:36.840 --> 00:02:42.780
term is pretty easy for you--
minus 1/3 x to the third.

00:02:42.780 --> 00:02:46.250
That's the antiderivative
of minus x squared, minus

00:02:46.250 --> 00:02:48.200
1/3 x to the third.

00:02:48.200 --> 00:02:56.120
And we're going to have to
evaluate this thing at 0 and 1

00:02:56.120 --> 00:02:57.770
and subtract the difference.

00:02:57.770 --> 00:03:01.040
Subtract this expression
evaluated at 0 from this

00:03:01.040 --> 00:03:04.030
expression evaluated at 1.

00:03:04.030 --> 00:03:05.080
I think I'm running
out of space.

00:03:05.080 --> 00:03:07.070
What happens when x equals 1?

00:03:07.070 --> 00:03:08.810
1 to the 3/2 is 1.

00:03:08.810 --> 00:03:09.755
1 to the third is 1.

00:03:09.755 --> 00:03:11.130
So it's 2/3 minus 1/3.

00:03:11.130 --> 00:03:12.370
Well that's easy.

00:03:12.370 --> 00:03:15.180
It's 1/3.

00:03:15.180 --> 00:03:17.030
I just put 1 in for x.

00:03:17.030 --> 00:03:19.470
And then when x is equal to 0
what is this expression equal?

00:03:19.470 --> 00:03:20.700
Well that's easy too.

00:03:20.700 --> 00:03:22.530
That's 0.

00:03:22.530 --> 00:03:23.720
So there you go.

00:03:23.720 --> 00:03:26.520
1/3 minus 0 or 1/3.

00:03:26.520 --> 00:03:27.610
That's kind of neat.

00:03:31.030 --> 00:03:35.940
You know I find is this of
exciting because if just my

00:03:35.940 --> 00:03:38.940
intuition I was like, oh I have
these two curves, and I mean

00:03:38.940 --> 00:03:43.320
they do intersect at the nice
integer number, but you know

00:03:43.320 --> 00:03:45.830
what it's probably going to be
some really messy number

00:03:45.830 --> 00:03:49.560
of what the areas between
these two curves, right.

00:03:49.560 --> 00:03:51.990
Who knows, maybe it'll involve
some you know-- a circle

00:03:51.990 --> 00:03:54.750
involves pi, which is this
really messy number, so maybe

00:03:54.750 --> 00:03:57.100
all curves have these
kind of messy areas.

00:03:57.100 --> 00:04:00.830
But this one it's just one of
those neat things about math.

00:04:00.830 --> 00:04:05.360
The area between the square
root of x and x squared is 1/3,

00:04:05.360 --> 00:04:07.000
which is a pretty clean number.

00:04:10.820 --> 00:04:13.100
Actually let me do one more
problem since I have time.

00:04:18.260 --> 00:04:19.250
It's a bit of a trick problem.

00:04:19.250 --> 00:04:24.970
I mean, you might actually find
this easy, but let's figure out

00:04:24.970 --> 00:04:38.310
the area between f of x
is equal to-- I don't

00:04:38.310 --> 00:04:41.130
know, x to the fifth.

00:04:41.130 --> 00:04:42.120
I'm going to do
something simple.

00:04:42.120 --> 00:04:43.645
Let me draw it actually.

00:04:56.930 --> 00:04:58.730
OK, I'm going to
draw the x-axis.

00:05:04.660 --> 00:05:06.990
x to the fifth is going
to go up super fast,

00:05:06.990 --> 00:05:07.780
something like that.

00:05:07.780 --> 00:05:09.220
It's going to go up real fast.

00:05:09.220 --> 00:05:10.640
Let's say I wanted to figure
out the areas-- this side's

00:05:10.640 --> 00:05:13.930
going to go real fast too--
between that, and instead of

00:05:13.930 --> 00:05:19.180
figuring out the area between
x-axis and that, f of x, I want

00:05:19.180 --> 00:05:28.730
to figure out the area between
f of x and-- Instead

00:05:28.730 --> 00:05:31.530
of figuring out this
bottom area, right?

00:05:31.530 --> 00:05:34.380
Like the normal problems we've
done, we would figure out this

00:05:34.380 --> 00:05:36.950
type of area, you know,
between two points.

