Geometric precision practice

WEBVTT
Kind: captions
Language: en

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- [Voiceover] A lot of
what geometry is about

00:00:02.625 --> 00:00:05.003
is proving things about the world.

00:00:05.003 --> 00:00:08.477
And in order to really prove
things about the world,

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we have to be very careful, very precise,

00:00:11.036 --> 00:00:12.702
very exact with our language

00:00:12.702 --> 00:00:14.064
so that we know what we're proving

00:00:14.064 --> 00:00:16.056
and we know what we're assuming

00:00:16.056 --> 00:00:19.104
and what type of deductions we are making

00:00:19.104 --> 00:00:20.871
as we prove things.

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And so, to get some practice being precise

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and exact with our language,

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I'm going to go through some exercises

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from the Geometric Definitions Exercise

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on Khan Academy.

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So this first one says,
"Three students attempt

00:00:36.688 --> 00:00:39.817
"to define what it means for lines l and m

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"to be perpendicular.

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"Can you match the teacher's
comments to the definitions?"

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All right. So looks like
three different students

00:00:47.030 --> 00:00:50.505
attempt definitions of what
it means to be perpendicular,

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and then there's these teacher's comments

00:00:52.178 --> 00:00:54.088
that we can move around.

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So we're gonna, I guess,
pretend that we're the teacher.

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So Ruby's definition
for being perpendicular:

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"l and m," lines l and
m, "are perpendicular

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"if they never meet."

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Well, that's not true.

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In fact, perpendicular lines
for sure will intersect.

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So, in fact, they
intersect at right angles.

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So that is not going to be,

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that's not going to be correct.

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And so, actually this looks right.

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"Were you thinking of parallel lines?"

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Because that's looks like
what she was trying to define.

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If things are on the same
plane and they never intersect,

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then you are talking about parallel lines.

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I also got Shriya's definition.

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"l and m are perpendicular
if the meet at one point

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"and one of the angles at
their point of intersection

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"is a right angle."
Well, that seems spot on.

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So let's see.

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I would say, "Woohoo! Nice Work!

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"I couldn't have said it better myself."

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Now let's just make sure this comment

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matches for this definition.

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Abhishek says, "l and m are perpendicular

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"if they meet at a single point,

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"such that the two lines make a 'T'."

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Well, that's, in a hand-wavy
way, kind of right.

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When you imagine perpendicular lines,

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you could imagine them kind of forming

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a cross or, I guess, part of ...

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You know, a T would be part of it.

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But I think this comment is spot-on.

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The teacher's comment, "Your
definition is kind of correct,

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"but it lacks mathematical precision."

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You know, what does he mean by a T?

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What does it mean to "make a T"?

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Shriya's definition is much more precise.

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They're perpendicular "if
they meet at one point

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"and one of their angles at
their point of intersection

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"is a right angle," is a 90 degree angle.

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Let's check our answers.

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Let's do a few more of these.

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This is actually a lot of fun.

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So once again, we're
gonna have three students

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attempting to define, but
now they're going to define

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"an object called an angle."

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"Can you match the teacher's
comments to the definitions?"

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So Ruby, oh, it's three,
same three students.

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Ruby says, "The amount of turn
between two straight lines

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"that have a common vertex."

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Well, this is kind of getting there.

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The definition of an angle,
we typically talk about

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two rays with a common vertex.

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She's talking about two
lines with a common vertex.

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And she's talking about
the amount of turn.

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So she's really talk
about more of, kind of,

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the measure of an angle.

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So let's see what comment here.

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So, "You seem to be getting at the idea

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"of the measure of an angle,

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and "not the definition
of the angle itself."

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So this is actually right.

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I would put this one right here.

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We just got lucky.

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This was already aligned.

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So Shriya's definition: "Two
lines that come together."

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So, once again, this is kind of ...

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The definition of an angle is
two rays with a common vertex.

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So, "Two lines that come together,"

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this is just intersecting lines.

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Now, when that happens, you
might be forming some angles,

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but I would just say, "Were you thinking

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of intersecting lines?"

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And let's see what Abhishek says.

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"A figure composed of two rays
sharing a common endpoint.

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"The common endpoint is
known as the vertex."

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Yup. That's a good definition of an angle.

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So Abhishek got it this time.

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Let's do another one.

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So three students are
now attempting to define

00:04:05.645 --> 00:04:08.347
"what it means for two
lines to be parallel."

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So now let's match the teacher's comments.

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So Daniela says, "Two lines are parallel

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"if they are distinct
and one can be translated

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"on top of the other."

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All right.

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So that actually seems pretty interesting.

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That's actually not the first way

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that I would have defined parallel lines.

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I would have said, "Hey, if
they're on the same plane

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"and they don't intersect,
then they are parallel."

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But this is, this seems pretty good

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because, if you're translating something,

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you're not, you aren't going to rotate it,

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you're not going to change its direction,

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I guess, one way to think about it.

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And so, if you're translating one ...

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If you can trans-

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If they're two different lines,

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but you can shift them without
changing their direction,

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which is what translation is all about,

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on top of each other, that
actually feels pretty good.

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So, I'll put that right over there.

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So Ori says, "Two lines are parallel

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"if they are close together
but don't intersect."

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So, if you're trying to
define parallel lines,

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parallel lines, it doesn't matter

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if they're close together or not.

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It's, they just have to be

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in the same plane and not intersect.

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They can be very far apart and
they could still be parallel.

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So this isn't an incorrect statement.

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You could have two lines
that are close together

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and don't intersect on the same plane,

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and they are going to be parallel.

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But this isn't a good definition,

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because you can also have parallel lines

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that are far apart.

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And so, actually I'd go with
this statement right here:

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"Part of your definition is correct,

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"but the other part is not.

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"Parallel lines don't have
to be close together."

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So this isn't a good
definition of parallel lines.

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So let's see. Kaori:
"Two lines are parallel

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"as long as they aren't perpendicular."

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Well, that's just not true,

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because you can intersect,

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you can have two lines that intersect

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at non-right angles, and
they're not parallel,

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and they're also not perpendicular.

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So this is, you know, "Sorry,
your definition is incorrect."

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This is actually a lot of fun,

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pretending to be the teacher.

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Let's do another one.

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All right.

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So "Three students attempt to define

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"what a line segment is."

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And we have a depiction of a
line segment right over here.

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We have point P, point
Q, and the line segment

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is all the points in between P and Q.

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So, so let's match the teacher's comments

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to the definitions.

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Ivy's definition: "All of the points

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"in line with P and Q,
extending infinitely

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"in both directions."

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Well, that would be the
definition of a line.

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That would be the line P, Q.

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That would be, if you're
extending infinitely

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in both directions, so ...

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I would say, "Are you thinking of a line

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"instead of a line segment?"

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Ethan's definition: "The
exact distance from P to Q.

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Well, that's just a ...

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that's the length of a line segment.

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That's not exactly what a line segment is.

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And see, Ebuka's definition.

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"The points P and Q, which
are called endpoints,

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"and all of the points in a straight line

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"between points P and Q.

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Yep. That looks like a good definition

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for a line segment.

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So we can just check our,

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we can just check our answer.

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So, looking good.

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Let's do one more of this.

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I'm just really enjoying
pretending to be a teacher.

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All right. "Three
students attempt to define

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"what a circle is."

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Define what a circle is.

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"Can you match the teacher's
comments to the definitions?"

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Duru. "The set of all points in a plane

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"that are the same distance
away from some given point,

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"which we call the center."

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That just seems like a pretty
good definition of a circle.

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So, I would, you know,
"Stupendous! Well done."

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Oliver's definition.

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"The set of all points in 3D space

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"that are the same distance
from a center point."

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If we're talking about 3D space

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and the set of all points
that are equidistant

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from that point in 3D space,

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now we're talking about
a sphere, not a circle.

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And so, "You seem to be confusing

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"a circle with a sphere."

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And then, finally, "A
perfectly round shape."

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Well, that's kind of true.

00:08:06.118 --> 00:08:07.703
But if you're talk about three dimensions,

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you could be talking about a sphere.

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If you're talking about, if
you go beyond three dimensions,

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hypersphere, whatever else.

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In two dimensions, yeah,
perfectly round shape,

00:08:14.875 --> 00:08:16.683
most people would call it a circle.

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But that doesn't have a
lot of precision to it.

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It doesn't have, it doesn't give us a lot

00:08:20.238 --> 00:08:23.062
that we can work with from a
mathematical point of view.

00:08:23.062 --> 00:08:25.094
So I would say, actually,
what the teacher's saying:

00:08:25.094 --> 00:08:27.634
"Your definition needs
to be much more precise."

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Duru's definition is
much, much more precise.

00:08:30.742 --> 00:08:34.143
The set of all points that
are equidistant from ...

00:08:34.143 --> 00:08:36.228
in a plane, that are equidistant away

00:08:36.228 --> 00:08:38.648
from a given point,
which we call the center.

00:08:38.648 --> 00:08:42.451
So yep. Carlos could use a
little bit more precision.

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We're all done.


Analyzing polynomial manipulations

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:01.974
- [Voiceover] What I
hope to do in this video

00:00:01.974 --> 00:00:04.098
is give ourselves some practice

00:00:04.098 --> 00:00:06.931
at critically looking at how folks

00:00:06.931 --> 00:00:09.090
manipulate polynomials.

00:00:09.090 --> 00:00:11.377
And the reason why this
is useful is because

00:00:11.377 --> 00:00:13.676
it's useful to be able
to do this to yourself

00:00:13.676 --> 00:00:15.731
as you manipulate polynomials to say wait,

00:00:15.731 --> 00:00:17.309
what did I exactly do there?

00:00:17.309 --> 00:00:19.074
Or a lot of times if you're reading a math

00:00:19.074 --> 00:00:20.688
or a science book they're going to do

00:00:20.688 --> 00:00:22.278
some proof or something like that,

00:00:22.278 --> 00:00:23.082
and they're gonna say oh well you know,

00:00:23.082 --> 00:00:24.336
it's obviously from
this step to this step,

00:00:24.336 --> 00:00:25.520
and you're gonna try to follow it and

00:00:25.520 --> 00:00:27.238
you're just like, well
does that make sense?

00:00:27.238 --> 00:00:29.502
So it's a really useful
muscle to be able to see

00:00:29.502 --> 00:00:33.797
do these steps, or whoever
manipulated the polynomial,

00:00:33.797 --> 00:00:35.670
does it make sense to you?

00:00:35.670 --> 00:00:37.504
And especially if it's
you, it's super useful

00:00:37.504 --> 00:00:39.222
to be able to find if there are errors,

00:00:39.222 --> 00:00:40.685
and to correct them, it'll just

00:00:40.685 --> 00:00:43.065
give you a better critical
eye for this type of thing.

00:00:43.065 --> 00:00:44.446
So let's just start with this one.

00:00:44.446 --> 00:00:47.824
We have 4x minus three
times x minus two squared.

00:00:47.824 --> 00:00:50.262
And it looks like this
person over five steps

00:00:50.262 --> 00:00:52.166
tries to expand it out.

00:00:52.166 --> 00:00:53.165
And so what I encourage you to do,

00:00:53.165 --> 00:00:54.685
pause the video right now,

00:00:54.685 --> 00:00:56.520
and see if they did it correctly.

00:00:56.520 --> 00:00:57.646
And if they didn't do it correctly,

00:00:57.646 --> 00:01:00.339
try to identify on what
step they messed up.

00:01:00.339 --> 00:01:03.798
All right, so assuming you had a go at it,

00:01:03.798 --> 00:01:05.598
let's do this together.

00:01:05.598 --> 00:01:08.523
So as we go from the first expression to

00:01:08.523 --> 00:01:10.868
the second, to step one, what do they do?

00:01:10.868 --> 00:01:13.306
Well, they just expanded
out x minus two squared.

00:01:13.306 --> 00:01:16.464
X minus two squared is just x minus two

00:01:16.464 --> 00:01:18.217
times x minus two, they haven't done

00:01:18.217 --> 00:01:20.422
anything to the 4x minus three yet.

