Proof: perpendicular lines have negative reciprocal slope

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I'd
like to do with this video

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is use some geometric
arguments to prove that the

00:00:06.789 --> 00:00:09.105
slopes of perpendicular lines are negative

00:00:09.105 --> 00:00:10.910
reciprocals of each other.

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And so, just to start off.

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We have lines L and M and
we are going to assume

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that they are perpendicular.

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So, they intersect at a right angle.

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We see that depicted right over here.

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And so, I'm gonna now
construct some other lines here

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to help us make our geometric argument.

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So, let me draw a horizontal
line that intersects

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this point right over here.

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Let's call that point A.

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And so, let me see if I can do that.

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There you go.

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So, that's a horizontal
line that intersects at A.

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And now, I'm gonna drop
some verticals in that.

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So, I'm gonna drop a vertical
line right over here,

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and I'm gonna drop a vertical
line right over here.

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And so, that is 90 degrees,
and that is 90 degrees.

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And I've constructed it that way.

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This top line is perpendicular horizontal.

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And then, I've dropped to vertical things.

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So, there are at 90 degree angles.

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And let me know setup some points.

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So, that I already said, that's point A.

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Let's call this point B.

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Let's call this point C.

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Let's call this point D.

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And let's call this
point E, right over here.

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Now, let's think about
what the slop of line L is.

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So, slope of L is going to be what?

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Well, you view line L as
the line that connects

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point CA, so it's the
slope of CA, you could say.

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This is the same thing
as slope of line CA.

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L is line CA.

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And so, to find the
slope, that's change in Y

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over change in X.

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So, our change in Y is going to be CB.

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So, it's gonna be the length of CB.

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That is our change in Y.

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So, it is CB over our change in X,

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which is the length of segment of BA,

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which is the length of BA right over here.

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So, that is BA.

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Now, what is the slope of line M?

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So, slope of M.

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And we could also say slope
of, we could call line M

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line AE.

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Line AE, like that.

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Well, if we're going to go
between point A and point E,

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once again, it's just change
in Y over change in X.

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Well, what's our change in Y going to be.

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Well, our change in Y.

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Well, we're gonna go from
this level down to this level,

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as we go from A to E.

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We could have done it over here as well.

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We're gonna go from A to E.

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That is our change in Y.

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So, we might be tempted to say,

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well that's just gonna be
the length of segment DE.

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But remember, our Y is decreasing.

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So, we're gonna subtract
that length as we go from

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this Y level, to that Y level over there.

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And what is our change in X?

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So, our change in X, as we go form A to E,

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our change in X is going to
be the length of segment AD.

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So, AD.

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So, our slope of M is
going to be negative DE.

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That's going to be the
negative of this length,

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cause we're dropping by that much.

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That's our change in Y,

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over segment A, over segment AD.

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So, some of you might
already be quite inspired

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by what we've already written,

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because now we just have to established

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that these two triangles,

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triangle CBA and triangle ADE are similar,

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and then we're going to be able to show

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that these are the negative
reciprocal of each other.

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So, let's show that these
two triangles are similar.

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So, let's say that we have
this angle right over here.

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And let's say that angle has
measure X, just for kicks.

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And let's say that we have,
let me do another color for,

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let's say we have this
angle right over here.

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And let's say that the measure
that that has measure Y.

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Well, we know X + Y + 90 is equal to 180,

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because together, they are supplementary.

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So, I could write that X
+ 90 + Y is going to be

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equal to 180 degrees.

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If you want, you can subtract
90 from both sides of that,

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and you could say look, X
+ Y is going to be equal

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to 90 degrees.

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These are algebraically
equivalent statements.

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So, is equal to 90 degrees.

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And how can we use this to fill out

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some of the other angles
in these triangles.

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Well, let's see X + this angle down here

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has to be equal to 90 degrees.

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Or you could say X + 90
+ what is going to be

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equal to 180.

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I'm looking at triangle
CBA right over here.

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The interior angles of a
triangle add up to 180.

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So, X + 90 + what is equal to 180.

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Well, X + 90 + Y is equal to 180.

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We already established that.

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Similarly over here.

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Y + 90 + what is going to be equal to 180.

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Well, same argument.

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We already know.

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Y + 90 + X is equal to 180.

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So, Y + 90 + X is equal to 180.

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And so, notice we have now established

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that triangle ABC and triangle EDA,

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all of their interior angles,

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their corresponding interior
angles are the same,

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or that their three
different angle measures

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all correspond to each other.

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They both have an angle of X.

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They both have a measure of X,

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they both have an angle of measure Y,

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and they're both right triangles.

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So, just by angle, angle, angle,

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so we could say by angle, angle, angle,

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one of our similarity postulates.

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We know that triangle EDA

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is similar to triangle ABC.

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And so, that tells us that
the ratio of corresponding

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sides are going to be the same.

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And so, for example, we know.

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Let's find the ration
of corresponding sides.

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We know that the ratio
of let's say CB to BA,

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so let's write this down.

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We know that the ratio, so this tells us

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that the ratio of corresponding sides

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are going to be the same.

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So, the ratio of CB over BA

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is going to be equal to...

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Well, the corresponding side to CB,

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it's the side opposite the X degree angle

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right over here.

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So, the corresponding
side to CB is side AD.

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So, that's going to be equal to AD over,

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what's the corresponding side to BA?

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Well, BA is opposite the Y degree angle.

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So, over here the
corresponding side is DE.

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AD over DE.

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Let me do that same color.

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Over DE.

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And so, this right over,
we saw from the beginning,

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this is the slope of L.

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So, slope of L.

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And how does this relate
to the slope of M?

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Notice, the slope of M is
the negative reciprocal

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of this.

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You take the reciprocal,
you're going to get DE over AD,

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and then you have to take
this negative right over here.

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So, we could write this
as the negative reciprocal

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of slope of M.

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Negative reciprocal of M's slope.

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And there you have it.

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We've just shown that if we

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assume L and M are perpendicular,

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and we setup these similar triangles,

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and we're able to show,

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that the slope of L is
the negative reciprocal

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of the slope of M.


Proof: parallel lines have the same slope

WEBVTT
Kind: captions
Language: en

00:00:00.311 --> 00:00:02.028
- [Voiceover] What I wanna
do in this video is prove

00:00:02.028 --> 00:00:05.952
that parallel lines have the same slope.

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So let's draw some parallel lines here.

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So, that's one line and then
let me draw another line

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that is parallel to that.

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I'm claiming that these
are parallel lines.

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And now, I'm gonna draw
some transversals here.

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So first let me draw a
horizontal transversal.

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So, just like that.

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And then let me do a vertical transversal.

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So,

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just like that.

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And I'm assuming that the
green one is horizontal

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and the blue one is vertical.

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So we assume that they are
perpendicular to each other,

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that these intersect at right angles.

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And from this, I'm gonna figure out,

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I'm gonna use some parallel
line angle properties

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to establish that this triangle

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and this triangle are similar

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and then use that to establish
that both of these lines,

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both of these yellow
lines have the same slope.

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So actually let me label some points here.

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So let's call that point
A, point B, point C,

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point D, and point E.

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So, let's see.

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First of all we know that angle CED

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is going to be congruent to angle AEB,

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because they're both right angles.

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So that's a right angle and
then that is a right angle

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right over there.

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We also know some things
about corresponding angles

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for where our transversal
intersects parallel lines.

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This angle corresponds
to this angle if we look

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at the blue transversal as it
intersects those two lines.

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And so they're going to
be, they're going to have

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the same measure, they're
going to be congruent.

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Now this angle on one side of this point B

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is going to also be congruent to that,

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because they are vertical angles.

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We've seen that multiple times before.

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And so we know that this angle, angle ABE

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is congruent to angle ECD.

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Sometimes this is called
alternate interior angles

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of a transversal and parallel lines.

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Well, if you look at triangle
CED and triangle ABE,

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we see they already have
two angles in common,

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so if they have two angles in common,

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well, then their third
angle has to be in common.

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So, because this third
angle's just gonna be

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180 minus these other two,
and so this third angle

00:02:20.031 --> 00:02:22.960
is just gonna be 180
minus this, the other two.

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And so just like that, we
notice we have all three angles

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are the same in both of these triangles,

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well, they're not all the same,

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but all of the corresponding angles,

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I should say, are the same.

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This blue angle has the same
measure as this blue angle,

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this magenta angle has the same measure

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as this magenta angle,
and then the other angles

00:02:42.355 --> 00:02:44.698
are right angles, these
are right triangles here.

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So we could say triangle AEB,

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triangle AEB

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is similar, similar

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similar to triangle DEC,

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triangle DEC

00:03:06.809 --> 00:03:10.700
by, and we could say
by angle, angle, angle,

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all the corresponding
angles are congruent,

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so we are dealing with similar triangles.

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And so we know similar
triangles are a ratio

00:03:17.119 --> 00:03:20.800
of corresponding sides
are going to be the same.