00:05:36.950 --> 00:05:49.710
Let's say I want to figure out
the area inside of the curve

00:05:49.710 --> 00:05:56.090
where the height here is 32.

00:05:56.090 --> 00:05:59.790
So I want to figure out this
area inside the curve.

00:05:59.790 --> 00:06:02.070
How do we do that?

00:06:02.070 --> 00:06:05.010
Well one way we could do it is
just like we did the last one,

00:06:05.010 --> 00:06:07.970
we can figure out some function
that's essentially a line, a

00:06:07.970 --> 00:06:11.030
horizontal line that goes
straight across here.

00:06:11.030 --> 00:06:14.800
And we'll essentially just
be figuring out the area

00:06:14.800 --> 00:06:16.050
between the two functions.

00:06:16.050 --> 00:06:18.460
So what's a function that's
a line that just goes

00:06:18.460 --> 00:06:20.820
straight at y equals 32?

00:06:20.820 --> 00:06:23.000
I think I just gave
you the answer.

00:06:23.000 --> 00:06:25.720
Exactly.

00:06:25.720 --> 00:06:28.580
Let me stick with
that greenish color.

00:06:28.580 --> 00:06:32.200
So we could say g of
x is equal to 32.

00:06:32.200 --> 00:06:33.415
It's just a constant function.

00:06:33.415 --> 00:06:35.510
It just goes straight across.

00:06:35.510 --> 00:06:38.310
And then we need to figure out
what the area is between the

00:06:38.310 --> 00:06:40.050
two, so we need to figure out
what are these two points.

00:06:42.990 --> 00:06:47.240
So when does x to
the fifth equal 32?

00:06:47.240 --> 00:06:50.000
I mean you could solve it
algebraically, you know,

00:06:50.000 --> 00:06:53.954
you could say x to
the fifth equals 32.

00:06:53.954 --> 00:07:03.420
x is equal to 2, and
actually, you know what?

00:07:03.420 --> 00:07:05.020
I made a mistake.

00:07:05.020 --> 00:07:07.760
Let's say that this is not
equal to x to the fifth.

00:07:07.760 --> 00:07:10.950
Let's say that f of x is
equal to x to the absolute

00:07:10.950 --> 00:07:13.270
value of x to the fifth.

00:07:13.270 --> 00:07:15.590
Because the mistake, obviously
x to the fifth is not a

00:07:15.590 --> 00:07:17.420
parabola looking thing.
x to the fifth goes

00:07:17.420 --> 00:07:18.630
negative like this.

00:07:18.630 --> 00:07:22.070
But I have committed so much to
this cup shape that I'll make

00:07:22.070 --> 00:07:24.350
it the absolute value
of x to the fifth.

00:07:24.350 --> 00:07:27.290
So if I say the absolute value
of x to the fifth is equal to

00:07:27.290 --> 00:07:30.170
32, I think you see where
I realized my mistake.

00:07:30.170 --> 00:07:33.280
But if I say the absolute value
of x to the fifth is 32,

00:07:33.280 --> 00:07:35.800
there's two places
where that's true.

00:07:35.800 --> 00:07:38.430
It's x is equal to
plus or minus 2.

00:07:38.430 --> 00:07:39.670
These are the two points.

00:07:39.670 --> 00:07:41.795
I should have done something
with an even exponent so I

00:07:41.795 --> 00:07:43.520
could have had this cup shape,
but anyway the absolute

00:07:43.520 --> 00:07:46.280
value solved my problem.

00:07:46.280 --> 00:07:48.740
So what is this area?

00:07:48.740 --> 00:07:51.180
We know it's between negative
2 and 2, so we just set up

00:07:51.180 --> 00:07:52.300
the indefinite integral.

00:07:52.300 --> 00:07:59.210
It's the indefinite integral
for minus 2 to 2 of the top

00:07:59.210 --> 00:08:04.350
function, the top boundary, 32,
minus the bottom boundary.

00:08:07.250 --> 00:08:11.270
Well, this will be a little bit
tricky, but minus the absolute

00:08:11.270 --> 00:08:16.360
value of x to the fifth dx.