00:01:20.422 --> 00:01:22.245
Now what do they do in this step,

00:01:22.245 --> 00:01:23.905
so that seems correct.

00:01:23.905 --> 00:01:26.436
So in step two, it looks
like they're just trying

00:01:26.436 --> 00:01:28.456
to multiply x minus two times x minus two.

00:01:28.456 --> 00:01:31.846
So you have x times x,
which would be x squared.

00:01:31.846 --> 00:01:34.620
You have x times negative two,

00:01:34.620 --> 00:01:37.406
which would be negative 2x.

00:01:37.406 --> 00:01:39.740
You have negative two times x,

00:01:39.740 --> 00:01:41.702
which would be negative 2x.

00:01:41.702 --> 00:01:44.174
Then you have negative
two times negative two,

00:01:44.174 --> 00:01:45.881
which would be positive four.

00:01:45.881 --> 00:01:48.147
So it looks like they
multiplied this out correctly,

00:01:48.147 --> 00:01:50.574
so step two we're still doing good.

00:01:50.574 --> 00:01:52.605
All right, now what do
they do in step three?

00:01:52.605 --> 00:01:54.010
And this whole time, 4x minus three,

00:01:54.010 --> 00:01:55.414
they haven't really touched it yet.

00:01:55.414 --> 00:01:56.610
So they're just trying to simplify it,

00:01:56.610 --> 00:01:59.048
and all they did is they
added these two middle terms.

00:01:59.048 --> 00:02:02.136
Minus negative 2x minus 2x is going to be

00:02:02.136 --> 00:02:05.317
negative 4x, so this still,

00:02:05.317 --> 00:02:07.035
this still looks correct.

00:02:07.035 --> 00:02:08.776
The x squared didn't change,
the plus four didn't change,

00:02:08.776 --> 00:02:10.750
they just added these middle two terms.

00:02:10.750 --> 00:02:13.327
Now as we go to this next step four,

00:02:13.327 --> 00:02:15.880
well now they're trying to multiply

00:02:15.880 --> 00:02:18.264
these two expressions,
so they're doing some

00:02:18.264 --> 00:02:20.180
algebraic multiplication.

00:02:20.180 --> 00:02:21.968
So let's see if we can figure this out.

00:02:21.968 --> 00:02:24.301
So we have 4x times,

00:02:24.301 --> 00:02:25.485
let me do this in a new color,

00:02:25.485 --> 00:02:27.366
I'm getting bored of that magenta.

00:02:27.366 --> 00:02:29.885
All right, so we have,

00:02:29.885 --> 00:02:34.041
we have 4x times x squared,

00:02:34.041 --> 00:02:37.257
which is indeed 4x to the third power.

00:02:37.257 --> 00:02:40.565
Then you have 4x times negative 4x,

00:02:40.565 --> 00:02:43.583
which is gonna be negative 16x squared,

00:02:43.583 --> 00:02:44.965
so they did that right.

00:02:44.965 --> 00:02:47.763
Then you have 4x times four,
which is going to be 16x,

00:02:47.763 --> 00:02:49.783
and they wrote that right over there.

00:02:49.783 --> 00:02:53.149
Then, you're going to have negative three

00:02:53.149 --> 00:02:56.551
times x squared, which
is negative 3x squared,

00:02:56.551 --> 00:02:58.234
we see that right over there.

00:02:58.234 --> 00:03:00.370
Then you're going to have negative three

00:03:00.370 --> 00:03:02.924
times negative 4x, which is going to be

00:03:02.924 --> 00:03:05.594
positive 12x,

00:03:05.594 --> 00:03:08.926
and they wrote negative 12x.

00:03:08.926 --> 00:03:11.584
So they forgot, they saw
a negative, negative,

00:03:11.584 --> 00:03:13.186
but they still put a negative there.

00:03:13.186 --> 00:03:15.311
Negative three times negative 4x,

00:03:15.311 --> 00:03:17.470
a negative times a negative
is gonna be a positive.

00:03:17.470 --> 00:03:19.165
Positive 12x.

00:03:19.165 --> 00:03:20.512
So they made an error here, and then

00:03:20.512 --> 00:03:23.402
they said negative three
times positive four,

00:03:23.402 --> 00:03:25.852
which is indeed negative
12, so this part is right.

00:03:25.852 --> 00:03:28.371
So the error, this thing should read

00:03:28.371 --> 00:03:31.064
positive 12x.

00:03:31.064 --> 00:03:33.723
So the error they made is in step four.

00:03:33.723 --> 00:03:36.265
Step four is the error,
and then that ended up

00:03:36.265 --> 00:03:38.088
giving them the wrong answer here,

00:03:38.088 --> 00:03:40.305
because they did a minus 12x instead of,

00:03:40.305 --> 00:03:42.429
if this was a negative
12x, then negative 12x

00:03:42.429 --> 00:03:44.728
plus 16x got you this 4x.

00:03:44.728 --> 00:03:47.270
But we know it's supposed to be plus 12x.

00:03:47.270 --> 00:03:50.981
So it really should be 28.

00:03:53.411 --> 00:03:56.812
This should be 28x right over here.

00:03:56.812 --> 00:03:58.449
But they really messed up.

00:03:58.449 --> 00:04:00.794
If you take that error,
they did this step right.

00:04:00.794 --> 00:04:03.011
But step four is where they
actually made the error.

00:04:03.011 --> 00:04:04.718
So let's keep going, let's give ourselves

00:04:04.718 --> 00:04:07.133
a little bit more practice at

00:04:07.133 --> 00:04:10.720
looking at ways to manipulate polynomials

00:04:10.720 --> 00:04:12.252
and see if they're valid.

00:04:12.252 --> 00:04:14.713
So here, this comes from an
exercise on Kahn Academy.

00:04:14.713 --> 00:04:17.929
Let's see which of these
are valid identities.

00:04:17.929 --> 00:04:20.634
Which of these are valid statements?

00:04:20.634 --> 00:04:23.757
So this first one, 2x plus y, times 4x,

00:04:23.757 --> 00:04:26.311
2x plus y times 4x minus 2y is

00:04:26.311 --> 00:04:28.110
all of this business right over here.

00:04:28.110 --> 00:04:30.049
Le'ts just multiply it out.

00:04:30.049 --> 00:04:35.029
2x times 4x is going to be 8x squared.

00:04:35.029 --> 00:04:39.127
2x times negative 2y is going to be

00:04:39.127 --> 00:04:42.755
negative 4xy.

00:04:42.755 --> 00:04:46.120
And then, let me switch colors,

00:04:46.120 --> 00:04:48.279
y times 4x is going to be

00:04:48.279 --> 00:04:51.886
plus 4xy.

00:04:51.886 --> 00:04:54.771
And then y times negative 2y

00:04:56.493 --> 00:05:01.119
is going to be minus 2y squared.

00:05:01.119 --> 00:05:04.578
And so let's see, these
two, did I do that right?

00:05:04.578 --> 00:05:07.571
Let's see, 2x,

00:05:07.571 --> 00:05:09.976
times negative 2y is negative 4xy.

00:05:09.976 --> 00:05:13.559
And then I had 4x times y is positive 4xy.

00:05:13.559 --> 00:05:16.013
So these two are gonna cancel out.

00:05:16.013 --> 00:05:19.507
So this is already, this is looking shady.

00:05:19.507 --> 00:05:21.794
So all we're left with is 8x squared

00:05:21.794 --> 00:05:24.255
minus 2y squared, if we factor a two out,

00:05:25.702 --> 00:05:27.486
it's going to be two times

00:05:27.486 --> 00:05:31.314
4x squared minus y squared.

00:05:31.314 --> 00:05:35.991
So this is not a true
statement right over here.

00:05:35.991 --> 00:05:37.874
Now let's try this one.

00:05:39.828 --> 00:05:43.484
N plus two squared minus n
squared is equal to this.

00:05:43.484 --> 00:05:45.528
But what's n plus two squared?

00:05:45.528 --> 00:05:48.000
That's going to be n squared plus

00:05:48.000 --> 00:05:51.599
plus 4n, it's gonna be 2n plus 2n,

00:05:51.599 --> 00:05:54.583
it's gonna be plus 4n plus four,

00:05:54.583 --> 00:05:57.020
and then we're gonna
subtract out an n squared,

00:05:57.020 --> 00:05:59.818
these cancel, so we're
gonna have 4n plus four,

00:05:59.818 --> 00:06:02.883
which is equal to four times n plus one.

00:06:02.883 --> 00:06:05.205
So this one right over here works out.

00:06:05.205 --> 00:06:07.596
This is a true, I guess in
the language of this question,

00:06:07.596 --> 00:06:10.046
it is a valid identity, or you could say

00:06:10.046 --> 00:06:12.484
it's a true statement,
this equation is true.

00:06:12.484 --> 00:06:14.898
And then we have this
last one right over here.

00:06:14.898 --> 00:06:17.255
And once again, let's see
if we can multiply it out.

00:06:17.255 --> 00:06:19.484
Let me go down here into the black space.

00:06:19.484 --> 00:06:23.942
So if I have a times 2a,
that's going to be 2a squared.

00:06:23.942 --> 00:06:26.798
And then a times one,

00:06:26.798 --> 00:06:29.676
is going to be plus a.

00:06:29.676 --> 00:06:33.530
And then if I have b times 2a,

00:06:33.530 --> 00:06:36.769
it's going to be plus 2ab,

00:06:36.769 --> 00:06:39.810
and then finally if I have b times one,

00:06:39.810 --> 00:06:41.923
it's going to be plus b.

00:06:41.923 --> 00:06:45.394
And then out here, we are subtracting a b.

00:06:45.394 --> 00:06:47.507
So over here we're going to subtract a b.

00:06:47.507 --> 00:06:50.293
These characters are going to cancel out.

00:06:50.293 --> 00:06:52.987
And then we're left with 2a squared

00:06:52.987 --> 00:06:55.285
plus a plus 2ab.

00:06:55.285 --> 00:06:56.783
And it looks like they factor out an a,

00:06:56.783 --> 00:06:59.012
so let's see if we can
factor out an a ourselves.

00:06:59.012 --> 00:07:01.647
So if we factor out an a,
we're gonna be left with

00:07:01.647 --> 00:07:05.018
this first term is going to be 2a,

00:07:05.018 --> 00:07:07.360
this right over here,
if we factor out an a,

00:07:07.360 --> 00:07:10.227
is going to be plus one, and
this if we factor out an a

00:07:10.227 --> 00:07:12.332
is going to be 2b.

00:07:14.621 --> 00:07:16.197
And that's exactly what
they wrote over here,

00:07:16.197 --> 00:07:17.335
they just wrote it in a different order.

00:07:17.335 --> 00:07:21.946
A times 2a, plus 2b plus one.

00:07:23.115 --> 00:07:25.398
So this is legit.

00:07:25.398 --> 00:07:27.522
So hopefully that gave
us some good practice

00:07:27.522 --> 00:07:29.542
at critically looking at whether

00:07:29.542 --> 00:07:31.202
people are making valid statements.

00:07:31.202 --> 00:07:32.398
And this is going to be the most useful

00:07:32.398 --> 00:07:35.035
to figure out if you are
making valid statements.

00:07:35.035 --> 00:07:37.053
All right, hopefully you enjoyed that.


Ellipse focus intuition exercise

WEBVTT
Kind: captions
Language: en

00:00:00.817 --> 00:00:01.885
- [Voiceover] I thought
it would be interesting

00:00:01.885 --> 00:00:04.786
to get a little bit more
intuition on ellipses

00:00:04.786 --> 00:00:08.688
and their foci, or fo-si,
sometimes I like to

00:00:08.688 --> 00:00:12.657
call them focuses, but
the correct word is foci

00:00:12.657 --> 00:00:15.792
or fo-si, but either
pronunciation is apparently okay.

00:00:15.792 --> 00:00:16.606
So let's work with this.

00:00:16.606 --> 00:00:18.579
This is interactive
exercise on Khan Academy

00:00:18.579 --> 00:00:19.578
as you see over here,

00:00:19.578 --> 00:00:20.994
so let's see if we can
work through this together

00:00:20.994 --> 00:00:23.200
and learn a little bit
more about ellipses.