00:03:20.800 --> 00:03:23.996
So we could say that
the ratio of let's say

00:03:23.996 --> 00:03:28.163
the ratio of BE, the ratio of
BE, let me write this down,

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this is this side right
over here, the ratio of BE

00:03:34.720 --> 00:03:36.387
to AE, to AE, to AE,

00:03:40.764 --> 00:03:44.931
is going to be equal to, so
that side over that side,

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well what is the corresponding side?

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The corresponding side to BE is side CE.

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So that's going to be the same

00:03:51.623 --> 00:03:54.873
as the ratio between CE and DE, and DE.

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And this just comes out of similar,

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the similarity of the triangles, CE to DE.

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So once again, once we established

00:04:07.613 --> 00:04:09.269
these triangles are similar,
we can say the ratio

00:04:09.269 --> 00:04:12.406
of corresponding sides
are going to be the same.

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Now what is the ratio between BE and AE?

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The ratio between BE and AE.

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Well that is the slope of
this top line right over here.

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We could say that's the slope of line AB,

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slope of line connecting,

00:04:34.423 --> 00:04:35.256
connecting

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A to B.

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All right, let me just use,
I could write it like this,

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that is slope of, slope of A,

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slope of line AB.

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Remember slope is, when
you're going from A to B,

00:04:52.270 --> 00:04:55.103
it's change in y over change in x.

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So when you're going from A
to B, your change in x is AE,

00:05:01.717 --> 00:05:04.550
and your change in y is BE, or EB,

00:05:05.487 --> 00:05:06.585
however you want to refer to it.

00:05:06.585 --> 00:05:09.428
So this right over here is change in y,

00:05:09.428 --> 00:05:12.142
and this over here is change in x.

00:05:12.142 --> 00:05:13.991
Well, now let's look at
this second expression

00:05:13.991 --> 00:05:17.324
right over here, CE over DE, CE over DE.

00:05:19.713 --> 00:05:22.544
Well, now, this is going to be change in y

00:05:22.544 --> 00:05:26.044
over change in x between point C and D.

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So this is, this right over
here, this is the slope of

00:05:32.005 --> 00:05:33.422
line, of line CD.

00:05:35.293 --> 00:05:38.601
And so just like that, by
establishing similarity,

00:05:38.601 --> 00:05:39.571
we were able see the ratio

00:05:39.571 --> 00:05:41.465
of corresponding sides are congruent,

00:05:41.465 --> 00:05:44.425
which shows us that the
slopes of these two lines

00:05:44.425 --> 00:05:45.788
are going to be the same.

00:05:45.788 --> 00:05:47.121
And we are done.


Points inside/outside/on a circle

WEBVTT
Kind: captions
Language: en

00:00:00.413 --> 00:00:02.886
- [Voiceover] A circle is
centered at the point C,

00:00:02.886 --> 00:00:04.374
which has the coordinates negative one,

00:00:04.374 --> 00:00:05.668
comma, negative three.

00:00:05.668 --> 00:00:08.034
And has a radius of six.

00:00:08.034 --> 00:00:09.699
Where does the point P, which has

00:00:09.699 --> 00:00:12.784
the coordinates negative six,
comma, negative six, lie?

00:00:12.784 --> 00:00:13.919
We have three options.

00:00:13.919 --> 00:00:17.833
Inside the circle, on the
circle, or outside the circle.

00:00:17.833 --> 00:00:19.529
And the key realization here is

00:00:19.529 --> 00:00:21.382
just what a circle is all about.

00:00:21.382 --> 00:00:26.066
If we have the point C, which
is the center of a circle,

00:00:26.066 --> 00:00:29.725
a circle of radius six, so
let me draw that radius.

00:00:29.725 --> 00:00:31.579
So we say that is its radius.

00:00:31.579 --> 00:00:32.662
Is six units.

00:00:33.688 --> 00:00:36.931
The circle will look something like this.

00:00:36.931 --> 00:00:39.506
Remember, the circle is
a set of all points that

00:00:39.506 --> 00:00:43.263
are exactly six units
away from that center.

00:00:43.263 --> 00:00:46.149
So that's the definition of a circle,

00:00:46.149 --> 00:00:47.283
it's a set of all points that are

00:00:47.283 --> 00:00:50.230
exactly six units away from the center.

00:00:50.230 --> 00:00:54.397
So if, for example, P is
less than six units away,

00:00:55.486 --> 00:00:57.388
it's going to be inside the circle.

00:00:57.388 --> 00:00:59.447
If it's exactly six units away,

00:00:59.447 --> 00:01:01.661
it's going to be on the circle,

00:01:01.661 --> 00:01:03.412
and if it's more than six units away,

00:01:03.412 --> 00:01:06.345
it's going to be outside of the circle.

00:01:06.345 --> 00:01:08.816
So the key is, is let's
find the distance between

00:01:08.816 --> 00:01:10.206
these two points.

00:01:10.206 --> 00:01:12.368
If the distance is less than six, inside,

00:01:12.368 --> 00:01:14.840
distance equals six, we're on the circle,

00:01:14.840 --> 00:01:18.091
distance more than six, we
are outside of the circle.

00:01:18.091 --> 00:01:19.268
So let's do that.

00:01:19.268 --> 00:01:21.229
So if we wanted to find,
well there's different

00:01:21.229 --> 00:01:23.540
notations for the distance.

00:01:23.540 --> 00:01:24.778
Well, I'll just write D, or I could

00:01:24.778 --> 00:01:27.506
write the distance between C and

00:01:27.506 --> 00:01:29.673
P is going to be equal to.

00:01:30.699 --> 00:01:32.035
And the distance formula
comes straight out of

00:01:32.035 --> 00:01:33.581
the Pythagorean Theorem.

00:01:33.581 --> 00:01:35.694
But it's going to be the square root of

00:01:35.694 --> 00:01:39.656
our change in X squared plus
our change in Y squared.

00:01:39.656 --> 00:01:41.716
So what is our change in X?

00:01:41.716 --> 00:01:43.671
So our change in X, if we view C as

00:01:43.671 --> 00:01:45.478
our starting point and P as our end point,

00:01:45.478 --> 00:01:46.815
and we could do it either way,

00:01:46.815 --> 00:01:49.439
our change in X, our change in X is

00:01:49.439 --> 00:01:52.106
negative six minus negative one.

00:01:53.608 --> 00:01:57.164
So negative six minus negative one.

00:01:57.164 --> 00:01:58.602
And we're going to square it.

00:01:58.602 --> 00:02:01.794
So what we have inside
here, that is change in X.

00:02:01.794 --> 00:02:04.524
So we're taking our change in X squared,

00:02:04.524 --> 00:02:07.563
and then plus our change in Y squared.

00:02:07.563 --> 00:02:09.930
So we are going, we're going from

00:02:09.930 --> 00:02:11.493
negative three to negative six.

00:02:11.493 --> 00:02:14.993
So our change in Y is negative
six minus negative three.

00:02:14.993 --> 00:02:17.669
Negative six minus negative three,

00:02:17.669 --> 00:02:19.833
and we're going to square everything.

00:02:19.833 --> 00:02:22.921
So that is our change in
Y inside the parentheses,

00:02:22.921 --> 00:02:24.568
and we're going to square it.

00:02:24.568 --> 00:02:27.147
So this is equal to, this is equal to

00:02:27.147 --> 00:02:30.897
negative six, negative
six plus positive one,

00:02:32.345 --> 00:02:33.273
is one way to think about it,

00:02:33.273 --> 00:02:36.416
so this is negative five squared.

00:02:36.416 --> 00:02:40.122
And then this is negative six plus three.

00:02:40.122 --> 00:02:43.059
So plus negative three squared.

00:02:43.059 --> 00:02:44.653
And once again, you can see,

00:02:44.653 --> 00:02:47.297
our change in X is negative five.

00:02:47.297 --> 00:02:51.209
We go five lower in X, and
we're going three lower in Y.

00:02:51.209 --> 00:02:53.219
So we're changing Y, is negative three.

00:02:53.219 --> 00:02:56.719
So this is equal to the square root of 25,

00:02:58.053 --> 00:03:00.423
square root of 25 plus nine.

00:03:00.423 --> 00:03:03.892
Square root of 25 plus
nine, which is equal to

00:03:03.892 --> 00:03:05.725
the square root of 34.

00:03:06.669 --> 00:03:10.017
Now, the key is, is the square root of 34

00:03:10.017 --> 00:03:13.757
less than six, greater
than six, or equal to six?

00:03:13.757 --> 00:03:16.954
Well, we know that six is equal to

00:03:16.954 --> 00:03:19.425
the square root of 36,
so the square root of

00:03:19.425 --> 00:03:21.995
34 is less than the square root of 36,

00:03:21.995 --> 00:03:24.105
so I could write the square root of 34 is

00:03:24.105 --> 00:03:26.319
less than the square root of 36,

00:03:26.319 --> 00:03:29.824
and so the square root
of 34 is less than six.

00:03:29.824 --> 00:03:32.654
These are the square root of 36, is six.

00:03:32.654 --> 00:03:34.473
And so since the distance between

00:03:34.473 --> 00:03:37.890
C and P is less than six, we are going to

00:03:38.904 --> 00:03:41.487
be on the inside of the circle.