00:08:16.360 --> 00:08:18.660
And actually instead of doing
this, I think you could see

00:08:18.660 --> 00:08:21.160
that there's symmetry here,
so we could just figure out

00:08:21.160 --> 00:08:23.690
this area and multiply by 2.

00:08:23.690 --> 00:08:27.223
This problem's a little hairy,
just because I had a bad

00:08:27.223 --> 00:08:29.160
choice of initial function.

00:08:29.160 --> 00:08:32.410
Not exactly what I wanted,
but we'll work on forward.

00:08:32.410 --> 00:08:38.640
So instead of doing that, let's
do the integral from 0 to 2 of

00:08:38.640 --> 00:08:45.020
32 minus x to the fifth dx.

00:08:45.020 --> 00:08:47.340
And then multiply that by 2.

00:08:47.340 --> 00:08:48.060
So what is that?

00:08:48.060 --> 00:08:56.060
That's 32x minus x to
the sixth over 6.

00:08:56.060 --> 00:09:05.720
And we're going to evaluate it
from 2 and 0 at 64 minus 64/6.

00:09:05.720 --> 00:09:13.860
2 to the sixth is 64, and
then 32 times 0 is 0 and the

00:09:13.860 --> 00:09:20.370
next is 6, that's 0, so the
answer is 64 minus 64/6.

00:09:20.370 --> 00:09:21.880
I'm about to run out of time.

00:09:21.880 --> 00:09:23.320
That's just a fraction
problem there.

00:09:23.320 --> 00:09:25.230
Oh, and that's half
of it, right?

00:09:25.230 --> 00:09:27.370
So we want to
multiply that by 2.

00:09:27.370 --> 00:09:34.290
So if we multiply that by
2, we get 128 minus 128/6.

00:09:34.290 --> 00:09:35.980
I haven't figured
out what it is.

00:09:35.980 --> 00:09:36.850
Well I guess we could
figure it out.

00:09:36.850 --> 00:09:45.840
It's 128 times 1 minus
1/6 or 128 times 5/6.

00:09:45.840 --> 00:09:48.280
And I don't know what that is.

00:09:48.280 --> 00:09:51.100
I can multiply if I wanted to,
but I have 10 seconds left

00:09:51.100 --> 00:09:52.400
so I'll leave you there.

00:09:52.400 --> 00:09:53.365
Hope I didn't confuse you.

00:09:53.365 --> 00:09:54.170
See you soon.


Indefinite Integration (part V)

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:03.830
I'm not going to do
a presentation on a

00:00:03.830 --> 00:00:04.710
type of integral.

00:00:04.710 --> 00:00:07.680
I guess if you have this in
your tool kit-- and actually

00:00:07.680 --> 00:00:12.910
you have it beyond the exam on
this type of integral, and you

00:00:12.910 --> 00:00:15.290
actually keep it and you retain
it, then you, I think, will

00:00:15.290 --> 00:00:17.890
become an integration jock.

00:00:17.890 --> 00:00:20.290
But anyway, let me show you
what I'm talking about.

00:00:20.290 --> 00:00:23.020
So let's just remember
what the product rule

00:00:23.020 --> 00:00:24.320
of differentiation was.

00:00:24.320 --> 00:00:29.990
So let's say I had two
functions, let's say f of x

00:00:29.990 --> 00:00:36.505
times g of x, and I wanted to
take the derivative of this.

00:00:39.700 --> 00:00:42.930
f of x times g of x.

00:00:42.930 --> 00:00:45.260
Well the chain rule just told
us that this is just the same

00:00:45.260 --> 00:00:47.420
thing as, let's say, the
derivative of the first

00:00:47.420 --> 00:00:58.590
function f prime of x times the
second function g of x plus now

00:00:58.590 --> 00:01:04.350
the first function f of x times
the derivative of the

00:01:04.350 --> 00:01:05.996
second function.

00:01:05.996 --> 00:01:10.210
And I'll show you where I'm
going with this in a second.

00:01:10.210 --> 00:01:14.160
Now, if we were to integrate
both sides of this equation,

00:01:14.160 --> 00:01:17.340
because we're still doing
algebra on some levels,

00:01:17.340 --> 00:01:18.780
anything you do to one side
of the equation, you

00:01:18.780 --> 00:01:20.140
can do to the other.