00:00:23.200 --> 00:00:26.381
So they tell us the
equation of an ellipse is

00:00:26.381 --> 00:00:29.051
X minus X one squared over A squared

00:00:29.051 --> 00:00:32.464
plus Y minus one squared
over B squared equals one.

00:00:32.464 --> 00:00:33.555
And this is true.

00:00:33.555 --> 00:00:35.691
This is the equation of an ellipse.

00:00:35.691 --> 00:00:38.407
This would be an ellipse that
is centered at X equals X one,

00:00:38.407 --> 00:00:40.034
Y equals Y one.

00:00:40.034 --> 00:00:43.099
The major radius would be A,

00:00:43.099 --> 00:00:45.630
the minor radius would be B.

00:00:45.630 --> 00:00:48.299
So then they say find the
foci of the ellipse bellow

00:00:48.299 --> 00:00:51.295
by moving the orange points
to their correct positions.

00:00:51.295 --> 00:00:53.859
Then use that information
to find the values of

00:00:53.859 --> 00:00:56.218
X one, Y one, A, and B.

00:00:56.218 --> 00:00:58.710
So they want us to move these
to the correct position,

00:00:58.710 --> 00:01:01.218
the correct position is where
the sum of the distance,

00:01:01.218 --> 00:01:03.262
so if we move it to the correct position,

00:01:03.262 --> 00:01:05.908
the sum of the length of that red line

00:01:05.908 --> 00:01:07.488
and the length of that green line

00:01:07.488 --> 00:01:09.137
should be constant.

00:01:09.137 --> 00:01:10.344
Now you can see, they're clearly not

00:01:10.344 --> 00:01:12.201
in the correct position right over here,

00:01:12.201 --> 00:01:14.408
because up here when I
was moving it around,

00:01:14.408 --> 00:01:16.590
you see that look, the
distance to focus one

00:01:16.590 --> 00:01:18.866
from this point on the ellipse is 11.7,

00:01:18.866 --> 00:01:19.725
that's the green line.

00:01:19.725 --> 00:01:20.792
The distance to focus two is six.

00:01:20.792 --> 00:01:22.697
You add them together, it's 17.8.

00:01:22.697 --> 00:01:25.390
But when you move it around, that changes.

00:01:25.390 --> 00:01:27.595
And so, we would have to find,

00:01:27.595 --> 00:01:29.570
we would have to find a different,

00:01:29.570 --> 00:01:31.938
we could just keep trying different foci

00:01:31.938 --> 00:01:34.724
until that sum of these, the red length

00:01:34.724 --> 00:01:36.443
and the green length is constant.

00:01:36.443 --> 00:01:38.927
Now in my mind, that
is a hard way to do it.

00:01:38.927 --> 00:01:40.877
An easier way to do it would be,

00:01:40.877 --> 00:01:45.149
well let's figure out the
center of the ellipse first,

00:01:45.149 --> 00:01:48.424
which we could do just literally
by inspecting this drawing,

00:01:48.424 --> 00:01:51.187
and then figure out what the
length of the major radius

00:01:51.187 --> 00:01:52.394
and the minus radius is

00:01:52.394 --> 00:01:55.319
and then use that
information to find the foci.

00:01:55.319 --> 00:01:56.736
So let's see if I can put these dots,

00:01:56.736 --> 00:01:58.570
let me put them first at the center.

00:01:58.570 --> 00:01:59.940
Alright, so let's see.

00:01:59.940 --> 00:02:02.007
So that's center,

00:02:02.007 --> 00:02:02.981
and let me,

00:02:02.981 --> 00:02:04.515
that looks pretty good,

00:02:04.515 --> 00:02:06.395
that looks like the center.

00:02:06.395 --> 00:02:07.881
And I can even confirm.

00:02:07.881 --> 00:02:10.412
So in the Y direction, that is

00:02:10.412 --> 00:02:13.408
if I'm at the top of the ellipse I am

00:02:13.408 --> 00:02:15.769
10 from that center and
if I'm at the bottom

00:02:15.769 --> 00:02:19.555
of the ellipse, I'm 10 from that center.

00:02:19.555 --> 00:02:21.923
And so that does look like the center.

00:02:21.923 --> 00:02:26.071
And it also tells us, it tells us that the

00:02:26.071 --> 00:02:29.000
major radius is 10

00:02:29.000 --> 00:02:31.991
and it's along the y-axis.

00:02:31.991 --> 00:02:36.566
So the major radius is 10
and it's along the y-axis.

00:02:36.566 --> 00:02:38.771
And we also know the Y
coordinate of the center,

00:02:38.771 --> 00:02:40.334
it's Y is equal to four.

00:02:40.977 --> 00:02:42.452
Y is equal to four.

00:02:42.452 --> 00:02:46.108
And then the X coordinate of
the center is X equals five.

00:02:47.436 --> 00:02:50.084
So once again, where
I put this orange dot,

00:02:50.084 --> 00:02:51.245
that is the center of the ellipse.

00:02:51.245 --> 00:02:54.705
It's at X equals five, Y equals four,

00:02:54.705 --> 00:02:56.353
and that's why I put X equals five,

00:02:56.353 --> 00:02:59.697
or X one equals five
and Y one equals four.

00:02:59.697 --> 00:03:02.855
And you see from the equation of ellipse,

00:03:02.855 --> 00:03:05.844
that's what defines the
center of the ellipse.

00:03:05.844 --> 00:03:08.723
And then A is going to be,

00:03:08.723 --> 00:03:09.930
A, we can see from here,

00:03:09.930 --> 00:03:12.228
this is the radius in the X direction,

00:03:12.228 --> 00:03:13.412
I guess is one way to think about it.

00:03:13.412 --> 00:03:14.504
And that's gonna be the minor radius,

00:03:14.504 --> 00:03:15.665
it's the shorter radius.

00:03:15.665 --> 00:03:17.127
And you see that has length eight.

00:03:17.127 --> 00:03:19.845
we're going from X equals
five to X equals 13.

00:03:19.845 --> 00:03:24.094
Or X equals negative
three to X equals five.

00:03:24.094 --> 00:03:25.603
So it has a length eight.

00:03:25.603 --> 00:03:28.555
So this thing right over
here, A is going to be eight.

00:03:29.387 --> 00:03:30.698
So this is eight.

00:03:30.698 --> 00:03:31.672
And we already said,

00:03:31.672 --> 00:03:34.506
the major radius is going to
be in the vertical direction

00:03:34.506 --> 00:03:36.479
and it is of length 10.

00:03:36.479 --> 00:03:37.687
So just like that we figured out

00:03:37.687 --> 00:03:39.195
we could write the equation down,

00:03:39.195 --> 00:03:40.822
I don't have my pen out,
but you could just say

00:03:40.822 --> 00:03:42.256
this is the equation,
this ellipse is gonna be

00:03:42.256 --> 00:03:46.852
X minus five squared over eight squared,

00:03:46.852 --> 00:03:50.638
plus Y minus four squared

00:03:50.638 --> 00:03:53.169
over 10 squared equals one.

00:03:53.169 --> 00:03:54.190
But now let's actually figure out

00:03:54.190 --> 00:03:55.746
where we need to put these,

00:03:55.746 --> 00:03:57.487
where we need to actually put these foci

00:03:57.487 --> 00:03:59.159
and then figure out that, hey look,

00:03:59.159 --> 00:04:00.715
if we put them in the right place

00:04:00.715 --> 00:04:04.221
the sum of the red and
green lines is going to be,

00:04:04.221 --> 00:04:06.288
is going to be a constant.

00:04:06.288 --> 00:04:08.075
And clearly where they are
right now it's not a constant.

00:04:08.075 --> 00:04:11.204
You see that sum of the
distances is at 20.6

00:04:11.204 --> 00:04:14.572
and then it goes to 20 point,
it's clearly not constant.

00:04:14.572 --> 00:04:15.918
So let's see if we can do it.

00:04:15.918 --> 00:04:17.544
And one way to do it is just to remember,

00:04:17.544 --> 00:04:20.098
well there's a formula
and we talk about it,

00:04:20.098 --> 00:04:21.863
we explore the intuition behind it

00:04:21.863 --> 00:04:23.674
in a previous video,

00:04:23.674 --> 00:04:25.717
but the formula for the foci ellipse

00:04:25.717 --> 00:04:28.688
is you find the difference
between the larger

00:04:28.688 --> 00:04:30.570
of these two values and the smaller

00:04:30.570 --> 00:04:31.940
of these two values.

00:04:31.940 --> 00:04:34.424
So in this case you'd find
the difference between

00:04:34.424 --> 00:04:36.979
10 squared and eight squared.

00:04:36.979 --> 00:04:40.321
Well that's going to be 100 minus 64,

00:04:40.321 --> 00:04:43.590
which is 36, and then you
take the square root of that.

00:04:43.590 --> 00:04:45.797
So it's going to be equal to six.

00:04:45.797 --> 00:04:48.676
So the focal length is
going to be equal to six.

00:04:48.676 --> 00:04:51.022
So if we go back to the center,

00:04:51.022 --> 00:04:52.810
if you go back to the center,

00:04:52.810 --> 00:04:55.051
if we had figured out
our focal length is six,

00:04:55.051 --> 00:04:57.233
and remember, all I did is
I took the larger of these,

00:04:57.233 --> 00:04:58.812
I squared it, and from that I subtracted

00:04:58.812 --> 00:04:59.927
the other one squared.

00:04:59.927 --> 00:05:02.735
So 100 squared minus eight squared is,

00:05:02.735 --> 00:05:05.383
is 100 minus 64 which is 36

00:05:05.383 --> 00:05:08.077
and I took the square
root of that to get six.

00:05:08.077 --> 00:05:12.140
So our focal length along
the major axis, I could say,

00:05:12.140 --> 00:05:13.463
is going to be six.

00:05:13.463 --> 00:05:16.389
So it's going to be six from the center up

00:05:16.389 --> 00:05:17.899
and six from the center down.

00:05:17.899 --> 00:05:19.129
So six from the center up,

00:05:19.129 --> 00:05:20.940
one, two, three,

00:05:20.940 --> 00:05:23.123
four, five, six.

00:05:23.123 --> 00:05:24.794
And six from the center down,

00:05:24.794 --> 00:05:26.581
one, two, three,

00:05:26.581 --> 00:05:28.835
four, five, six.

00:05:28.835 --> 00:05:30.135
Now that's what the math tells us,

00:05:30.135 --> 00:05:32.480
but let's just make sure
that this actually holds.

00:05:32.480 --> 00:05:33.873
This actually holds.

00:05:33.873 --> 00:05:37.054
So now let's go to points on the ellipse

00:05:37.054 --> 00:05:39.817
and you see, it's calculating the distance

00:05:39.817 --> 00:05:42.139
to each of the foci,

00:05:42.139 --> 00:05:45.390
and the length of the
lines are the distances,

00:05:45.390 --> 00:05:46.736
and you see when you sum them up

00:05:46.736 --> 00:05:50.777
we are at a nice, constant 20,

00:05:50.777 --> 00:05:53.192
which is the really cool
thing about ellipses.

00:05:53.192 --> 00:05:55.003
That look, I can go anywhere around here

00:05:55.003 --> 00:05:56.670
and the sum of these two distances,

00:05:56.670 --> 00:05:59.339
of the green and the
red line, their lengths,

00:05:59.339 --> 00:06:00.478
is a constant 20.

00:06:00.478 --> 00:06:03.496
That's actually what defines an ellipse.

00:06:03.496 --> 00:06:05.818
In fact, you can draw an
ellipse on a chalk board,

00:06:05.818 --> 00:06:08.442
you could draw this
ellipse on a chalk board.

00:06:08.442 --> 00:06:10.787
If you had a string of
say length 20 centimeters,

00:06:10.787 --> 00:06:12.970
and you put one focus
there and one focus there

00:06:12.970 --> 00:06:15.454
and you kind of put your
chalk in the string,

00:06:15.454 --> 00:06:17.196
and you just tried to move it around,

00:06:17.196 --> 00:06:19.634
and so it would look something,

00:06:19.634 --> 00:06:21.955
it would look something like that.