00:03:42.402 --> 00:03:44.720
If I somehow got square root of 36 here,

00:03:44.720 --> 00:03:45.699
then we'd be on the circle,

00:03:45.699 --> 00:03:48.170
and if I somehow got
square root of 37 here,

00:03:48.170 --> 00:03:51.870
or something larger, we would
have been outside the circle.


Area of trapezoid on the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.890
- [Voiceover] So we have a trapezoid here

00:00:01.890 --> 00:00:04.510
on the coordinate plane
and what we want to do is

00:00:04.510 --> 00:00:08.652
find the area of this trapezoid,
just given this diagram.

00:00:08.652 --> 00:00:10.120
And, like always, pause this video and see

00:00:10.120 --> 00:00:11.879
if you can figure it out.

00:00:11.879 --> 00:00:14.271
Well, we know how to figure
out the area of a trapezoid.

00:00:14.271 --> 00:00:16.924
We have videos where
we derive this formula.

00:00:16.924 --> 00:00:19.246
But, the area of a
trapezoid, just put simply,

00:00:19.246 --> 00:00:23.777
is equal to the average of
the lengths of the bases,

00:00:23.777 --> 00:00:27.944
we could say base one plus
base two times the height.

00:00:29.285 --> 00:00:31.846
And, so what are our bases here?

00:00:31.846 --> 00:00:35.106
What are, and what is going
to be our height over here?

00:00:35.106 --> 00:00:38.139
Well, we could call base one,

00:00:38.139 --> 00:00:40.346
we could call that segment CL,

00:00:40.346 --> 00:00:43.268
so it would be the length of
segment CL, right over here.

00:00:43.268 --> 00:00:45.169
I'll do that in magenta.

00:00:45.169 --> 00:00:47.207
That is going to be base one.

00:00:47.207 --> 00:00:49.790
Base two, base two, that could,

00:00:51.290 --> 00:00:53.117
we'll do that in a different color.

00:00:53.117 --> 00:00:56.700
Base two would be the
length of segment OW,

00:00:58.577 --> 00:00:59.748
or b two would be the length

00:00:59.748 --> 00:01:03.011
of segment OW, right over there.

00:01:03.011 --> 00:01:06.526
And then, our height, our height h,

00:01:06.526 --> 00:01:07.759
well, that would just be an altitude

00:01:07.759 --> 00:01:09.739
and they did one in a dotted line, here.

00:01:09.739 --> 00:01:12.449
Notice it intersects the, base one,

00:01:12.449 --> 00:01:15.918
I guess you could say segment
CL at a right angle, here.

00:01:15.918 --> 00:01:19.188
And so, this would be,
this would be the height.

00:01:19.188 --> 00:01:21.416
So, if we know the
lengths of each of these,

00:01:21.416 --> 00:01:22.847
if we know each of these values,

00:01:22.847 --> 00:01:24.480
which are the lengths of these segments,

00:01:24.480 --> 00:01:28.349
then we can evaluate the area
of this actual trapezoid.

00:01:28.349 --> 00:01:30.838
And, once again, if this is
completely unfamiliar to you

00:01:30.838 --> 00:01:32.418
or of you're curious,
we have multiple videos

00:01:32.418 --> 00:01:33.518
talking about the proofs

00:01:33.518 --> 00:01:35.788
or how we came up with this formula.

00:01:35.788 --> 00:01:37.878
You an even break out,
break down a trapezoid

00:01:37.878 --> 00:01:39.269
into two triangles and a rectangle,

00:01:39.269 --> 00:01:40.965
which is one way to think about it.

00:01:40.965 --> 00:01:43.334
Anyway, let's see how we
could figure this out.

00:01:43.334 --> 00:01:45.694
So, the first one is,
what is b one going to be?

00:01:45.694 --> 00:01:48.444
B one is the length of segment CL

00:01:49.575 --> 00:01:53.906
and you could say, "We'll
look, we know what these,

00:01:53.906 --> 00:01:55.876
"the coordinates of these points are."

00:01:55.876 --> 00:01:57.922
You could say, "Let's use
the distance formula,"

00:01:57.922 --> 00:01:59.307
and you could say, "Well,
the distance formula is just

00:01:59.307 --> 00:02:01.618
"an application of the
Pythagorean Theorem."

00:02:01.618 --> 00:02:02.907
So, this is going to be

00:02:02.907 --> 00:02:06.477
the square root of our
change in x squared,

00:02:06.477 --> 00:02:09.258
so our change in x is going
to be this, right over here.

00:02:09.258 --> 00:02:11.056
And, notice, we're going
from x equals negative four

00:02:11.056 --> 00:02:13.928
to x equals eight, as w go from C to L.

00:02:13.928 --> 00:02:18.095
So, our change in x is equal
to eight minus negative four

00:02:20.250 --> 00:02:22.700
which is equal to 12.

00:02:22.700 --> 00:02:26.299
And, our change in in y, our change in y,

00:02:26.299 --> 00:02:30.290
we are going from y equals
negative one to y equals five.

00:02:30.290 --> 00:02:32.478
So, we could say our change in y

00:02:32.478 --> 00:02:35.828
equal to five minus negative one

00:02:35.828 --> 00:02:37.189
which, of course, is equal to six.

00:02:37.189 --> 00:02:41.450
And, you see that: one,
two, three, four, five, six.

00:02:41.450 --> 00:02:43.299
And, the segment that we care about,

00:02:43.299 --> 00:02:46.268
its length that we care about,
that's just the hypotenuse

00:02:46.268 --> 00:02:48.909
of this right triangle
that has one side 12

00:02:48.909 --> 00:02:50.687
and one side six.

00:02:50.687 --> 00:02:52.209
So, the length of that hypotenuse,

00:02:52.209 --> 00:02:53.942
from the Pythagorean Theorem,

00:02:53.942 --> 00:02:55.770
and, as I mentioned,
the distance formula is

00:02:55.770 --> 00:02:58.091
just an application of
the Pythagorean Theorem,

00:02:58.091 --> 00:02:59.724
this is going to be

00:02:59.724 --> 00:03:02.664
our change in x, squared, 12 squared,

00:03:02.664 --> 00:03:06.074
plus our change in y,
squared, so plus six squared,

00:03:06.074 --> 00:03:09.221
and this is going to be
equal to 144 plus 36.

00:03:09.221 --> 00:03:12.054
So, the square root of 144 plus 36

00:03:13.983 --> 00:03:16.816
is one hundred, one hundred and 80

00:03:18.079 --> 00:03:22.964
which is equal to, let's
see, 180 is 36 times five.

00:03:22.964 --> 00:03:27.131
So, that is six square roots,
six square roots of five.

00:03:28.770 --> 00:03:30.332
Oh, let me not skip some steps.

00:03:30.332 --> 00:03:33.901
So, this is the square
root of 36 times five

00:03:33.901 --> 00:03:36.577
which is equal to, square root of 36 is 6,

00:03:36.577 --> 00:03:38.910
so six square roots of five.

00:03:40.350 --> 00:03:42.950
Now, let's figure out b two.

00:03:42.950 --> 00:03:46.759
So, b two, once again, change in x squared

00:03:46.759 --> 00:03:48.828
plus the square root
of change in x squared

00:03:48.828 --> 00:03:50.418
plus change in y squared.

00:03:50.418 --> 00:03:53.068
Well, let's see, if we're going from,

00:03:53.068 --> 00:03:55.298
we could set up a right
triangle, if you like,

00:03:55.298 --> 00:03:58.279
like this, to figure those things out.

00:03:58.279 --> 00:04:02.446
So, our change in x, we're
going from x is at negative two,

00:04:03.889 --> 00:04:07.000
x is going from negative
two to positive four.

00:04:07.000 --> 00:04:09.036
So, our change in x is six.

00:04:09.036 --> 00:04:11.953
Our change in y, we are going from,

00:04:13.958 --> 00:04:18.125
we are going from y equals, y
equals five to y equals eight.

00:04:20.048 --> 00:04:23.139
So, our change in y is equal to three.

00:04:23.139 --> 00:04:25.217
So, just applying the Pythagorean Theorem

00:04:25.217 --> 00:04:27.055
to find the length of
the hypotenuse, here,

00:04:27.055 --> 00:04:28.209
is going to be the square root

00:04:28.209 --> 00:04:31.109
of change in x squared, six squared,

00:04:31.109 --> 00:04:34.176
plus change in y squared,
plus three squared

00:04:34.176 --> 00:04:36.213
which is going to be equal to,

00:04:36.213 --> 00:04:39.463
it's going to be equal to 36 plus nine,

00:04:42.780 --> 00:04:45.613
which is 45, so, square root of 45

00:04:47.580 --> 00:04:50.948
which is equal to the square
root of nine times five

00:04:50.948 --> 00:04:55.501
which is equal to three
square roots of five.

00:04:55.501 --> 00:04:56.990
And so, we only have
one left to figure out.

00:04:56.990 --> 00:04:58.717
We have to figure out h.

00:04:58.717 --> 00:05:01.889
We have to figure out the length of h.

00:05:01.889 --> 00:05:05.128
So, h is going to be equal to,

00:05:05.128 --> 00:05:07.860
and so, what is our, if
we're going from W to N,

00:05:07.860 --> 00:05:09.810
our change in x is two.