00:01:20.140 --> 00:01:22.160
So if we were to integrate both
sides-- well if you integrate

00:01:22.160 --> 00:01:27.040
this side, you're taking the
integral of a derivative, you

00:01:27.040 --> 00:01:29.430
just get back to what you took
the derivative originally of,

00:01:29.430 --> 00:01:35.915
so this just becomes f of x
times g of x, and then we have

00:01:35.915 --> 00:01:39.000
to integrate the right hand
side, well that just becomes--

00:01:39.000 --> 00:01:40.990
and we're doing the indefinite
integral, kind of the

00:01:40.990 --> 00:01:43.360
antiderivative, but we can
use this tool when we do

00:01:43.360 --> 00:01:45.540
definite integrals as well.

00:01:45.540 --> 00:01:55.960
So that's the integral of f
prime of x g of x dx plus

00:01:55.960 --> 00:02:04.840
the integral of f of x
g prime of x d of x.

00:02:04.840 --> 00:02:07.580
And now this might seem a
little bit arbitrary, and it is

00:02:07.580 --> 00:02:11.560
a little bit arbitrary, let me
take-- well, I could take

00:02:11.560 --> 00:02:16.290
either of these-- but let me
just take this one and move it

00:02:16.290 --> 00:02:17.200
to this side of the equation.

00:02:17.200 --> 00:02:19.010
So I'm going to subtract
this term from both

00:02:19.010 --> 00:02:20.210
sides of this equation.

00:02:20.210 --> 00:02:25.120
And so, we could say this-- so
let me change colors, because

00:02:25.120 --> 00:02:28.330
this could get confusing-- this
term right here, we could say

00:02:28.330 --> 00:02:43.785
that term f of x g prime of x d
of x is equal to this term--

00:02:43.785 --> 00:02:47.770
let me switch back to the
yellow-- is equal to f of

00:02:47.770 --> 00:02:56.260
x g of x minus this term.

00:02:56.260 --> 00:02:58.105
Because I put it onto this
side of the equation.

00:03:07.640 --> 00:03:08.930
So what did I just do?

00:03:08.930 --> 00:03:12.300
It looks like I just-- well, I
am just essentially playing

00:03:12.300 --> 00:03:14.170
with the product rule
from differentiation.

00:03:14.170 --> 00:03:15.020
That's all I did.

00:03:15.020 --> 00:03:16.880
And you probably wondering,
well, Sal, this is all nice

00:03:16.880 --> 00:03:20.910
and it looks fancy, but what
good is this going to do me?

00:03:20.910 --> 00:03:26.140
Well, what I essentially just
did is I kind of proved this--

00:03:26.140 --> 00:03:29.950
you could call this a formula,
but I often forget it,

00:03:29.950 --> 00:03:32.450
especially once I haven't done
it a long time, and then I

00:03:32.450 --> 00:03:35.350
actually just reprove it to
myself just by remembering the

00:03:35.350 --> 00:03:39.250
product rule-- but this is
called integration by parts.

00:03:39.250 --> 00:03:42.030
And I'll show you
where this is useful.

00:03:42.030 --> 00:03:44.130
Let's say we want to take
the indefinite integral

00:03:44.130 --> 00:03:58.590
of x cosine of x d of x.

00:03:58.590 --> 00:04:02.580
Well, everything we have in our
integration tool kit so far I

00:04:02.580 --> 00:04:04.130
don't think help
us here, right?

00:04:04.130 --> 00:04:06.680
Because we don't have a
function and its derivative, so

00:04:06.680 --> 00:04:10.540
we can't do substitution or--
which is the same thing as the

00:04:10.540 --> 00:04:13.550
reverse chain rule-- this
isn't a simple polynomial.

00:04:13.550 --> 00:04:17.550
So if you encounter this when
you're doing integrals, kind of

00:04:17.550 --> 00:04:21.480
the last tool kit-- and this is
pretty sophisticated-- is to

00:04:21.480 --> 00:04:22.770
do integration by parts.

00:04:22.770 --> 00:04:24.795
And so how can we use this
for integration by parts?