00:06:21.955 --> 00:06:24.620
Anyway, hopefully you
found that as fun as I did.


Factoring and completing square of higher degree expressions

WEBVTT
Kind: captions
Language: en

00:00:00.408 --> 00:00:01.774
- [Voiceover] What I
hope to do in this video

00:00:01.774 --> 00:00:04.309
is leverage some of our
experience and knowledge

00:00:04.309 --> 00:00:06.073
from the quadratic world

00:00:06.073 --> 00:00:09.974
to factor, and rewrite,
and re-express expressions

00:00:09.974 --> 00:00:11.542
that are higher degree,

00:00:11.542 --> 00:00:13.643
higher degree polynomials.

00:00:13.643 --> 00:00:15.234
So what am I talking about?

00:00:15.234 --> 00:00:17.672
So let's say that I have the expression

00:00:17.672 --> 00:00:19.495
a to the sixth power,

00:00:19.495 --> 00:00:23.477
plus 2a to the third power b squared,

00:00:23.477 --> 00:00:25.427
plus b to the fourth,

00:00:25.427 --> 00:00:26.937
and I want to factor this.

00:00:26.937 --> 00:00:29.444
I want to write this as a
product of two expressions.

00:00:29.444 --> 00:00:31.267
How do I do that?

00:00:31.267 --> 00:00:33.543
And at first, it might
seem a little intimidating.

00:00:33.543 --> 00:00:35.400
I have a to the sixth power!

00:00:35.400 --> 00:00:36.770
This is a sixth degree term,

00:00:36.770 --> 00:00:37.943
you have b to the fourth power,

00:00:37.943 --> 00:00:39.533
how do I handle it?

00:00:39.533 --> 00:00:41.844
But a clue, you might say,
"Well, this is kind of

00:00:41.844 --> 00:00:44.328
"looking like when I took a square of

00:00:44.328 --> 00:00:47.800
"a expression with two variables."

00:00:47.800 --> 00:00:50.969
So let's just take a little
bit of a detour here.

00:00:50.969 --> 00:00:54.069
If I took x plus y, squared,

00:00:54.069 --> 00:00:55.369
what is this going to be equal to?

00:00:55.369 --> 00:00:59.967
Well, this is x plus y, times x plus y,

00:00:59.967 --> 00:01:01.975
which is, of course, going to be equal to,

00:01:01.975 --> 00:01:04.506
x times x is x squared,

00:01:04.506 --> 00:01:06.271
then you have x times y is xy,

00:01:06.271 --> 00:01:07.989
but then you have y times x as well,

00:01:07.989 --> 00:01:10.439
so you have 2xy's, and this is all review.

00:01:10.439 --> 00:01:12.190
If this isn't review, I
encourage you to watch

00:01:12.190 --> 00:01:17.068
the videos on multiplying expressions,

00:01:17.068 --> 00:01:19.170
and then, or multiplying
binomials, I guess you could say,

00:01:19.170 --> 00:01:20.807
and then you finally have y times y,

00:01:20.807 --> 00:01:22.966
so you'd have plus y squared.

00:01:22.966 --> 00:01:23.802
So when you write it like this,

00:01:23.802 --> 00:01:27.738
you see well it looks like
there might be a parallel.

00:01:27.738 --> 00:01:29.305
This a to the sixth term looks like it

00:01:29.305 --> 00:01:32.034
might parallel to this x squared term,

00:01:32.034 --> 00:01:34.100
and this b to the fourth
looks like it might

00:01:34.100 --> 00:01:35.644
parallel to this, 'cause notice,

00:01:35.644 --> 00:01:37.990
if I have something squared here,

00:01:37.990 --> 00:01:40.300
and then I have that
something in this middle term,

00:01:40.300 --> 00:01:41.705
and I have something squared here,

00:01:41.705 --> 00:01:42.889
and I have that something
in that middle term,

00:01:42.889 --> 00:01:43.969
and then I have a two,

00:01:43.969 --> 00:01:46.140
well then, that means that
this thing is a perfect square.

00:01:46.140 --> 00:01:47.974
It's x plus y, squared,

00:01:47.974 --> 00:01:49.600
and we do have that.

00:01:49.600 --> 00:01:51.808
Notice, a to the sixth,

00:01:52.270 --> 00:01:53.244
that's the same thing as

00:01:53.244 --> 00:01:57.376
a to the third, squared,

00:01:57.376 --> 00:01:59.523
and then we have an a to the third

00:01:59.523 --> 00:02:02.138
right here, sitting in this middle term.

00:02:02.138 --> 00:02:02.730
So let me write that.

00:02:02.730 --> 00:02:04.240
We have that a to the third there,

00:02:04.240 --> 00:02:08.756
and of course we have this plus two there,

00:02:08.756 --> 00:02:11.022
and then b to the fourth,

00:02:11.438 --> 00:02:15.908
well, that's the same thing
as b squared, squared,

00:02:15.908 --> 00:02:18.566
and of course, we also have a b squared

00:02:18.566 --> 00:02:20.169
in this middle term.

00:02:20.169 --> 00:02:21.736
So when you write it this way,

00:02:21.736 --> 00:02:25.335
it has the exact same pattern
as we had right over here.

00:02:25.335 --> 00:02:28.346
This thing is nothing but,

00:02:29.654 --> 00:02:32.138
let me write it this way,

00:02:32.138 --> 00:02:36.944
as a to the third, let
me do it in those colors,

00:02:36.944 --> 00:02:41.653
as a to the third, plus b squared,

00:02:44.038 --> 00:02:48.473
and then that whole quantity squared,

00:02:48.473 --> 00:02:50.470
and you can see, look, if
we were to multiply it out,

00:02:50.470 --> 00:02:51.166
and actually, let me just do it,

00:02:51.166 --> 00:02:53.109
I'll do it all in white,
so it makes sense to you,

00:02:53.109 --> 00:02:55.300
or, so actually, it might
make a little less sense,

00:02:55.300 --> 00:02:55.938
'cause I'm not gonna take--

00:02:55.938 --> 00:02:57.471
Actually, let me switch colors,

00:02:57.471 --> 00:03:00.693
so it's gonna be a to the third,

00:03:05.676 --> 00:03:07.664
plus b squared,

00:03:08.957 --> 00:03:12.524
times a to the third, plus b squared.

00:03:21.540 --> 00:03:23.502
Well, a to the third times a to the third

00:03:23.502 --> 00:03:26.335
is a to the sixth.

00:03:26.335 --> 00:03:28.669
a to the third plus b
squared, that's gonna be

00:03:28.669 --> 00:03:31.664
a to the third, b squared,

00:03:33.034 --> 00:03:34.776
and then you're gonna have b
squared times a to the third,

00:03:34.776 --> 00:03:36.633
well that's just another
a to the third, b squared,

00:03:36.633 --> 00:03:38.642
so actually, let's just write this as

00:03:38.642 --> 00:03:43.130
2a to the third,

00:03:44.807 --> 00:03:46.363
b squared,

00:03:47.303 --> 00:03:48.975
and then finally, b
squared times b squared

00:03:48.975 --> 00:03:52.167
is going to be b to the fourth power.

00:03:52.167 --> 00:03:54.501
So once again, if you
see something like this,

00:03:54.501 --> 00:03:56.103
you're like, okay, it seems daunting,

00:03:56.103 --> 00:03:59.022
but it seems close to a
pattern that I've seen

00:03:59.022 --> 00:04:01.142
when I've been dealing with quadratics.

00:04:01.142 --> 00:04:03.743
This looks close to a
perfect square quadratic,

00:04:03.743 --> 00:04:04.811
and if you actually write this out,

00:04:04.811 --> 00:04:05.809
it's actually a pretty big hit,

00:04:05.809 --> 00:04:08.340
you see, well look, if
x is a to the third,

00:04:08.340 --> 00:04:11.788
and y is b squared, then these
are the exact same thing.

00:04:11.788 --> 00:04:14.540
So this is a factored out version of it.

00:04:14.540 --> 00:04:16.339
Let's do another one of these.

00:04:16.339 --> 00:04:21.274
So let's say, let me draw
a little dotted line here,

00:04:21.274 --> 00:04:23.271
and let's say that we have

00:04:23.271 --> 00:04:25.767
an eighth degree polynomial.

00:04:25.767 --> 00:04:28.135
We have x to the eighth,

00:04:28.135 --> 00:04:32.942
minus 12x to the fourth, minus two,

00:04:32.942 --> 00:04:35.333
and we want to complete the square here,

00:04:35.333 --> 00:04:36.889
and you're probably used
to completing the square

00:04:36.889 --> 00:04:40.894
with second degree
polynomials, or quadratics,

00:04:40.894 --> 00:04:42.636
and all of a sudden you
have an eighth degree,

00:04:42.636 --> 00:04:44.935
but there's no reason why
you can't do that here,

00:04:44.935 --> 00:04:47.338
'cause we could say this is
going to be the same thing as

00:04:47.338 --> 00:04:52.168
x to the eighth, minus
12x to the fourth power.

00:04:52.168 --> 00:04:56.208
Whoops, my primitive brain
wrote x squared there.

00:04:56.208 --> 00:04:59.169
x to the eighth, minus 12x to the fourth,

00:04:59.169 --> 00:05:01.374
and let me create some space
to add and subtract something,

00:05:01.374 --> 00:05:04.242
and then let me put
the minus two out here,

00:05:04.242 --> 00:05:05.635
and we just have to realize that, look,

00:05:05.635 --> 00:05:06.703
for it to be a perfect square,

00:05:06.703 --> 00:05:10.500
that this is x to the fourth,
squared right over here.

00:05:10.500 --> 00:05:12.834
We have an x to the fourth here,

00:05:12.834 --> 00:05:14.505
and so, just like we
would complete the square

00:05:14.505 --> 00:05:17.605
if we were doing it with a
second degree polynomial,

00:05:17.605 --> 00:05:21.135
or quadratic, you would take
half the coefficient here,

00:05:21.135 --> 00:05:24.978
which is six, and then we
would want to add that,

00:05:24.978 --> 00:05:26.603
so we would want to add,

00:05:26.603 --> 00:05:28.089
well, we wouldn't add
half that coefficient,

00:05:28.089 --> 00:05:29.877
we would add the square
of half that coefficient.

00:05:29.877 --> 00:05:31.676
This is just how you complete the square,

00:05:31.676 --> 00:05:34.449
if this was an x squared,
and this was just an x,

00:05:34.449 --> 00:05:36.373
and the key here is just realize

00:05:36.373 --> 00:05:38.968
that x to the eighth, is
x to the fourth, squared.

00:05:38.968 --> 00:05:39.769
When you did it with quadratics,

00:05:39.769 --> 00:05:42.207
you just had an x here, and
you just had the x squared.

00:05:42.207 --> 00:05:43.507
Now you have an x to the fourth here,

00:05:43.507 --> 00:05:44.959
and you have x to the fourth squared,

00:05:44.959 --> 00:05:46.410
which is x to the eighth.

00:05:46.410 --> 00:05:48.267
But you take half of this coefficient,

00:05:48.267 --> 00:05:51.030
which is six.

00:05:51.030 --> 00:05:51.971
You could do it as negative six,

00:05:51.971 --> 00:05:52.934
since you have this out here,

00:05:52.934 --> 00:05:56.011
but then you square it, and you get 36.

00:05:56.011 --> 00:05:59.076
But you can't just add 36
willy, nilly to an expression.

00:05:59.076 --> 00:06:00.365
That changes its value.

00:06:00.365 --> 00:06:01.305
In order to not change its value,

00:06:01.305 --> 00:06:04.103
you also need to subtract 36.

00:06:04.103 --> 00:06:05.810
But the whole point of doing it is,

00:06:05.810 --> 00:06:09.037
this part of it now is a perfect square.

00:06:09.037 --> 00:06:12.636
It's a perfect square
eighth degree polynomial.

00:06:12.636 --> 00:06:14.144
So this is going to be the same thing as

00:06:14.144 --> 00:06:16.129
x to the fourth,

00:06:16.914 --> 00:06:19.774
minus, let me write it this way,

00:06:19.774 --> 00:06:21.132
x to the fourth,

00:06:21.132 --> 00:06:26.008
minus six, squared.