00:05:09.810 --> 00:05:11.520
Change in x is equal to two.

00:05:11.520 --> 00:05:14.790
We're going from X equals
four to x equals six.

00:05:14.790 --> 00:05:16.141
If you want to do that purely numerically,

00:05:16.141 --> 00:05:17.940
we would say, "Okay, our
endpoint, our x value is six.

00:05:17.940 --> 00:05:19.351
"Our starting point, our x-value is four.

00:05:19.351 --> 00:05:21.400
"Six minus four is two."

00:05:21.400 --> 00:05:22.790
You see that visually, here.

00:05:22.790 --> 00:05:24.601
So, it's going to be the square root

00:05:24.601 --> 00:05:26.968
of two squared plus our,

00:05:26.968 --> 00:05:29.771
let me write that radical
a little bit better.

00:05:29.771 --> 00:05:33.090
So, it's the square root
of our change in x squared

00:05:33.090 --> 00:05:35.318
plus our change in y squared.

00:05:35.318 --> 00:05:38.002
Our change in y is negative four.

00:05:38.002 --> 00:05:40.031
Change in y is negative four

00:05:40.031 --> 00:05:41.410
but we're going to square it,

00:05:41.410 --> 00:05:43.428
so it's going to become a positive 16.

00:05:43.428 --> 00:05:44.750
So, this is going to be equal

00:05:44.750 --> 00:05:46.602
to the square root of four plus 16.

00:05:46.602 --> 00:05:48.102
Square root of 20,

00:05:49.220 --> 00:05:53.490
which is equal to the square
root of four times five,

00:05:53.490 --> 00:05:55.992
which is equal to two times
the square root of five.

00:05:55.992 --> 00:05:58.221
It's nice that the square
root of five keeps popping up.

00:05:58.221 --> 00:05:59.240
And so, now we just substitute

00:05:59.240 --> 00:06:01.042
into our original expression.

00:06:01.042 --> 00:06:04.731
And so, our Area of our
trapezoid is going to be

00:06:04.731 --> 00:06:08.719
one half times six square roots of five,

00:06:08.719 --> 00:06:10.800
six square roots of five,

00:06:10.800 --> 00:06:13.319
plus three square roots of five,

00:06:13.319 --> 00:06:16.530
plus three square roots of five,

00:06:16.530 --> 00:06:18.501
let me close that parentheses,

00:06:18.501 --> 00:06:20.819
times two square roots of five,

00:06:20.819 --> 00:06:24.021
times two square roots of five.

00:06:24.021 --> 00:06:26.401
And, let's see how we can simplify this.

00:06:26.401 --> 00:06:27.411
So, six square roots of five

00:06:27.411 --> 00:06:28.450
plus three square roots of five,

00:06:28.450 --> 00:06:32.021
that is nine square roots of five.

00:06:32.021 --> 00:06:33.840
Let's see, the one half times the two,

00:06:33.840 --> 00:06:35.811
those cancel out to just be one.

00:06:35.811 --> 00:06:36.644
And so, we're left with

00:06:36.644 --> 00:06:39.387
nine square roots of five
times the square root of five.

00:06:39.387 --> 00:06:40.838
Well, the square root of five
times the square root of five

00:06:40.838 --> 00:06:43.198
is just going to be five.

00:06:43.198 --> 00:06:45.996
So, this is equal to nine times five

00:06:45.996 --> 00:06:50.163
which is equal to 45 square
units or units squared.


Worked example: divergent geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.345 --> 00:00:03.153
- [Voiceover] So we've got
this infinite series here,

00:00:03.153 --> 00:00:04.986
and let's see, it looks
like a geometric series.

00:00:04.986 --> 00:00:07.323
We're gonna go from this
term to the second term,

00:00:07.323 --> 00:00:10.406
we are multiplying by negative three,

00:00:11.328 --> 00:00:13.453
and then to go to the next
term we're gonna multiply

00:00:13.453 --> 00:00:15.781
by negative three again.

00:00:15.781 --> 00:00:18.562
So it looks like we have a
common ratio of negative three.

00:00:18.562 --> 00:00:20.909
So we could actually rewrite this series

00:00:20.909 --> 00:00:23.492
as being equal to negative 0.5,

00:00:26.168 --> 00:00:30.070
I could say times negative
three to the zero power,

00:00:30.070 --> 00:00:32.820
negative three to the zero power,

00:00:34.213 --> 00:00:37.494
plus negative zero, or maybe I can just

00:00:37.494 --> 00:00:41.661
keep writing this way, minus
0.5 times negative three

00:00:43.986 --> 00:00:48.256
to the first power, times
negative three to the first power,

00:00:48.256 --> 00:00:49.089
minus 0.5,

00:00:50.732 --> 00:00:51.565
minus 0.5

00:00:52.821 --> 00:00:56.236
times negative three to the second power,

00:00:56.236 --> 00:00:58.997
negative three to the second power.

00:00:58.997 --> 00:01:01.302
And we're just gonna keep going like that.

00:01:01.302 --> 00:01:03.087
And we could just say we're
just gonna keep having

00:01:03.087 --> 00:01:05.385
minus 0.5 times negative three to each

00:01:05.385 --> 00:01:08.229
or to a higher and
higher and higher powers.

00:01:08.229 --> 00:01:10.653
Or we could write this in sigma notation.

00:01:10.653 --> 00:01:12.799
This is equal to the same thing,

00:01:12.799 --> 00:01:17.748
as the sum from, let's say
n equals zero to infinity,

00:01:17.748 --> 00:01:19.754
we're just gonna keep
going on and on forever.

00:01:19.754 --> 00:01:22.603
And it's gonna be this
first, it's gonna be,

00:01:22.603 --> 00:01:24.337
you can kind of think the
thing we're multiplying

00:01:24.337 --> 00:01:25.768
by negative three to some power,

00:01:25.768 --> 00:01:28.268
so it's gonna be negative 0.5,

00:01:30.200 --> 00:01:31.915
actually, I'm just gonna
do that yellow color.

00:01:31.915 --> 00:01:33.248
So it's gonna be

00:01:34.610 --> 00:01:37.443
negative 0.5 times negative three,

00:01:39.604 --> 00:01:43.604
negative, I'm gonna do
that blue color, so times

00:01:44.814 --> 00:01:48.015
negative three to the nth power.

00:01:48.015 --> 00:01:50.085
Here this is when n is
zero, here's n is one,

00:01:50.085 --> 00:01:52.367
here is n is equal to two.

00:01:52.367 --> 00:01:54.531
So we've been able to rewrite
this in different ways,

00:01:54.531 --> 00:01:57.301
but let's actually see
if we can evaluate this.

00:01:57.301 --> 00:02:00.936
And so we have a common
ratio of negative three.

00:02:00.936 --> 00:02:03.277
So our r here is negative three.

00:02:03.277 --> 00:02:05.435
And the first thing that
you should think about is

00:02:05.435 --> 00:02:08.055
well, in order for this to converge,

00:02:08.055 --> 00:02:12.012
our common ratio, the
magnitude of the common ratio,

00:02:12.012 --> 00:02:14.352
or the absolute value of the common ratio

00:02:14.352 --> 00:02:17.852
needs to be less than one for convergence.

00:02:21.359 --> 00:02:24.289
And what is the absolute
value of negative three?

00:02:24.289 --> 00:02:26.990
Well, the absolute value of
negative three is equal to three

00:02:26.990 --> 00:02:29.876
which is definitely not less than one.

00:02:29.876 --> 00:02:32.354
So this thing will not converge.

00:02:32.354 --> 00:02:34.771
This thing will not converge.

00:02:36.252 --> 00:02:38.304
And even if you look at
this, it makes sense,

00:02:38.304 --> 00:02:40.548
'cause the magnitudes
of each of these terms

00:02:40.548 --> 00:02:42.314
are getting larger and larger and larger.

00:02:42.314 --> 00:02:44.305
We're flipping between
adding and subtracting.

00:02:44.305 --> 00:02:46.521
But we're adding and
subtracting larger and larger

00:02:46.521 --> 00:02:48.636
and larger and larger and larger values.

00:02:48.636 --> 00:02:51.821
Intuitively, when things
converge you're kind of,

00:02:51.821 --> 00:02:55.960
each successive term tends
to get diminishingly small

00:02:55.960 --> 00:02:57.481
or maybe it cancels out in some type

00:02:57.481 --> 00:02:58.987
of an interesting way.

00:02:58.987 --> 00:03:03.154
But because the absolute
value of the common ratio is

00:03:04.738 --> 00:03:07.815
greater than or equal to
one in this situation,

00:03:07.815 --> 00:03:11.232
this is not going to converge to a value.


Worked example: convergent geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.355 --> 00:00:01.869
- [Voiceover] Let's get
some practice taking sums

00:00:01.869 --> 00:00:04.755
of infinite geometric series.

00:00:04.755 --> 00:00:06.798
So we have one over here.