00:04:24.795 --> 00:04:27.750
Well, integration by parts
tells us that if we have an

00:04:27.750 --> 00:04:29.910
integral where we have a
function and then the

00:04:29.910 --> 00:04:33.480
derivative of another function,
then we could use this formula

00:04:33.480 --> 00:04:35.590
to hopefully simplify it.

00:04:35.590 --> 00:04:38.250
So what I'm going to do-- and
you might view this as, well,

00:04:38.250 --> 00:04:39.580
Sal, how did you
know to do this?

00:04:39.580 --> 00:04:42.370
And I'll tell you my thought
process actually after

00:04:42.370 --> 00:04:44.270
I show you what I did.

00:04:44.270 --> 00:04:47.400
We always, in math in general,
you always want to simplify.

00:04:47.400 --> 00:04:50.800
You always want to move from
something that's complicated

00:04:50.800 --> 00:04:52.440
to something that's simpler.

00:04:52.440 --> 00:04:57.800
So in this situation, we could
assume that x is f of x, and we

00:04:57.800 --> 00:05:00.730
could assume that g prime of x
is cosine of x, or we could

00:05:00.730 --> 00:05:02.420
assume the other way around.

00:05:02.420 --> 00:05:08.070
The reason why I'm going to
assume that x is f of x, and

00:05:08.070 --> 00:05:12.450
I'm going to assume g prime of
x is cosine of x, is because

00:05:12.450 --> 00:05:15.540
later we want to take the
derivative of f of x.

00:05:15.540 --> 00:05:18.400
The derivative of f of x
simplifies things a lot.

00:05:18.400 --> 00:05:22.730
And we also want to take the
integral of g prime of x.

00:05:22.730 --> 00:05:25.430
We want to take the
antiderivative of g prime of x.

00:05:25.430 --> 00:05:29.680
And the antiderivative of
cosine of x is sine of x,

00:05:29.680 --> 00:05:31.280
which is just as complicated.

00:05:31.280 --> 00:05:33.440
It's not making it any
more complicated.

00:05:33.440 --> 00:05:36.510
And actually, try it the other
way around, and you'll see that

00:05:36.510 --> 00:05:38.380
if you took the antiderivative
of x, you would get something

00:05:38.380 --> 00:05:39.390
that's more complicated.

00:05:39.390 --> 00:05:40.740
You get x squared over 2.

00:05:40.740 --> 00:05:44.020
So that's the intuition, and
let me just solve through it,

00:05:44.020 --> 00:05:46.920
and hopefully it'll make
a little bit more sense.

00:05:46.920 --> 00:05:51.540
So if I assume that f of x is
x, and g prime of x is cosine

00:05:51.540 --> 00:05:55.170
of x, then f of x-- this yellow
term, let me write it in

00:05:55.170 --> 00:05:58.695
yellow, just for fun-- so f of
x-- so I'm saying

00:05:58.695 --> 00:06:00.120
that f of x is x.

00:06:03.510 --> 00:06:05.300
So that's x.

00:06:05.300 --> 00:06:10.580
And let's say g of x is
cosine of x, right?

00:06:10.580 --> 00:06:13.690
I'm sorry, g prime of
x is cosine of x.

00:06:13.690 --> 00:06:16.840
So the derivative of g
of x is cosine of x.

00:06:16.840 --> 00:06:18.340
So what's g of x?

00:06:18.340 --> 00:06:21.250
It was the antiderivative
of cosine of x.

00:06:21.250 --> 00:06:22.350
So that's sine of x.

00:06:24.970 --> 00:06:26.310
I hope you understand
what I'm saying.

00:06:26.310 --> 00:06:27.230
This can be a little confusing.

00:06:27.230 --> 00:06:29.340
Let me write it
separately here.

00:06:29.340 --> 00:06:32.120
Actually, well I'm running out
of space, but I'll write it in

00:06:32.120 --> 00:06:34.340
the corner right down here.

00:06:34.340 --> 00:06:38.600
See, I'm saying that f of x
is x and I'm saying that

00:06:38.600 --> 00:06:43.300
g of x is sine of x.

00:06:43.300 --> 00:06:45.500
And the reason why I knew g of
x is sine of x is because I

00:06:45.500 --> 00:06:51.630
said the derivative, I said g
prime of x, is cosine of x.