00:06:26.008 --> 00:06:28.400
Maybe I should have written this in blue.

00:06:28.400 --> 00:06:29.944
The colors are the hard part.

00:06:29.944 --> 00:06:33.032
x to the fourth, minus six,

00:06:33.032 --> 00:06:35.470
and then the whole thing squared,

00:06:35.470 --> 00:06:36.805
and then, of course, I have to worry about

00:06:36.805 --> 00:06:38.845
these two characters out here.

00:06:39.276 --> 00:06:41.207
If I subtract 36 and then subtract two,

00:06:41.207 --> 00:06:44.109
well, that's the same
thing as subtracting 38.

00:06:44.109 --> 00:06:45.340
So just like that, this is just like

00:06:45.340 --> 00:06:47.964
completing the square with
a second degree polynomial,

00:06:47.964 --> 00:06:51.203
but I just did it with an
eighth degree polynomial.

00:06:51.203 --> 00:06:52.236
Let's do one more,

00:06:52.236 --> 00:06:54.930
just to really exercise this muscle.

00:06:54.930 --> 00:06:56.834
This, in some ways, isn't a new skill.

00:06:56.834 --> 00:06:58.703
This is just looking at an old skill,

00:06:58.703 --> 00:07:01.373
and applying it to slightly hairier,

00:07:01.373 --> 00:07:02.801
slightly more complex things,

00:07:02.801 --> 00:07:05.971
or maybe realizing that they
aren't that more complex.

00:07:05.971 --> 00:07:09.137
So let's say that we had

00:07:11.137 --> 00:07:14.535
25x to the sixth,

00:07:14.535 --> 00:07:18.822
minus 16y squared.

00:07:21.561 --> 00:07:23.070
So let's do this at the top.

00:07:23.070 --> 00:07:24.777
What are other ways we can write this?

00:07:24.777 --> 00:07:25.814
Can we factor things?

00:07:25.814 --> 00:07:27.633
What are other ways could we write it?

00:07:27.633 --> 00:07:29.841
Well, one way that we could write it,

00:07:29.841 --> 00:07:31.848
we could say that, look,

00:07:33.848 --> 00:07:35.034
25 is five squared,

00:07:35.034 --> 00:07:37.438
and x to the sixth is x
to the third, squared,

00:07:37.438 --> 00:07:41.977
so we could rewrite
this right over here as

00:07:41.977 --> 00:07:44.444
five squared, times

00:07:44.444 --> 00:07:47.681
x to the third, squared,

00:07:47.681 --> 00:07:50.534
and then we have minus,

00:07:50.534 --> 00:07:52.298
16 is four squared.

00:07:52.298 --> 00:07:53.796
Let me do this in this color.

00:07:53.796 --> 00:07:57.198
16 is four squared, y squared,

00:07:57.198 --> 00:07:59.276
so we could write it like this if we like.

00:07:59.276 --> 00:08:00.309
But we could also write it,

00:08:00.309 --> 00:08:02.237
we have five to the second power,

00:08:02.237 --> 00:08:04.071
we have x to the third,
to the second power.

00:08:04.071 --> 00:08:05.267
Well we could use our exponent properties,

00:08:05.267 --> 00:08:07.458
say, well this is going
to be the same thing as

00:08:07.458 --> 00:08:11.606
5x to the third, to the second power.

00:08:11.606 --> 00:08:12.639
If I have a product of two things,

00:08:12.639 --> 00:08:13.835
and I raise it to the power,

00:08:13.835 --> 00:08:15.089
that's the same thing as

00:08:15.089 --> 00:08:16.470
taking the product of each of them,

00:08:16.470 --> 00:08:17.840
raised to the power,

00:08:17.840 --> 00:08:19.942
or, if I take the product and
then raise it to the power,

00:08:19.942 --> 00:08:20.627
that's the same thing as

00:08:20.627 --> 00:08:22.055
raising them each to the power first,

00:08:22.055 --> 00:08:23.367
and then taking the product.

00:08:23.367 --> 00:08:25.004
It's one of our exponent properties.

00:08:25.004 --> 00:08:27.091
Now we could do the same thing

00:08:27.445 --> 00:08:29.455
for the second term.

00:08:29.455 --> 00:08:32.009
Four squared times y
squared is the same thing as

00:08:32.009 --> 00:08:34.436
4y, squared,

00:08:34.436 --> 00:08:35.562
and when you write it like this,

00:08:35.562 --> 00:08:36.665
and this is another legitimate way

00:08:36.665 --> 00:08:39.800
to rewrite this higher degree expression,

00:08:39.800 --> 00:08:43.039
you now see that this is
a difference of squares,

00:08:43.039 --> 00:08:44.200
and if it's a difference of squares,

00:08:44.200 --> 00:08:46.296
we can rewrite it as,

00:08:46.681 --> 00:08:48.708
let me do it like this.

00:08:48.708 --> 00:08:50.391
We can rewrite it as

00:08:50.391 --> 00:08:52.553
5x to the third

00:08:53.677 --> 00:08:55.550
plus 4y,

00:08:58.058 --> 00:09:01.468
times 5x to the third

00:09:01.468 --> 00:09:03.390
minus 4y.

00:09:05.560 --> 00:09:07.149
Once again, this comes straight out of

00:09:07.149 --> 00:09:09.034
when you first learned

00:09:09.373 --> 00:09:11.775
a squared, minus b squared is equal to

00:09:11.775 --> 00:09:15.374
a plus b, times a minus b.

00:09:15.374 --> 00:09:18.334
In this case, a is 5x to the third power,

00:09:18.334 --> 00:09:20.761
and b is 4y,

00:09:20.761 --> 00:09:22.932
and you see that right over here.

00:09:22.932 --> 00:09:24.801
So once again, whole point here is

00:09:24.801 --> 00:09:27.402
just apply some of our quadratic toolkit,

00:09:27.402 --> 00:09:29.375
but applying it to higher
degree polynomials,

00:09:29.375 --> 00:09:32.851
and seeing that it's really,
on some levels, the same thing.


Understanding function inverses example

WEBVTT
Kind: captions
Language: en

00:00:00.050 --> 00:00:02.880
This right over here is our

00:00:02.886 --> 00:00:07.191
understanding inverses of functions
exercise on Khan Academy.

00:00:07.191 --> 00:00:11.845
It's a good exercise to make sure
you understand inverses of functions.

00:00:11.845 --> 00:00:15.444
It's an interactive one
where we can move this line around

00:00:15.444 --> 00:00:18.435
and it tells us 'the graph of h(x)
is the green',

00:00:18.435 --> 00:00:24.225
so that's this dotted green line,
'the dashed line segment shown below'.

00:00:24.225 --> 00:00:25.927
So that's this.

00:00:25.927 --> 00:00:30.027
'Drag the endpoints of the segment below
to graph h inverse of (x).

00:00:30.027 --> 00:00:32.042
There's a couple of ways to tackle it.

00:00:32.042 --> 00:00:34.635
Perhaps the simplest one
is we say, okay, look,

00:00:34.635 --> 00:00:38.279
h(x), what does h(x)
map from and to?

00:00:38.279 --> 00:00:41.122
So h(x), this point
shows that h(x),

00:00:41.122 --> 00:00:45.579
if you input -8 into h of (x),
h of -8 is 1,

00:00:45.579 --> 00:00:48.443
so it's mapping from -8 to 1.

00:00:48.443 --> 00:00:55.133
Well, the inverse of that, then,
should map from 1 to -8.

00:00:55.135 --> 00:00:59.825
So let's put that point on the graph,
and let's go on the other end.

00:01:02.133 --> 00:01:03.919
On the other end of h of x,

00:01:03.919 --> 00:01:08.499
we see that when you input 3 into h of x,
when x is equal to 3,

00:01:08.512 --> 00:01:10.633
h of x is equal to -4.

00:01:10.633 --> 00:01:14.665
So this point shows us
that it's mapping from 3 to -4.

00:01:14.665 --> 00:01:19.409
So the inverse of that would map
from -4 to 3.

00:01:19.409 --> 00:01:24.259
If you input -4 it should output 3.

00:01:26.233 --> 00:01:31.133
Since we took
the two end points of this line

00:01:31.133 --> 00:01:33.507
and found the inverse mapping of it,

00:01:33.507 --> 00:01:36.952
what I have just done here
is that I have graphed the inverse.

00:01:36.952 --> 00:01:38.584
Another way to think about the inverse

00:01:38.584 --> 00:01:43.314
is if you were to draw the line y = x,

00:01:48.474 --> 00:01:51.139
these things should be reflections
around the line y =x

00:01:51.139 --> 00:01:56.779
because one way to think about it is,
you're swapping the xs for the ys.

00:01:56.779 --> 00:02:01.237
If you were to draw the line y = x,

00:02:01.237 --> 00:02:04.424
if you flipped it around,
the line y = x, the green line,

00:02:04.424 --> 00:02:06.100
you would actually get the old line.

00:02:06.100 --> 00:02:09.087
This would flip over there
and this would flip over there.

00:02:09.087 --> 00:02:15.477
But either way, we're done.
We have graphed h inverse of x.


Understanding inverse functions

WEBVTT
Kind: captions
Language: en

00:00:00.463 --> 00:00:03.293
You may by now be familiar with the notion

00:00:03.300 --> 00:00:06.157
of evaluating a function
with a particular value,

00:00:06.157 --> 00:00:09.265
so for example,
if this table is our function definition,

00:00:09.265 --> 00:00:12.935
if someone were to say,
"Well, what is f of -9?"

00:00:12.937 --> 00:00:16.007
you could say, okay, if we input -9
into our function,

00:00:16.014 --> 00:00:21.264
if x is -9, this table tell us
that f of x is going to be equal to 5.

00:00:21.264 --> 00:00:24.711
You might already have experience
with doing composite functions,

00:00:24.711 --> 00:00:31.521
where you say, f of f of -9 plus 1.

00:00:33.447 --> 00:00:36.033
So this is interesting,
it seems very daunting,

00:00:36.033 --> 00:00:39.670
but you say, well we know
what f of -9 is, this is going to be 5,

00:00:39.670 --> 00:00:41.690
so it's going to be f of 5 plus 1.

00:00:41.690 --> 00:00:44.884
So this is going to be equal to f of 6,

00:00:44.884 --> 00:00:49.974
and if we look at our table,
f of 6 is equal to -7.

00:00:49.985 --> 00:00:52.078
So all of that is review so far,

00:00:52.078 --> 00:00:57.128
but what I want to now do is
start evaluating the inverse of functions.

00:00:57.128 --> 00:00:59.273
This function f is invertable,

00:00:59.273 --> 00:01:04.203
because it's a one-to-one mapping
between the xs and the f of xs.

00:01:04.217 --> 00:01:09.637
No two xs map to the same f of x,
so this is an invertable function.

00:01:09.639 --> 00:01:12.410
With that in mind,
let's see if we can evaluate

00:01:12.410 --> 00:01:17.980
something like f inverse of 8.

00:01:17.987 --> 00:01:19.425
What is that going to be?

00:01:19.425 --> 00:01:24.315
I encourage you to pause the video
and try to think about it.

00:01:24.315 --> 00:01:28.404
So f of x, just as a reminder
of what functions do,

00:01:28.404 --> 00:01:33.284
f of x is going to map from this domain,
from a value in its domain

00:01:33.284 --> 00:01:38.430
to a corresponding value in the range.

00:01:38.441 --> 00:01:43.561
So this is what f does,
this is domain...

00:01:43.561 --> 00:01:47.494
and this right over here is the range.

00:01:47.494 --> 00:01:51.742
Now f inverse, if you pass it,
the value and the range,

00:01:51.742 --> 00:01:56.042
it'll map it back
to the corresponding value

00:01:56.042 --> 00:01:58.986
in the domain.

00:01:58.986 --> 00:02:01.783
But how do we think about it like this?