00:00:06.798 --> 00:00:08.224
And just to make sure that we're dealing

00:00:08.224 --> 00:00:09.673
with a geometric series,

00:00:09.673 --> 00:00:12.082
let's make sure we have a common ratio.

00:00:12.082 --> 00:00:15.234
So let's see, to go from the
first term to the second term

00:00:15.234 --> 00:00:18.986
we multiply by 1/3,
then go to the next term

00:00:18.986 --> 00:00:21.879
we are going to multiply by 1/3 again,

00:00:21.879 --> 00:00:23.439
and we're going to keep doing that.

00:00:23.439 --> 00:00:26.590
So we could rewrite the series as

00:00:26.590 --> 00:00:29.757
eight plus eight times 1/3,

00:00:31.146 --> 00:00:33.813
eight times 1/3, plus

00:00:36.525 --> 00:00:39.531
eight times 1/3 squared,

00:00:39.531 --> 00:00:42.712
eight times 1/3 squared.

00:00:42.712 --> 00:00:46.903
Each successive term we
multiply by 1/3 again.

00:00:46.903 --> 00:00:48.312
And so when you look at
it this way, you're like,

00:00:48.312 --> 00:00:51.351
okay, we could write
this in sigma notation.

00:00:51.351 --> 00:00:55.193
This is going be equal to, so
this, the first thing we wrote

00:00:55.193 --> 00:00:58.259
is equal to this, which is equal to,

00:00:58.259 --> 00:01:00.342
this is equal to the sum.

00:01:01.413 --> 00:01:03.970
And we could start at zero or at one,

00:01:03.970 --> 00:01:05.648
depending on how we like to do it.

00:01:05.648 --> 00:01:09.484
We could say from k is equal to zero.

00:01:09.484 --> 00:01:11.430
And this is an infinite series right here,

00:01:11.430 --> 00:01:13.674
we're just gonna keep on going forever,

00:01:13.674 --> 00:01:18.091
so to infinity of, well
what's our first term?

00:01:18.091 --> 00:01:20.562
Our first term is eight.

00:01:20.562 --> 00:01:23.982
So it's going to be eight
times our common ratio,

00:01:23.982 --> 00:01:28.149
times our common ratio 1/3 to the k power.

00:01:29.935 --> 00:01:31.937
Now, let me just verify
that this indeed works.

00:01:31.937 --> 00:01:33.868
And I always do this
just as a reality check,

00:01:33.868 --> 00:01:35.501
and I encourage you to do the same.

00:01:35.501 --> 00:01:38.463
So, when k equals zero, that
should be the first term

00:01:38.463 --> 00:01:39.296
right over here.

00:01:39.296 --> 00:01:41.526
You get eight times 1/3 to the zero power,

00:01:41.526 --> 00:01:42.833
which is indeed eight.

00:01:42.833 --> 00:01:44.063
When k is equal to one,

00:01:44.063 --> 00:01:46.007
that's gonna be our second term here.

00:01:46.007 --> 00:01:48.586
That's gonna be eight times
1/3 to the first power.

00:01:48.586 --> 00:01:49.756
That's what we have here.

00:01:49.756 --> 00:01:52.298
And so when k is equal to two,

00:01:52.298 --> 00:01:54.402
that is this term right over here.

00:01:54.402 --> 00:01:57.750
So these are all
describing the same thing.

00:01:57.750 --> 00:01:59.281
So now that we've seen that we can write

00:01:59.281 --> 00:02:02.533
a geometric series in multiple
ways, let's find the sum.

00:02:02.533 --> 00:02:07.166
Well, we've seen before and
we prove it in other videos,

00:02:07.166 --> 00:02:11.083
if you have a sum from
k equal zero to infinity

00:02:12.580 --> 00:02:15.080
and you have your first term a

00:02:16.339 --> 00:02:18.256
times r to the k power,

00:02:19.441 --> 00:02:22.941
r to the k power, assuming this converges,

00:02:23.827 --> 00:02:26.358
so, assuming that the absolute value

00:02:26.358 --> 00:02:29.601
of your common ratio is less than one,

00:02:29.601 --> 00:02:32.148
this is what needs to
be true for convergence,

00:02:32.148 --> 00:02:35.747
this is going to be equal to,
this is going to be equal to

00:02:35.747 --> 00:02:38.247
our first term which is a over

00:02:40.549 --> 00:02:42.799
one minus our common ratio,

00:02:44.488 --> 00:02:46.498
one minus our common ratio.

00:02:46.498 --> 00:02:47.786
And if this looks unfamiliar to you,

00:02:47.786 --> 00:02:48.881
I encourage you to watch the video

00:02:48.881 --> 00:02:52.115
where we find the formula,
we derive the formula

00:02:52.115 --> 00:02:54.580
for the sum of an
infinite geometric series.

00:02:54.580 --> 00:02:56.702
But just applying that over here,

00:02:56.702 --> 00:02:58.990
we are going to get, we are going to get,

00:02:58.990 --> 00:03:01.398
this is going to be
equal to our first term

00:03:01.398 --> 00:03:02.648
which is eight,

00:03:03.928 --> 00:03:05.261
so that is eight

00:03:06.539 --> 00:03:07.789
over one minus,

00:03:10.027 --> 00:03:14.027
one minus our common ratio, over 1/3.

00:03:15.909 --> 00:03:17.233
And we know this is going to converge,

00:03:17.233 --> 00:03:19.075
because our common ratio, the magnitude,

00:03:19.075 --> 00:03:22.783
the absolute value of 1/3
is indeed less than one.

00:03:22.783 --> 00:03:26.268
And so this is all going to converge to,

00:03:26.268 --> 00:03:29.518
this is going to converge to eight over

00:03:30.690 --> 00:03:35.034
one minus 1/3 is 2/3,

00:03:35.034 --> 00:03:39.217
which is the same thing
as eight times 3/2,

00:03:39.217 --> 00:03:41.598
which is, let's see this could become,

00:03:41.598 --> 00:03:44.228
divide eight by two, that becomes four,

00:03:44.228 --> 00:03:46.561
and so this will become 12.


Writing geometric series in sigma notation

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.992
- [Voiceover] So we have
sum here of two plus six

00:00:01.992 --> 00:00:03.325
plus 18 plus 54.

00:00:04.335 --> 00:00:07.061
And we can obviously just
evaluate it, add up these numbers.

00:00:07.061 --> 00:00:10.099
But what I want to do is
I wanna use it as practice

00:00:10.099 --> 00:00:14.255
for rewriting a series like
this using sigma notation.

00:00:14.255 --> 00:00:16.262
So let's just think about
what's happening here.

00:00:16.262 --> 00:00:17.623
Let's see if we can see any pattern

00:00:17.623 --> 00:00:19.391
from one term to the next.

00:00:19.391 --> 00:00:21.812
Let's see, to go from two to six,

00:00:21.812 --> 00:00:23.597
we could say we are adding four,

00:00:23.597 --> 00:00:25.261
but then when we go from six to 18,

00:00:25.261 --> 00:00:29.203
we're not adding four
now, we are now adding 12,

00:00:29.203 --> 00:00:31.453
so it's not an arithmetic series.

00:00:31.453 --> 00:00:33.669
Let's see, maybe it's a geometric.

00:00:33.669 --> 00:00:36.463
So to go from two to
six, what are we doing?

00:00:36.463 --> 00:00:39.213
Well, we're multiplying by three.

00:00:40.481 --> 00:00:43.661
So, let me write that,
we're multiplying by three.

00:00:43.661 --> 00:00:45.291
To go to six to 18, what are we doing?

00:00:45.291 --> 00:00:46.669
Well, we're multiplying by three.

00:00:46.669 --> 00:00:49.629
To go to 18 to 54, we're
multiplying by three.

00:00:49.629 --> 00:00:52.912
So it looks like this is
indeed a geometric series,

00:00:52.912 --> 00:00:55.022
and we have a common ratio of three.

00:00:55.022 --> 00:00:57.807
So let's rewrite this
using sigma notation.

00:00:57.807 --> 00:01:01.533
So this is going to be the
sum, and we could start,

00:01:01.533 --> 00:01:03.992
well, there's a bunch of
ways that we could write it.

00:01:03.992 --> 00:01:08.159
We could write it as, let's
start with k equaling zero.

00:01:09.015 --> 00:01:12.808
And so we have our
first term which is two,

00:01:12.808 --> 00:01:17.071
so it's two times our common
ratio to the kth power.

00:01:17.071 --> 00:01:21.073
So times our common ratio
three to the k power.

00:01:21.073 --> 00:01:23.224
So before I even write how
many terms we have here

00:01:23.224 --> 00:01:26.529
or how high we go with our k,
let's see if this makes sense.

00:01:26.529 --> 00:01:29.973
When k is equal to zero, this
is gonna be two times three

00:01:29.973 --> 00:01:31.686
to the zeroth power.

00:01:31.686 --> 00:01:34.458
So that's two times one,

00:01:34.458 --> 00:01:36.409
so that's this first term right there.

00:01:36.409 --> 00:01:38.392
When k is equal to one,
that'll be two times three

00:01:38.392 --> 00:01:39.402
to the first power.

00:01:39.402 --> 00:01:40.638
Well, that's gonna be six.