00:06:51.630 --> 00:06:55.360
So you know, if the function--
if the derivative of a function

00:06:55.360 --> 00:06:57.970
is cosine, then we know the
function itself is sine.

00:06:57.970 --> 00:06:59.450
That's just something
you memorize.

00:06:59.450 --> 00:07:01.880
I haven't proven it to you yet,
but it's usually in the inside

00:07:01.880 --> 00:07:02.920
cover of your calculus book.

00:07:02.920 --> 00:07:04.540
But let's just move forward.

00:07:04.540 --> 00:07:11.460
And then this is minus
the integral of the

00:07:11.460 --> 00:07:12.750
derivative of f of x.

00:07:12.750 --> 00:07:14.045
Well what's the
derivative of f of x?

00:07:14.045 --> 00:07:16.600
Well, we said f of
x is x, right?

00:07:16.600 --> 00:07:21.870
So the derivative is
just 1 times g of x.

00:07:21.870 --> 00:07:23.926
Well, I already said that
g of x is sine of x.

00:07:27.470 --> 00:07:30.430
And I think you would agree
that we've now simplified this

00:07:30.430 --> 00:07:32.340
a good bit, because this is
just the integral of

00:07:32.340 --> 00:07:33.860
sine of x, right?

00:07:33.860 --> 00:07:39.290
So this is just equal to x sine
of x-- this is just this first

00:07:39.290 --> 00:07:42.860
term right here-- minus-- and
what's the integral

00:07:42.860 --> 00:07:44.540
of sine of x?

00:07:44.540 --> 00:07:49.960
Well, the derivative of-- well,
let me make it even simpler.

00:07:49.960 --> 00:07:53.470
Well we can make this a minus,
we can make this a minus sign

00:07:53.470 --> 00:07:55.930
of x and make this a plus.

00:07:55.930 --> 00:07:57.010
And now it's really easy.

00:07:57.010 --> 00:07:59.120
What's the antiderivative
of minus sine of x?

00:07:59.120 --> 00:08:00.580
We can ignore this 1.

00:08:00.580 --> 00:08:02.800
What's the antiderivative
of minus sine of x?

00:08:02.800 --> 00:08:06.510
Well, yeah, it's
just cosine of x.

00:08:06.510 --> 00:08:10.810
And we should never
forget the plus c.

00:08:10.810 --> 00:08:16.010
So we just used the product
rule to derive this formula for

00:08:16.010 --> 00:08:18.240
integration by parts, and in a
lot of calculus books they

00:08:18.240 --> 00:08:21.280
do this u and v and dvd.

00:08:21.280 --> 00:08:22.740
This is the same exact thing.

00:08:22.740 --> 00:08:27.350
I like this more, because
it naturally makes

00:08:27.350 --> 00:08:27.970
more sense to me.

00:08:27.970 --> 00:08:31.570
It's easier for me to read,
and I can derive at any time,

00:08:31.570 --> 00:08:32.820
just from the product rule.

00:08:32.820 --> 00:08:34.500
So I don't have to
necessarily memorize it.

00:08:34.500 --> 00:08:36.290
You might want to memorize it
for the exam, because it's

00:08:36.290 --> 00:08:38.920
faster for when you
take the AP exam.

00:08:38.920 --> 00:08:43.910
But when you do calculus-- it's
been about fifteen years since

00:08:43.910 --> 00:08:47.430
I learned it-- and I just
remember integration by parts

00:08:47.430 --> 00:08:50.750
is just really a derivation of
the product rule, and that

00:08:50.750 --> 00:08:51.590
gets me back to the format.

00:08:51.590 --> 00:08:52.920
I don't even have
to look it up.

00:08:52.920 --> 00:08:54.260
And then I can use it.

00:08:54.260 --> 00:08:57.930
So that's an introduction
to integration by parts.

00:08:57.930 --> 00:09:00.890
In the next presentation, I
will do a bunch of examples--

00:09:00.890 --> 00:09:03.330
well, as many as I can fit in
to ten minutes of actually

00:09:03.330 --> 00:09:07.930
using integration by parts to
solve fairly fancy integrals.

00:09:07.930 --> 00:09:09.850
I'll see you in the
next presentation.