00:02:01.783 --> 00:02:05.034
Well, f inverse of 8,
this is whatever maps to 8,

00:02:05.034 --> 00:02:08.854
so if this was 8, we'd have to say,
well, what mapped to 8?

00:02:08.854 --> 00:02:14.563
We see here f of 9 is 8,

00:02:14.563 --> 00:02:19.003
so f inverse of 8
is going to be equal to 9.

00:02:24.368 --> 00:02:30.048
If it makes it easier,
we could construct a table,

00:02:32.691 --> 00:02:37.613
where I could say x and f inverse of x,

00:02:37.613 --> 00:02:41.734
and what I'd do is swap
these two columns.

00:02:41.734 --> 00:02:47.584
f of x goes from -9 to 5,
f inverse of x goes from 5 to -9.

00:02:47.584 --> 00:02:51.709
All I did was swap these two.
Now we're mapping from this to that.

00:02:51.709 --> 00:02:54.812
So f inverse of x is going to map
from 7 to -7.

00:02:54.812 --> 00:02:58.143
Notice, instead of mapping
from this thing to that thing,

00:02:58.143 --> 00:03:01.677
we're now going to map
from that thing to this thing.

00:03:01.677 --> 00:03:06.086
So f inverse is going to map
from 13 to 5.

00:03:06.086 --> 00:03:10.710
It's going to map from -7 to 6.

00:03:10.710 --> 00:03:14.987
It's going to map from 8 to 9,

00:03:14.987 --> 00:03:21.137
and it's going to map from 12 to 11.

00:03:21.137 --> 00:03:24.735
Looks like I got all of them, yep.

00:03:24.735 --> 00:03:27.981
So all I did was swap these columns.

00:03:27.981 --> 00:03:30.457
The f inverse maps from this column
to that column.

00:03:30.457 --> 00:03:33.706
So I just swapped them out.
Now it becomes a little clearer.

00:03:33.706 --> 00:03:35.458
You see it right here,

00:03:35.458 --> 00:03:40.080
f inverse of 8,
if you input 8 into f inverse,

00:03:40.080 --> 00:03:41.677
you get 9.

00:03:41.677 --> 00:03:44.527
Now we can use that
to start doing fancier things.

00:03:44.535 --> 00:03:51.956
We can evaluate something like
f of f inverse of 7.

00:03:54.966 --> 00:04:01.827
f of f inverse of 7.

00:04:01.827 --> 00:04:04.027
What is this going to be?

00:04:04.027 --> 00:04:06.166
Let's first evaluate f inverse of 7.

00:04:06.166 --> 00:04:11.626
f inverse of 7 maps from 7 to -7.

00:04:11.633 --> 00:04:16.673
So this is going to be f
of this stuff in here,

00:04:18.271 --> 00:04:22.991
f inverse of 7, you see,
is -7.

00:04:23.002 --> 00:04:28.332
And then to evaluate the function,
f of -7 is going to be 7.

00:04:28.332 --> 00:04:31.320
And that makes complete sense.

00:04:31.320 --> 00:04:34.620
We mapped from f inverse of 7
to -7

00:04:34.620 --> 00:04:40.168
and evaluating the function of that,
went back to 7.

00:04:40.168 --> 00:04:44.253
So let's do one more of these
just to really feel comfortable

00:04:44.253 --> 00:04:47.131
with mapping back-and-forth
between these two sets,

00:04:47.131 --> 00:04:51.299
between applying the function
and the inverse of the function.

00:04:51.299 --> 00:04:56.049
Let's evaluate f inverse
of f inverse of 13.

00:05:05.759 --> 00:05:08.761
f inverse of 13.

00:05:08.761 --> 00:05:10.027
What is that going to be?

00:05:10.027 --> 00:05:13.652
I encourage you to pause the video
and try to figure it out.

00:05:13.652 --> 00:05:16.178
What's f inverse of 13?

00:05:16.178 --> 00:05:20.840
That's, looking at this table right here,
f inverse goes from 13 to 5.

00:05:20.840 --> 00:05:26.890
You see it over here, f went from 5 to 13,
so f inverse is going to go from 13 to 5.

00:05:26.890 --> 00:05:31.091
So, f inverse of 13 is going to be 5,

00:05:31.091 --> 00:05:35.911
so this is the same thing
as f inverse of 5.

00:05:35.916 --> 00:05:42.336
And f inverse of 5? -9.
So this is going to be equal to -9.

00:05:42.336 --> 00:05:49.027
Once again, f inverse goes
from 5 to -9.

00:05:49.027 --> 00:05:53.969
So at first when you start doing
these functions and inverse of functions

00:05:53.969 --> 00:05:57.524
it looks a little confusing,
hey, I'm going back and forth,

00:05:57.524 --> 00:05:58.925
but you just have to remember

00:05:58.925 --> 00:06:01.960
a function maps from one set of numbers
to another set of numbers.

00:06:01.960 --> 00:06:04.799
The inverse of that function
goes the other way.

00:06:04.799 --> 00:06:09.067
If the function goes from 9 to 8,
the inverse is going to go from 8 to 9.

00:06:09.067 --> 00:06:12.807
So one way to think about it is,
you just switch these columns.

00:06:12.807 --> 00:06:17.667
Hopefully, that clarifies
more things than it confuses.


Product rule proof

WEBVTT
Kind: captions
Language: en

00:00:00.441 --> 00:00:01.857
- [Voiceover] What I
hope to do in this video

00:00:01.857 --> 00:00:05.781
is give you a satisfying
proof of the product rule.

00:00:05.781 --> 00:00:08.962
So let's just start with our
definition of a derivative.

00:00:08.962 --> 00:00:12.143
So if I have the function F of X,

00:00:12.143 --> 00:00:14.883
and if I wanted to take
the derivative of it,

00:00:14.883 --> 00:00:17.390
by definition, by definition,

00:00:17.390 --> 00:00:21.687
the derivative of F of X is the limit

00:00:21.687 --> 00:00:24.155
as H approaches zero,

00:00:24.155 --> 00:00:29.131
of F of X plus H minus F of X,

00:00:29.131 --> 00:00:32.782
all of that over, all of that over H.

00:00:32.782 --> 00:00:34.058
If we want to think of it visually,

00:00:34.058 --> 00:00:37.031
this is the slope of the
tangent line and all of that,

00:00:37.031 --> 00:00:38.540
but now I want to do something

00:00:38.540 --> 00:00:40.351
a little bit more interesting.

00:00:40.351 --> 00:00:42.967
I want to find the derivative

00:00:43.689 --> 00:00:46.452
with respect to X, not just of F of X,

00:00:46.452 --> 00:00:48.796
but the product of two functions,

00:00:48.796 --> 00:00:52.257
F of X times G of X.

00:00:52.257 --> 00:00:54.300
And if I can come up with
a simple thing for this,

00:00:54.300 --> 00:00:56.343
that essentially is the product rule.

00:00:56.343 --> 00:00:58.874
Well if we just apply the
definition of a derivative,

00:00:58.874 --> 00:01:00.679
that means I'm gonna take the limit

00:01:01.864 --> 00:01:04.442
as H approaches zero,

00:01:04.442 --> 00:01:06.710
and the denominator I'm gonna have at H,

00:01:07.572 --> 00:01:09.475
and the denominator,
I'm gonna write a big,

00:01:09.475 --> 00:01:11.147
it's gonna be a big rational expression,

00:01:11.147 --> 00:01:13.191
in the denominator I'm gonna have an H.

00:01:13.191 --> 00:01:16.093
And then I'm gonna evaluate
this thing at X plus H.

00:01:16.093 --> 00:01:20.686
So that's going to be F of X plus H,

00:01:20.686 --> 00:01:25.686
G of X plus H and from
that I'm gonna subtract

00:01:25.789 --> 00:01:27.601
this thing evaluated F of X.

00:01:27.601 --> 00:01:29.945
Or, sorry, this thing evaluated X.

00:01:29.945 --> 00:01:32.503
So that's gonna be F of X times G of X.

00:01:32.503 --> 00:01:34.825
And I'm gonna put a
big, awkward space here

00:01:34.825 --> 00:01:36.683
and you're gonna see why in a second.

00:01:36.683 --> 00:01:39.121
So if I just, if I evaluate this at X,

00:01:39.121 --> 00:01:40.878
this is gonna be minus

00:01:41.506 --> 00:01:44.711
F of X, G of X.

00:01:44.711 --> 00:01:48.240
All I did so far is I just
applied the definition

00:01:48.240 --> 00:01:50.841
of the derivative, instead
of applying it to F of X,

00:01:50.841 --> 00:01:53.511
I applied it to F of X times G of X.

00:01:53.511 --> 00:01:55.415
So you have F of X plus H, G of X plus H

00:01:55.415 --> 00:01:56.994
minus F of X, G of X,

00:01:56.994 --> 00:01:58.340
all of that over H.

00:01:58.340 --> 00:02:00.516
Limit as H approaches zero.

00:02:01.103 --> 00:02:04.326
Now why did I put this
big, awkward space here?

00:02:04.326 --> 00:02:06.114
Because just the way I've
written it write now,

00:02:06.114 --> 00:02:08.366
it doesn't seem easy to
algebraically manipulate.

00:02:08.366 --> 00:02:10.293
I don't know how to evaluate this limit,

00:02:10.293 --> 00:02:13.055
there doesn't seem to be
anything obvious to do.

00:02:13.055 --> 00:02:14.914
And what I'm about to show you,

00:02:14.914 --> 00:02:17.027
I guess you could view it
as a little bit of a trick.

00:02:17.027 --> 00:02:19.441
I can't claim that I would
have figured it out on my own.

00:02:19.441 --> 00:02:22.923
Maybe eventually if I
were spending hours on it.

00:02:22.923 --> 00:02:25.130
I'm assuming somebody was
fumbling with it long enough

00:02:25.130 --> 00:02:26.328
that said, "Oh wait, wait.

00:02:26.328 --> 00:02:29.740
"Look, if I just add and
subtract at the same term here,

00:02:29.740 --> 00:02:31.878
"I can begin to
algebraically manipulate it

00:02:31.878 --> 00:02:33.410
"and get it to what we all know

00:02:33.410 --> 00:02:35.918
"as the classic product rule."

00:02:35.918 --> 00:02:38.303
So what do I add and subtract here?

00:02:39.164 --> 00:02:41.363
Well let me give you a clue.

00:02:41.922 --> 00:02:44.430
So if we have plus,

00:02:44.430 --> 00:02:46.487
actually, let me change this,

00:02:46.487 --> 00:02:51.227
minus F of X plus H, G of X,

00:02:51.765 --> 00:02:53.127
I can't just subtract, if I subtract it

00:02:53.127 --> 00:02:55.078
I've got to add it too, so
I don't change the value

00:02:55.078 --> 00:02:56.030
of this expression.

00:02:56.030 --> 00:03:00.940
So plus F of X plus H, G of X.

00:03:01.556 --> 00:03:02.897
Now I haven't changed the value,

00:03:02.897 --> 00:03:05.288
I just added and
subtracted the same thing,

00:03:05.288 --> 00:03:07.773
but now this thing can be manipulated

00:03:07.773 --> 00:03:10.304
in interesting algebraic ways to get us to

00:03:10.304 --> 00:03:12.997
what we all love about the product rule.

00:03:12.997 --> 00:03:14.762
And at any point you get inspired,

00:03:14.762 --> 00:03:16.869
I encourage you to pause this video.

00:03:17.869 --> 00:03:20.446
Well to keep going, let's just keep

00:03:20.446 --> 00:03:22.768
exploring this expression.

00:03:22.768 --> 00:03:25.433
So all of this is going to be equal to,

00:03:25.433 --> 00:03:27.748
it's all going to be equal to the limit

00:03:28.631 --> 00:03:31.581
as H approaches zero.

00:03:31.581 --> 00:03:32.904
So the first thing I'm gonna do is

00:03:32.904 --> 00:03:35.962
I'm gonna look at, I'm
gonna look at this part,

00:03:36.781 --> 00:03:39.306
this part of the expression.

00:03:39.306 --> 00:03:42.325
And in particular,
let's see, I am going to

00:03:42.325 --> 00:03:45.761
factor out an F of X plus H.