00:01:40.638 --> 00:01:44.106
And then when k is, so
this is k equals zero.

00:01:44.106 --> 00:01:45.585
Let me do this in a different color.

00:01:45.585 --> 00:01:49.164
So this is k equals zero,
I say different color,

00:01:49.164 --> 00:01:51.014
and then I do the same color.

00:01:51.014 --> 00:01:55.181
All right, so this is k equals
zero, this is k equals one,

00:01:56.510 --> 00:02:00.571
this is k equals two, and then
this would be k equals three,

00:02:00.571 --> 00:02:03.554
which would be two times
three to the third power.

00:02:03.554 --> 00:02:06.730
So two times 27 is indeed equal to 54.

00:02:06.730 --> 00:02:10.688
So we're gonna go up
to k is equal to three.

00:02:10.688 --> 00:02:13.349
So that's one way that
we could write this.

00:02:13.349 --> 00:02:15.270
There's other ways that
you could write this.

00:02:15.270 --> 00:02:17.839
You could write it as,

00:02:17.839 --> 00:02:21.130
so we're gonna still do,
we have our first term

00:02:21.130 --> 00:02:24.176
right over here, but for
example, we could write it

00:02:24.176 --> 00:02:27.965
as our common ratio, and I'll
use a different index now,

00:02:27.965 --> 00:02:30.756
let's say to the n minus one power.

00:02:30.756 --> 00:02:32.371
And instead of starting at zero,

00:02:32.371 --> 00:02:34.121
I could start at n equals one,

00:02:34.121 --> 00:02:35.234
but notice it has the same effect.

00:02:35.234 --> 00:02:36.934
When you say n equals
one, it's one minus one,

00:02:36.934 --> 00:02:38.891
you get the zeroth power.

00:02:38.891 --> 00:02:43.313
And so we're increasing
all of the indexes by one,

00:02:43.313 --> 00:02:44.608
so instead of going from zero to three,

00:02:44.608 --> 00:02:46.769
we're going from one to four.

00:02:46.769 --> 00:02:49.079
And you could verify that this
is still going to work out,

00:02:49.079 --> 00:02:52.224
'cause when n is equal to four,

00:02:52.224 --> 00:02:54.476
it's three to the four minus one power,

00:02:54.476 --> 00:02:55.972
so it's still three to the third power,

00:02:55.972 --> 00:02:58.946
which is 27 times two which still 54.

00:02:58.946 --> 00:03:02.779
So this is n equals one,
that is n equals two,

00:03:04.449 --> 00:03:08.961
that is n equals three,
and that is n equals four.

00:03:08.961 --> 00:03:10.609
But either way, these are
ways that you could write it

00:03:10.609 --> 00:03:12.359
using sigma notation.


Writing arithmetic series in sigma notation

WEBVTT
Kind: captions
Language: en

00:00:00.148 --> 00:00:01.257
- [Voiceover] What I
want to do in this video

00:00:01.257 --> 00:00:04.638
is get some practice writing
series in sigma notation

00:00:04.638 --> 00:00:07.496
and I have a series in
front of us right over here.

00:00:07.496 --> 00:00:09.304
We have seven plus nine plus 11

00:00:09.304 --> 00:00:13.148
and we keep on adding
all the way up to 405.

00:00:13.148 --> 00:00:15.605
So first, let's just think
about what's going on here.

00:00:15.605 --> 00:00:18.940
How can we think about what
happens at each successive term?

00:00:18.940 --> 00:00:21.772
So we're at seven and
then we're going to nine

00:00:21.772 --> 00:00:23.160
and then we're going to 11.

00:00:23.160 --> 00:00:25.467
It looks like we're adding two every time

00:00:25.467 --> 00:00:28.714
so it looks like this
is an arithmetic series.

00:00:28.714 --> 00:00:31.964
So we add two and then we add two again

00:00:33.819 --> 00:00:36.109
and we're going to keep adding two

00:00:36.109 --> 00:00:38.776
all the way until we get to 405.

00:00:40.192 --> 00:00:41.947
So let's think about how many times

00:00:41.947 --> 00:00:44.498
we are going to add two to get to 200,

00:00:44.498 --> 00:00:47.975
sorry, how many times we have
to add two to get to 405.

00:00:47.975 --> 00:00:50.975
So 405 is seven plus two times what?

00:00:52.122 --> 00:00:53.700
So let me write this down.

00:00:53.700 --> 00:00:56.283
So if we wanted 405 is equal to

00:00:57.210 --> 00:00:58.960
seven plus two times,

00:01:00.324 --> 00:01:02.495
I'll just write two times x.

00:01:02.495 --> 00:01:04.329
I'm just trying to
figure out how many times

00:01:04.329 --> 00:01:07.996
do I have to add two
to seven to get to 405?

00:01:09.079 --> 00:01:11.757
And so that is going to
be equal to, let's see,

00:01:11.757 --> 00:01:13.393
so we subtract seven from both sides.

00:01:13.393 --> 00:01:15.810
We have 398 is equal to two x

00:01:17.934 --> 00:01:21.155
or let's see, divide both sides by two

00:01:21.155 --> 00:01:24.591
and we get this is going to be what?

00:01:24.591 --> 00:01:25.424
199?

00:01:27.287 --> 00:01:28.787
199 is equal to x.

00:01:30.844 --> 00:01:34.344
So we're essentially adding two 199 times.

00:01:35.736 --> 00:01:37.020
So this is the first
time we're adding two.

00:01:37.020 --> 00:01:38.111
This is two times.

00:01:38.111 --> 00:01:40.495
We're adding two times
one, adding two times two

00:01:40.495 --> 00:01:43.553
and here, we're adding two times 199

00:01:43.553 --> 00:01:45.919
to our original seven.

00:01:45.919 --> 00:01:48.899
So let's think about this a little bit.

00:01:48.899 --> 00:01:51.793
So this is going be a sum,

00:01:51.793 --> 00:01:52.710
a sum from,

00:01:53.832 --> 00:01:56.079
so there's a couple of ways
we could think about it.

00:01:56.079 --> 00:01:58.939
We could think about how
many times we've added two

00:01:58.939 --> 00:02:02.816
so we could start with
us adding two zero times,

00:02:02.816 --> 00:02:05.506
the number seven is when we
haven't added two at all,

00:02:05.506 --> 00:02:08.144
all the way to when we add two

00:02:08.144 --> 00:02:08.977
199 times.

00:02:11.827 --> 00:02:13.461
And let's think about this a little bit.

00:02:13.461 --> 00:02:14.685
This is going to be,

00:02:14.685 --> 00:02:18.352
we could write it as
seven plus two times k.

00:02:19.524 --> 00:02:20.988
Seven times two,

00:02:20.988 --> 00:02:22.648
seven plus two times k.

00:02:22.648 --> 00:02:26.208
When k equals zero, this
is just going to be seven.

00:02:26.208 --> 00:02:29.002
When k equals one, it's
seven times two plus one.

00:02:29.002 --> 00:02:30.191
Well, it's going to be nine.

00:02:30.191 --> 00:02:31.829
When k is equal to two,

00:02:31.829 --> 00:02:36.113
it's going to be seven plus
two times two which is 11.

00:02:36.113 --> 00:02:39.254
And all the way when k is equal to 199,

00:02:39.254 --> 00:02:43.422
it's going to be seven plus
two times 199 which is 398

00:02:43.422 --> 00:02:45.005
which would be 405.

00:02:45.872 --> 00:02:48.730
So that's one way that we could write it.

00:02:48.730 --> 00:02:51.504
Another way, we could also write it as,

00:02:51.504 --> 00:02:53.219
let me do this in a different color,

00:02:53.219 --> 00:02:55.297
we could, if we want to start our index at

00:02:55.297 --> 00:02:56.714
k is equal to one

00:02:58.676 --> 00:03:01.261
then let's see, it's
going to be the first term

00:03:01.261 --> 00:03:03.344
is going to be seven plus

00:03:05.214 --> 00:03:07.047
two times k minus one,

00:03:09.661 --> 00:03:11.539
times k minus one.

00:03:11.539 --> 00:03:13.060
Notice, the first term works out

00:03:13.060 --> 00:03:14.674
because we're not adding two at all

00:03:14.674 --> 00:03:17.424
so one minus one is equal to zero

00:03:18.763 --> 00:03:20.642
so you're just going to get seven.

00:03:20.642 --> 00:03:23.131
Then when k is equal to
two, the second term,

00:03:23.131 --> 00:03:25.334
we're going to add two one time

00:03:25.334 --> 00:03:28.158
because two minus one is two

00:03:28.158 --> 00:03:29.836
so that gives us that one.

00:03:29.836 --> 00:03:33.653
And so how many total terms
are we going to have here?

00:03:33.653 --> 00:03:34.744
Well, one way to think about is

00:03:34.744 --> 00:03:37.165
I just shifted the indices up by one

00:03:37.165 --> 00:03:40.729
so we're going to go
from k equals one to 200.

00:03:40.729 --> 00:03:41.793
And you can verify this.