00:03:45.761 --> 00:03:48.850
So if you factor out an F of X plus H,

00:03:48.850 --> 00:03:51.196
this part right over here is going to be

00:03:51.196 --> 00:03:53.722
F of X plus H,

00:03:53.722 --> 00:03:56.346
F of X plus H,

00:03:56.346 --> 00:03:57.599
times

00:03:57.599 --> 00:04:00.889
you're going to be left
with G of X plus H.

00:04:01.749 --> 00:04:06.749
G of, that's a slightly
different shade of green,

00:04:07.703 --> 00:04:10.977
G of X plus H, that's that there,

00:04:10.977 --> 00:04:13.182
minus G of X,

00:04:13.182 --> 00:04:15.289
minus G of X,

00:04:15.289 --> 00:04:17.325
oops, I forgot the parentheses.

00:04:18.489 --> 00:04:20.206
Oops, it's a different color.

00:04:20.206 --> 00:04:22.715
I got a new software program
and it's making it hard

00:04:22.715 --> 00:04:23.735
for me to change colors.

00:04:23.735 --> 00:04:25.964
My apologies, this is not
a straightforward proof

00:04:25.964 --> 00:04:28.495
and the least I could do is
change colors more smoothly.

00:04:28.495 --> 00:04:31.305
Alright, (laughing) G of X plus H

00:04:31.305 --> 00:04:35.312
minus G of X, that's that
one right over there,

00:04:35.870 --> 00:04:38.789
and then all of that over this H.

00:04:39.393 --> 00:04:41.647
All of that over H.

00:04:41.647 --> 00:04:43.597
So that's this part here

00:04:43.597 --> 00:04:47.258
and then this part over here

00:04:48.652 --> 00:04:51.463
this part over here, and
actually it's still over H,

00:04:51.463 --> 00:04:53.731
so let me actually circle it like this.

00:04:53.731 --> 00:04:57.050
So this part over here

00:04:58.562 --> 00:05:00.393
I can write as.

00:05:01.278 --> 00:05:03.615
So then we're going to have plus...

00:05:04.593 --> 00:05:06.674
actually here let me,

00:05:06.674 --> 00:05:11.334
let me factor out a G of X here.

00:05:12.217 --> 00:05:14.424
So plus G of X

00:05:14.424 --> 00:05:19.270
plus G of X times this F of X plus H.

00:05:20.433 --> 00:05:23.561
Times F of X plus H

00:05:24.608 --> 00:05:26.227
minus this F of X.

00:05:28.133 --> 00:05:31.082
Minus that F of X.

00:05:31.082 --> 00:05:33.054
All of that over H.

00:05:33.054 --> 00:05:35.167
All of that over H.

00:05:35.167 --> 00:05:36.747
Now we know from our limit properties,

00:05:36.747 --> 00:05:39.254
the limit of all of this business,

00:05:39.254 --> 00:05:40.851
well that's just going
to be the same thing

00:05:40.851 --> 00:05:44.543
as the limit of this as H approaches zero

00:05:44.543 --> 00:05:47.144
plus the limit of this
as H approaches zero.

00:05:47.144 --> 00:05:48.677
And then the limit of the product

00:05:48.677 --> 00:05:51.092
is going to be the same thing
as the product of the limits.

00:05:51.092 --> 00:05:53.483
So if I used both of
those limit properties,

00:05:53.483 --> 00:05:57.698
I can rewrite this whole
thing as the limit,

00:05:58.578 --> 00:05:59.970
let me give myself some real estate,

00:05:59.970 --> 00:06:03.783
the limit as H approaches
zero of F of X plus H,

00:06:04.956 --> 00:06:09.310
of F of X plus H times,

00:06:10.218 --> 00:06:11.835
times the limit

00:06:12.813 --> 00:06:16.435
as H approaches zero,
of all of this business,

00:06:16.435 --> 00:06:20.655
G of X plus H minus G of X,

00:06:21.612 --> 00:06:23.535
minus G of X,

00:06:23.535 --> 00:06:25.207
all of that over H,

00:06:25.207 --> 00:06:27.134
I think you might see where this is going.

00:06:27.134 --> 00:06:28.614
Very exciting.

00:06:29.573 --> 00:06:30.886
Plus,

00:06:30.886 --> 00:06:33.318
plus the limit,

00:06:34.324 --> 00:06:36.334
let me write that a
little bit more clearly.

00:06:36.334 --> 00:06:41.334
Plus the limit as H
approaches zero of G of X,

00:06:42.970 --> 00:06:45.377
our nice brown colored G of X,

00:06:46.261 --> 00:06:48.768
times, now that we have our product here,

00:06:48.768 --> 00:06:49.946
the limit,

00:06:50.574 --> 00:06:51.589
the limit

00:06:52.544 --> 00:06:56.432
as H approaches zero of F of X plus H.

00:06:57.549 --> 00:07:02.237
Of F of X plus H minus F of X,

00:07:03.654 --> 00:07:05.884
minus F of X,

00:07:05.884 --> 00:07:07.254
all of that,

00:07:07.254 --> 00:07:08.461
all of that over H.

00:07:08.461 --> 00:07:10.343
And let me put the parentheses
where they're appropriate.

00:07:10.343 --> 00:07:11.456
So that,

00:07:11.456 --> 00:07:12.664
that,

00:07:12.664 --> 00:07:14.126
that,

00:07:14.126 --> 00:07:15.335
that.

00:07:15.335 --> 00:07:16.936
And all I did here, the limit,

00:07:16.936 --> 00:07:18.585
the limit of this sum,

00:07:18.585 --> 00:07:19.955
that's gonna be the sum of the limits,

00:07:19.955 --> 00:07:21.232
that's gonna be the limit of this

00:07:21.232 --> 00:07:22.835
plus the limit of that,

00:07:22.835 --> 00:07:24.691
and then the limit of
the products is gonna be

00:07:24.691 --> 00:07:26.550
the same thing as the
product of the limits.

00:07:26.550 --> 00:07:28.791
So I just used those
limit properties here.

00:07:28.791 --> 00:07:30.439
But now let's evaluate them.

00:07:30.439 --> 00:07:31.550
What's the limit,

00:07:31.550 --> 00:07:33.315
and I'll do them in different colors,

00:07:33.315 --> 00:07:34.545
what's this thing right over here?

00:07:34.545 --> 00:07:37.564
The limit is H approaches
zero of F of X plus H.

00:07:37.564 --> 00:07:38.972
Well that's just going to be

00:07:39.671 --> 00:07:40.757
F of X.

00:07:41.519 --> 00:07:43.655
Now, this is the exciting part,

00:07:43.655 --> 00:07:44.932
what is this?

00:07:44.932 --> 00:07:47.138
The limit is H approaches
zero of G of X plus H

00:07:47.138 --> 00:07:49.064
minus G of X over H.

00:07:49.064 --> 00:07:49.854
Well that's just our,

00:07:49.854 --> 00:07:51.503
that's the definition of our derivative.

00:07:51.503 --> 00:07:54.242
That's the derivative of G.

00:07:54.242 --> 00:07:55.938
So this is going to be,

00:07:55.938 --> 00:07:58.817
this is going to be the
derivative of G of X,

00:07:58.817 --> 00:08:01.015
which is going to be G prime of X.

00:08:01.922 --> 00:08:04.152
G prime of X.

00:08:04.152 --> 00:08:06.056
So you're multiplying these two

00:08:06.056 --> 00:08:08.308
and then you're going to have plus,

00:08:08.308 --> 00:08:10.955
what's the limit of H
approaches zero of G of X?

00:08:10.955 --> 00:08:12.581
Well there's not even any H in here,

00:08:12.581 --> 00:08:14.832
so this is just going to be G of X.

00:08:14.832 --> 00:08:17.171
So plus G of X

00:08:18.775 --> 00:08:20.859
times the limit,

00:08:21.766 --> 00:08:24.181
so let's see, this one is in brown,

00:08:24.181 --> 00:08:25.968
and the last one I'll do in yellow.

00:08:25.968 --> 00:08:28.499
Times the limit as H approaches zero,

00:08:28.499 --> 00:08:30.076
and we're getting very close,

00:08:30.076 --> 00:08:31.983
the drum roll should be starting,

00:08:31.983 --> 00:08:33.607
limit is H approaches zero of F of X

00:08:33.607 --> 00:08:35.441
plus H minus F of X over H.

00:08:35.441 --> 00:08:37.135
Well that's the definition
of the derivative

00:08:37.135 --> 00:08:38.158
of F of X.

00:08:38.158 --> 00:08:40.247
This is F prime of X.

00:08:40.247 --> 00:08:42.476
Times F prime of X.

00:08:42.476 --> 00:08:43.800
So there you have it.

00:08:43.800 --> 00:08:46.749
The derivative of F of
X times G of X is this.

00:08:46.749 --> 00:08:48.351
And if I wanted to write
it a little bit more

00:08:48.351 --> 00:08:50.714
condensed form, it is equal to,

00:08:50.714 --> 00:08:54.261
it is equal to F of X

00:08:54.261 --> 00:08:57.710
times the derivative
of G with respect to X

00:08:58.609 --> 00:09:01.542
times the derivative
of G with respect to X

00:09:01.542 --> 00:09:03.370
plus G of X,

00:09:04.092 --> 00:09:08.240
plus G of X times the derivative
of F with respect to X.

00:09:08.846 --> 00:09:10.402
F with respect to X.

00:09:10.402 --> 00:09:11.935
Or another way to think about it,

00:09:11.935 --> 00:09:14.209
this is the first function
times the derivative

00:09:14.209 --> 00:09:16.230
of the second plus the second function

00:09:16.230 --> 00:09:17.971
times the derivative of the first.

00:09:17.971 --> 00:09:21.176
This is the proof, or a proof,

00:09:21.176 --> 00:09:24.563
there's actually others
of the product rule.


Graphing piecewise function

WEBVTT
Kind: captions
Language: en

00:00:00.233 --> 00:00:01.938
- [Voiceover] So, I
have this somewhat hairy

00:00:01.938 --> 00:00:03.630
function definition here,

00:00:03.630 --> 00:00:05.184
and I want to see if we can graph it.

00:00:05.184 --> 00:00:06.611
And this is a piecewise function.

00:00:06.611 --> 00:00:09.600
It's defined as a different,
essentially different lines.

00:00:09.600 --> 00:00:11.200
You see this right over here,
even with all the decimals

00:00:11.200 --> 00:00:13.366
and the negative signs,
this is essentially a line.

00:00:13.366 --> 00:00:15.713
It's defined by this line
over this interval for x,

00:00:15.713 --> 00:00:17.833
this line over this interval of x,

00:00:17.833 --> 00:00:20.048
and this line over this interval of x.

00:00:20.048 --> 00:00:21.473
Well, let's see if we can graph it.

00:00:21.473 --> 00:00:23.467
I encourage you, especially
if you have some graph paper,

00:00:23.467 --> 00:00:25.333
to see if you can graph
this on your own first

00:00:25.333 --> 00:00:27.069
before I work through it.

00:00:27.300 --> 00:00:29.158
So, let's think about this first interval.

00:00:29.158 --> 00:00:31.340
When negative 10 is
less than or equal to x,

00:00:31.340 --> 00:00:33.773
which is less than negative two,

00:00:33.773 --> 00:00:38.628
then our function is defined
by negative 0.125x plus 4.75.

00:00:38.628 --> 00:00:43.266
So this is going to be a
line, a downward sloping line,

00:00:43.266 --> 00:00:45.100
and the easiest way I can
think about graphing it is

00:00:45.100 --> 00:00:49.767
let's just plot the endpoints
here, and then draw the line.

00:00:49.767 --> 00:00:52.100
So, when x is equal to 10,

00:00:52.138 --> 00:00:57.138
sorry, when x is equal to
negative 10, so we would have

00:00:57.334 --> 00:01:00.366
negative zero, actually
let me write it this way.

00:01:00.463 --> 00:01:02.433
Let me do it over here where I do the,

00:01:02.433 --> 00:01:06.600
so we're going to have negative 0.125

00:01:06.600 --> 00:01:11.284
times negative 10 plus 4.75.

00:01:11.284 --> 00:01:13.400
That is going to be equal to, let's see,

00:01:13.400 --> 00:01:15.866
the negative times a
negative is a positive,

00:01:15.866 --> 00:01:20.866
and then 10 times this is
going to be 1.25 plus 4.75.

00:01:22.266 --> 00:01:24.967
That is going to be equal to six.

00:01:24.967 --> 00:01:29.032
So, we're going to have the
point negative 10 comma six.

00:01:29.032 --> 00:01:32.836
And that point, and it
includes, so x is defined there,

00:01:32.836 --> 00:01:34.600
it's less than or equal to,

00:01:34.600 --> 00:01:36.800
and then we go all the
way to negative two.

00:01:36.800 --> 00:01:41.800
So, when x is equal to negative
two, we have negative 0.125

00:01:41.834 --> 00:01:46.834
times negative two plus 4.75 is equal to,

00:01:48.161 --> 00:01:50.400
see negative times negative is positive,

00:01:50.400 --> 00:01:52.933
two times this is going to be point,

00:01:52.933 --> 00:01:57.033
is going to be positive 0.25 plus 4.75.

00:01:57.033 --> 00:01:59.580
It's going to be equal to positive five.

00:01:59.580 --> 00:02:03.285
Now, we might be tempted,
we might be tempted,

00:02:03.285 --> 00:02:05.701
to just circle in this dot over here,

00:02:05.701 --> 00:02:08.979
but remember, this interval
does not include negative two.

00:02:08.981 --> 00:02:11.633
It's up to and including,
it's up to negative two,

00:02:11.633 --> 00:02:12.866
not including.

00:02:12.866 --> 00:02:17.667
So, I'm gonna put a
little open circle there,

00:02:17.667 --> 00:02:19.434
and then I'm gonna draw the line.

00:02:19.434 --> 00:02:23.308
And then I'm gonna draw the line.

00:02:23.308 --> 00:02:28.066
I am going to draw my best
attempt, my best attempt,

00:02:28.066 --> 00:02:29.700
at the line.

00:02:29.700 --> 00:02:31.314
Now, let's do the next interval.

00:02:31.314 --> 00:02:33.945
The next interval, this one's
a lot more straightforward.

00:02:33.945 --> 00:02:37.100
We started x equals negative
two, when x equals negative two

00:02:37.100 --> 00:02:42.100
negative two plus seven is,
negative two plus seven is five.

00:02:43.000 --> 00:02:46.466
So, negative two, so
negative two comma five,

00:02:46.466 --> 00:02:48.433
so it actually includes
that point right over there.

00:02:48.433 --> 00:02:50.366
So we're actually able to fill it in,

00:02:50.366 --> 00:02:54.100
and then when x is negative one,

00:02:54.100 --> 00:02:59.100
negative one plus seven is
going to be positive six.

00:02:59.400 --> 00:03:02.501
Positive six, but we're not
including x equals negative one

00:03:02.501 --> 00:03:06.560
up to and including, so it's
going to be right over here.

00:03:06.560 --> 00:03:09.200
When x is negative one,
we are approaching,

00:03:09.200 --> 00:03:11.700
or as x approaches negative
one, we're approaching

00:03:11.700 --> 00:03:14.533
negative one plus seven is six.

00:03:14.533 --> 00:03:17.633
So, that's that interval right over there.

00:03:17.633 --> 00:03:19.700
And now let's look at this last interval.

00:03:19.700 --> 00:03:23.333
This last interval,
when x is negative one,

00:03:23.333 --> 00:03:24.750
you're going to have,

00:03:24.750 --> 00:03:27.256
well, this is just going
to be positive 12 over 11

00:03:27.256 --> 00:03:29.261
'cause we're multiplying
it by negative one,

00:03:29.261 --> 00:03:34.261
plus 54 over 11 which
is equal to 66 over 11

00:03:34.267 --> 00:03:36.485
which is equal to positive six.

00:03:36.485 --> 00:03:39.740
So, we're able to fill
in that right over there,

00:03:39.740 --> 00:03:42.500
and then when x is equal to 10,

00:03:42.500 --> 00:03:45.176
you have negative 120 over 11.

00:03:45.176 --> 00:03:46.655
I just multiplied this times 10,

00:03:46.655 --> 00:03:49.567
12 times 10 is 120, and
we have the negative,

00:03:49.567 --> 00:03:52.879
plus 54 over 11.

00:03:52.879 --> 00:03:54.565
So this is the same thing.

00:03:54.565 --> 00:03:55.894
This is going to be, what is this?

00:03:55.894 --> 00:04:00.894
This is negative 66
over 11, is that right?

00:04:01.296 --> 00:04:05.600
Let's see, if you, yeah,
that is negative 66 over 11,

00:04:05.600 --> 00:04:07.830
which is equal to negative six.

00:04:07.830 --> 00:04:09.838
So when x is equal to 10,

00:04:09.838 --> 00:04:13.730
our function is equal to negative six.

00:04:13.730 --> 00:04:16.861
And so this one actually
doesn't have any jumps in it.

00:04:16.861 --> 00:04:21.861
It could've, but we see,
so there we have it.

00:04:22.643 --> 00:04:25.562
We have graphed this function
that has been defined

00:04:25.562 --> 00:04:28.266
in a piecewise way.


Piecewise function formula from graph

WEBVTT
Kind: captions
Language: en

00:00:00.303 --> 00:00:01.974
- [Voiceover] By now we're used to seeing

00:00:01.974 --> 00:00:06.525
functions defined like h(y)=y^2

00:00:06.525 --> 00:00:11.192
or f(x)= to the square root of x.

00:00:11.192 --> 00:00:12.864
But what we're now going to explore

00:00:12.864 --> 00:00:15.886
is functions that are
defined piece by piece

00:00:15.886 --> 00:00:17.859
over different intervals
and functions like

00:00:17.859 --> 00:00:21.331
this you'll sometimes
view them as a piecewise,

00:00:21.331 --> 00:00:22.910
or these types of function definitions

00:00:22.910 --> 00:00:26.323
they might be called a
piecewise function definition.

00:00:26.323 --> 00:00:29.806
Let's take a look at this
graph right over here.

00:00:29.806 --> 00:00:31.779
This graph, you can see that the function

00:00:31.779 --> 00:00:34.960
is constant over this interval, 4x.

00:00:34.960 --> 00:00:37.166
And then it jumps up
in this interval for x,

00:00:37.166 --> 00:00:40.022
and then it jumps back down
for this interval for x.

00:00:40.022 --> 00:00:41.694
Let's think about how we would write this

00:00:41.694 --> 00:00:44.062
using our function notation.

00:00:44.062 --> 00:00:47.359
If we say that this right
over here is the x-axis

00:00:47.359 --> 00:00:52.142
and this is the y=f(x) axis.

00:00:52.142 --> 00:00:56.507
Then, let's see, our function
f(x) is going to be equal to,

00:00:56.507 --> 00:00:58.683
there's three different intervals.

00:00:58.683 --> 00:01:00.192
So let me give myself some space

00:01:00.192 --> 00:01:01.794
for the three different intervals.

00:01:01.794 --> 00:01:06.020
Now this first interval
is from, not including -9,

00:01:06.020 --> 00:01:07.575
and I have this open circle here.

00:01:07.575 --> 00:01:08.574
Not a closed in circle.

00:01:08.574 --> 00:01:12.219
So not including -9 but
x being greater than -9

00:01:12.219 --> 00:01:15.238
and all the way up to and including -5.

00:01:15.238 --> 00:01:19.580
I could write that as -9 is less than x,

00:01:19.580 --> 00:01:23.179
less than or equal to -5.

00:01:23.179 --> 00:01:25.175
That's this interval, and what is

00:01:25.175 --> 00:01:26.801
the value of the function
over this interval?

00:01:26.801 --> 00:01:29.587
Well we see, the value
of the function is -9.

00:01:29.587 --> 00:01:32.629
It's a constant -9 over that interval.

00:01:32.629 --> 00:01:33.906
It's a little confusing because the value

00:01:33.906 --> 00:01:36.599
of the function is actually also the value

00:01:36.599 --> 00:01:40.361
of the lower bound on this
interval right over here.

00:01:40.361 --> 00:01:43.913
It's very important to look at
this says, -9 is less than x,

00:01:43.913 --> 00:01:45.260
not less than or equal.

00:01:45.260 --> 00:01:46.839
If it was less than or
equal, then the function

00:01:46.839 --> 00:01:49.648
would have been defined at
x equals -9, but it's not.

00:01:49.648 --> 00:01:52.156
We have an open circle right over there.

00:01:52.156 --> 00:01:53.944
But now let's look at the next interval.

00:01:53.944 --> 00:01:58.944
The next interval is
from -5 is less than x,

00:01:59.767 --> 00:02:04.387
which is less than or equal to -1.

00:02:04.387 --> 00:02:08.149
Over that interval, the
function is equal to,

00:02:08.149 --> 00:02:09.844
the function is a constant 6.

00:02:09.844 --> 00:02:10.842
It jumps up here.

00:02:10.842 --> 00:02:12.792
Sometimes people call this a
step function, it steps up.

00:02:12.792 --> 00:02:15.486
It looks like stairs to some degree.

00:02:15.486 --> 00:02:18.992
Now it's very important
here, that at x equals -5,

00:02:18.992 --> 00:02:20.966
for it to be defined only one place.

00:02:20.966 --> 00:02:22.521
Here it's defined by this part.

00:02:22.521 --> 00:02:24.123
It's only defined over here.

00:02:24.123 --> 00:02:25.493
So that's why it's
important that this isn't

00:02:25.493 --> 00:02:28.140
a -5 is less than or equal to.

00:02:28.140 --> 00:02:30.625
Because then if you put
-5 into the function,

00:02:30.625 --> 00:02:32.505
this thing would be filled in,

00:02:32.505 --> 00:02:34.618
and then the function would
be defined both places

00:02:34.618 --> 00:02:35.826
and that's not cool for a function,

00:02:35.826 --> 00:02:37.498
it wouldn't be a function anymore.

00:02:37.498 --> 00:02:40.818
So it's very important that when you input

00:02:40.818 --> 00:02:45.640
- 5 in here, you know which
of these intervals you are in.

00:02:45.640 --> 00:02:48.357
You can't be in two of these intervals.

00:02:48.357 --> 00:02:49.703
If you are in two of these intervals,

00:02:49.703 --> 00:02:51.445
the intervals should
give you the same values

00:02:51.445 --> 00:02:52.768
so that the function maps,

00:02:52.768 --> 00:02:55.856
from one input to the same output.

00:02:55.856 --> 00:02:56.948
Now let's keep going.

00:02:56.948 --> 00:02:58.643
We have this last
interval where we're going

00:02:58.643 --> 00:03:03.217
from -1 to 9.

00:03:03.217 --> 00:03:06.073
>From -1 to +9.

00:03:06.073 --> 00:03:09.602
And x starts off with -1 less than x,

00:03:09.602 --> 00:03:11.645
because you have an open
circle right over here

00:03:11.645 --> 00:03:12.574
and that's good because

00:03:12.574 --> 00:03:15.194
X equals -1 is defined up here,

00:03:15.194 --> 00:03:18.955
all the way to x is
less than or equal to 9.

00:03:18.955 --> 00:03:23.082
Over that interval, what is
the value of our function?

00:03:23.082 --> 00:03:26.286
Well you see, the value of
our function is a constant -7.

00:03:26.286 --> 00:03:29.026
A constant -7 and we're done.

00:03:29.026 --> 00:03:31.023
We have just constructed a

00:03:31.023 --> 00:03:34.111
piece by piece definition
of this function.

00:03:34.111 --> 00:03:37.402
Actually, when you see this
type of function notation,

00:03:37.402 --> 00:03:38.419
it becomes a lot clearer why

00:03:38.419 --> 00:03:42.273
function notation is useful even.

00:03:42.273 --> 00:03:44.409
Hopefully you enjoyed that.

00:03:44.409 --> 00:03:48.089
I always find these piecewise
functions a lot of fun.