00:03:41.793 --> 00:03:43.680
When k is equal to 200,

00:03:43.680 --> 00:03:47.597
this is going to be 200
minus one which is 199.

00:03:48.838 --> 00:03:50.505
Two times 199 is 398

00:03:52.752 --> 00:03:54.835
plus seven is indeed 405.

00:03:55.675 --> 00:03:59.430
So when k equals 200, that
is our last term here.

00:03:59.430 --> 00:04:01.576
So either way, these are legitimate ways

00:04:01.576 --> 00:04:04.263
of expressing this arithmetic series

00:04:04.263 --> 00:04:06.263
in using sigma notation.


Evidence for evolution

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.670
- [Voiceover] We've done
many videos on Khan Academy

00:00:02.670 --> 00:00:05.209
on evolution and natural
selection explaining them,

00:00:05.209 --> 00:00:07.688
but I thought I would do
a video going a little bit

00:00:07.688 --> 00:00:09.893
more in depth in evidence for evolution

00:00:09.893 --> 00:00:11.383
and natural selection.

00:00:11.383 --> 00:00:14.147
And I started with this
quote "Nothing in biology

00:00:14.147 --> 00:00:16.745
makes sense except in
the light of evolution."

00:00:16.745 --> 00:00:19.496
This is by Theodus
Dobzhansky, who's a famous

00:00:19.496 --> 00:00:22.012
biologist, he's passed
away now, and what he's

00:00:22.012 --> 00:00:23.912
saying is absolutely
true and this is why it's

00:00:23.912 --> 00:00:26.454
so important to appreciate
the evidence for evolution

00:00:26.454 --> 00:00:28.938
and natural selection
and to understand them,

00:00:28.938 --> 00:00:32.841
because before the theory
of evolution, biology was

00:00:32.841 --> 00:00:35.678
just about observation
and classification without

00:00:35.678 --> 00:00:38.455
having a cohesive narrative
for how all of this

00:00:38.455 --> 00:00:39.667
came about.

00:00:39.667 --> 00:00:43.093
And since Darwin had
come up with this theory

00:00:43.093 --> 00:00:45.989
in the mid-19th century,
we've had far more tools

00:00:45.989 --> 00:00:49.502
to back it up beyond just
the observations we had

00:00:49.502 --> 00:00:50.682
up until that point.

00:00:50.682 --> 00:00:53.430
We have our tools around
dating and the fossil

00:00:53.430 --> 00:00:55.328
record, which gives us much more evidence.

00:00:55.328 --> 00:00:57.847
We have our tools of
microbiology and genetics,

00:00:57.847 --> 00:01:00.445
which gives us even
stronger evidence, so a lot

00:01:00.445 --> 00:01:02.065
of times people say "Oh,
it's a theory, evolution

00:01:02.065 --> 00:01:03.222
"is it just a theory?"

00:01:03.222 --> 00:01:06.196
Well, it's about as strong as theories get

00:01:06.196 --> 00:01:09.661
and without it, as Theodus
Dobzhansky said, biology

00:01:09.661 --> 00:01:12.735
as we know it and all of
the progress we've made

00:01:12.735 --> 00:01:14.551
in biology frankly wouldn't make any sense

00:01:14.551 --> 00:01:16.908
and probably would not have happened.

00:01:16.908 --> 00:01:19.719
Now I'm going to broadly
go into three types

00:01:19.719 --> 00:01:22.696
of evidence in this video for evolution

00:01:22.696 --> 00:01:24.056
and natural selection.

00:01:24.056 --> 00:01:26.068
The first is structural.

00:01:26.068 --> 00:01:28.830
And these are the types
of things that folks like

00:01:28.830 --> 00:01:31.457
Darwin would have observed,
that people have been

00:01:31.457 --> 00:01:33.565
observing in biology for a long time

00:01:33.565 --> 00:01:36.006
but evolution and natural
selection starts to make

00:01:36.006 --> 00:01:38.202
a lot more sense of it,
and here we're talking

00:01:38.202 --> 00:01:40.134
about the macro structure,
things that we can

00:01:40.134 --> 00:01:41.989
for the most part observe
with our eyes or with

00:01:41.989 --> 00:01:44.018
a very simple microscope.

00:01:44.018 --> 00:01:47.753
The next level is what
we've learned, really over

00:01:47.753 --> 00:01:51.673
the last 100 or so years,
at the micro level,

00:01:51.673 --> 00:01:53.560
in microbiology.

00:01:53.560 --> 00:01:56.893
Microbiology and especially in genetics.

00:01:58.512 --> 00:02:02.509
So this has really firmed
up the theory of evolution.

00:02:02.509 --> 00:02:04.927
And then the last dimension we'll look at

00:02:04.927 --> 00:02:08.975
is direct observation, direct observation,

00:02:08.975 --> 00:02:11.378
and this is really where
it goes beyond a theory.

00:02:11.378 --> 00:02:12.967
We are seeing it happen.

00:02:12.967 --> 00:02:15.114
A lot of times people
say "Oh, it's a theory,

00:02:15.114 --> 00:02:18.220
"you know, the theory
says it happened over tens

00:02:18.220 --> 00:02:20.817
"of millions of years but
no one was around to really

00:02:20.817 --> 00:02:23.619
"observe, even if we see
a lot of evidence, no one

00:02:23.619 --> 00:02:25.622
"knows if it for sure happened."

00:02:25.622 --> 00:02:28.138
But if you're directly
observing things, well you

00:02:28.138 --> 00:02:30.216
know it's for sure
happening and as we'll see,

00:02:30.216 --> 00:02:32.762
evolution does not only
occur over time scales

00:02:32.762 --> 00:02:35.025
of millions or tens of millions of years,

00:02:35.025 --> 00:02:37.516
it actually can occur
and we see it occurring

00:02:37.516 --> 00:02:41.217
all the time on scales well within a human

00:02:41.217 --> 00:02:43.391
observational capacity,
within just a matter

00:02:43.391 --> 00:02:45.677
of months or years.

00:02:45.677 --> 00:02:47.049
So let's go through each of these.

00:02:47.049 --> 00:02:48.547
So first let's talk about
structural and this is

00:02:48.547 --> 00:02:49.811
a very high-level overview.

00:02:49.811 --> 00:02:51.441
I encourage you to do more research on it.

00:02:51.441 --> 00:02:54.325
You will find loads and
loads and loads of any

00:02:54.325 --> 00:02:56.077
type of this evidence.

00:02:56.077 --> 00:02:57.999
So the first thing I want to talk about is

00:02:57.999 --> 00:03:02.278
homologous structures,
homologous structures,

00:03:02.278 --> 00:03:06.028
that you see throughout
the biological world.

00:03:07.022 --> 00:03:10.105
Hom-ol-o-gous, homologous structures.

00:03:11.198 --> 00:03:14.208
And the word homologous
means things that have

00:03:14.208 --> 00:03:17.326
similar structure,
similar position, similar

00:03:17.326 --> 00:03:21.620
ancestry but not necessarily
the exact same function.

00:03:21.620 --> 00:03:24.957
And here you see examples
of a, well as a human,

00:03:24.957 --> 00:03:26.498
we would consider a forearm.

00:03:26.498 --> 00:03:28.460
You see the human forearm
and wrist and then you

00:03:28.460 --> 00:03:30.941
see the homologous
structures in dogs and birds

00:03:30.941 --> 00:03:32.056
and whales.

00:03:32.056 --> 00:03:34.846
And even though this part
of those animals have

00:03:34.846 --> 00:03:37.498
very different functions,
a human does not walk

00:03:37.498 --> 00:03:40.760
on its hands for the most
part; a dog does walk

00:03:40.760 --> 00:03:44.009
on its front legs; a bird
isn't walking at all,

00:03:44.009 --> 00:03:47.642
it's using them to flap
its wings; and a whale,

00:03:47.642 --> 00:03:51.258
this is making up its actual fins,

00:03:51.258 --> 00:03:54.972
it's using them to propel or to control

00:03:54.972 --> 00:03:57.064
their movement inside of the water.

00:03:57.064 --> 00:03:58.830
And even though they have these very, very

00:03:58.830 --> 00:04:00.630
different functions and
at first when you look

00:04:00.630 --> 00:04:02.878
at a human and a bird and
a whale on the outside,

00:04:02.878 --> 00:04:05.461
they might look reasonably
different, when you look

00:04:05.461 --> 00:04:08.470
at these bone structures,
they are eerily similar,

00:04:08.470 --> 00:04:12.285
especially color-coded the way it is.

00:04:12.285 --> 00:04:15.495
So these are, this is
a very strong hint that

00:04:15.495 --> 00:04:18.242
maybe humans, dogs,
birds, and whales share

00:04:18.242 --> 00:04:22.132
a common ancestor, more
recently in the past

00:04:22.132 --> 00:04:26.124
than say other animals or
organisms, I should say,

00:04:26.124 --> 00:04:30.103
that don't have, whose
structures aren't as

00:04:30.103 --> 00:04:33.313
homologous as these are right over here.

00:04:33.313 --> 00:04:36.330
And if you were independently
trying to create

00:04:36.330 --> 00:04:39.685
structures for what each
of these different species

00:04:39.685 --> 00:04:42.958
are doing, it's not
obvious that you would have

00:04:42.958 --> 00:04:46.829
such homologous structures
actually be involved.

00:04:46.829 --> 00:04:49.388
Now these are all
species that exist today,

00:04:49.388 --> 00:04:51.226
these are all species
that exist on the planet

00:04:51.226 --> 00:04:53.691
at the same time, but
we also see structural

00:04:53.691 --> 00:04:57.108
evidence by going into the fossil record.

00:04:58.053 --> 00:05:00.566
In the last few hundred
years, or early in the last

00:05:00.566 --> 00:05:02.763
hundred years is where
we've gotten really good

00:05:02.763 --> 00:05:06.377
at it, we've gotten good
at looking at different

00:05:06.377 --> 00:05:09.801
layers of rock strata and
being able to date them

00:05:09.801 --> 00:05:11.716
and say "Okay, that layer
was laid down x-million

00:05:11.716 --> 00:05:13.988
"years ago, that layer
was laid down a little bit

00:05:13.988 --> 00:05:16.341
"more recent, this one
was even more recent,"

00:05:16.341 --> 00:05:18.791
and then looking at
fossils within that to say

00:05:18.791 --> 00:05:20.913
"Okay, 20 million years
ago there were species

00:05:20.913 --> 00:05:23.071
"around that looked something like that,

00:05:23.071 --> 00:05:25.132
"and then 10 million years
ago there were species

00:05:25.132 --> 00:05:26.268
"that looked like that."

00:05:26.268 --> 00:05:30.497
And one example is if you
look at a horse-like animal.

00:05:30.497 --> 00:05:32.368
So this is right over
here, we're talking about

00:05:32.368 --> 00:05:34.932
horses, zebras, donkeys,
mules, things like that,

00:05:34.932 --> 00:05:37.809
the modern ones, well this
is their bone structure

00:05:37.809 --> 00:05:40.368
but if you look at the
fossil record from 12 to 5

00:05:40.368 --> 00:05:45.128
million years ago, you see
fossils that look like this

00:05:45.128 --> 00:05:48.024
and they're very close
so you see, it's very

00:05:48.024 --> 00:05:52.034
believable that you could
have evolution from this

00:05:52.034 --> 00:05:55.254
to that, but then you
go further back and once

00:05:55.254 --> 00:05:59.557
again, it draws, it seems
like a very gradual process.

00:05:59.557 --> 00:06:02.218
And once again, this
happening over, these are from

00:06:02.218 --> 00:06:04.905
12 to 5 million years ago,
these are from 16 to 12

00:06:04.905 --> 00:06:06.913
million years ago, these are from over

00:06:06.913 --> 00:06:08.551
34 million years ago.

00:06:08.551 --> 00:06:12.612
And so you can see how this
is happening at a very,

00:06:12.612 --> 00:06:15.414
very gradual pace and
the mechanism, and we go

00:06:15.414 --> 00:06:17.756
into some depth in other
videos in Khan Academy,

00:06:17.756 --> 00:06:19.720
you have variation in species, you have

00:06:19.720 --> 00:06:21.320
the environment selecting for it.

00:06:21.320 --> 00:06:23.556
The environment might
change or different things

00:06:23.556 --> 00:06:25.812
happen so you have different
forms of selection,

00:06:25.812 --> 00:06:28.109
different types of combinations sprout up,

00:06:28.109 --> 00:06:30.066
they're more suitable for the environment,

00:06:30.066 --> 00:06:32.127
they start to reproduce
better, they become

00:06:32.127 --> 00:06:34.626
the dominant species or
they take over certain

00:06:34.626 --> 00:06:37.127
parts of a niche or an
ecosystem, and so you have

00:06:37.127 --> 00:06:41.089
this change, this heritable
change of traits over time.

00:06:41.089 --> 00:06:43.267
And so when you look at
the fossil record, it makes

00:06:43.267 --> 00:06:47.186
a lot of sense that, okay,
this is strong evidence

00:06:47.186 --> 00:06:49.720
for evolution, that the
animals that we see today

00:06:49.720 --> 00:06:52.982
weren't just put on, just
created all of a sudden

00:06:52.982 --> 00:06:55.808
and haven't changed
since then, but there's

00:06:55.808 --> 00:06:57.988
a constant change and
we can see it directly

00:06:57.988 --> 00:07:00.770
through the fossil record.

00:07:00.770 --> 00:07:04.042
Now the next point of evidence, I will put

00:07:04.042 --> 00:07:06.379
a bit of a caveat because
the gentleman who first

00:07:06.379 --> 00:07:09.018
created this, his name was Haeckel, he was

00:07:09.018 --> 00:07:11.061
a controversial figure,
he had some spurious

00:07:11.061 --> 00:07:15.111
theories, and even this
diagram that he created,

00:07:15.111 --> 00:07:17.748
it seems like he fudged a
little bit of the drawings

00:07:17.748 --> 00:07:20.917
in order to make a stronger argument,

00:07:20.917 --> 00:07:23.807
but even with modern
observations, these drawings

00:07:23.807 --> 00:07:25.728
are pretty close to being correct.

00:07:25.728 --> 00:07:27.086
And it's very, very compelling.

00:07:27.086 --> 00:07:29.623
It shows the embryonic
development of a whole

00:07:29.623 --> 00:07:33.677
series of species, from a fish on the left

00:07:33.677 --> 00:07:36.427
to a reptile to birds to mammals

00:07:37.805 --> 00:07:40.312
and another mammal, to non-human mammals

00:07:40.312 --> 00:07:41.926
and, of course, to humans.

00:07:41.926 --> 00:07:43.789
And you can see at the early stages,

00:07:43.789 --> 00:07:46.224
they look eerily similar.

00:07:46.224 --> 00:07:48.974
In fact, you see proto gill slits

00:07:52.087 --> 00:07:54.837
in all of these animals, which later

00:07:54.837 --> 00:07:57.143
differentiate into things
that are more suitable

00:07:57.143 --> 00:07:59.773
for what that animal actually becomes.

00:07:59.773 --> 00:08:02.660
And Haeckel, he's the
guy who claimed ontogeny

00:08:02.660 --> 00:08:04.822
recapitulates phylogeny,
which is a very fancy

00:08:04.822 --> 00:08:07.164
way of saying that your
embryonic development

00:08:07.164 --> 00:08:10.148
is telling the story of
the evolutionary path,

00:08:10.148 --> 00:08:12.587
which isn't true, but
you'll even hear people

00:08:12.587 --> 00:08:13.835
quote that today.

00:08:13.835 --> 00:08:15.841
But his drawings and his observations,

00:08:15.841 --> 00:08:19.400
this is compelling
evidence for life sharing

00:08:19.400 --> 00:08:23.245
a common ancestry, coming
from similar origins

00:08:23.245 --> 00:08:26.980
that got more and more
different over time through

00:08:26.980 --> 00:08:29.984
the process of natural selection.

00:08:29.984 --> 00:08:31.932
So everything I've talked
about so far has been

00:08:31.932 --> 00:08:34.123
kind of macro structure,
things we can observe.

00:08:34.123 --> 00:08:36.328
The next thing I'm gonna
talk about is, you can

00:08:36.328 --> 00:08:38.423
think about as micro
structures or processes,

00:08:38.423 --> 00:08:40.554
and this is microbiology.

00:08:40.554 --> 00:08:41.804
Microbiology...

00:08:44.608 --> 00:08:46.025
Biolo, biology...

00:08:48.060 --> 00:08:49.320
Microbiology.

00:08:49.320 --> 00:08:51.406
And the more we understand
about microbiology

00:08:51.406 --> 00:08:55.283
the more compelling case of evolution

00:08:55.283 --> 00:08:59.110
because when we look at
even one, all life forms

00:08:59.110 --> 00:09:01.693
that we know, they involve DNA.

00:09:02.936 --> 00:09:05.988
How the DNA gets replicated and translated

00:09:05.988 --> 00:09:09.523
and transcribed is very similar from one

00:09:09.523 --> 00:09:11.119
life form to another.

00:09:11.119 --> 00:09:13.202
The idea of DNA going to,

00:09:14.870 --> 00:09:17.703
DNA coding for proteins, proteins,

00:09:19.169 --> 00:09:22.588
that are made up of
amino acids is something

00:09:22.588 --> 00:09:24.745
that we see throughout biology.

00:09:24.745 --> 00:09:26.998
Amino acids, which once again hints

00:09:26.998 --> 00:09:28.392
at a common ancestry.

00:09:28.392 --> 00:09:30.700
And not only are those molecular and many

00:09:30.700 --> 00:09:33.556
of the very proteins
are very, very similar,

00:09:33.556 --> 00:09:35.569
more similar than if you
look at the macro level

00:09:35.569 --> 00:09:37.696
or even at the structural
level between different

00:09:37.696 --> 00:09:41.062
species, and not just do
they share these common

00:09:41.062 --> 00:09:43.477
microstructures and
processes, but the actual

00:09:43.477 --> 00:09:47.354
information stored in
things like DNA also are

00:09:47.354 --> 00:09:50.224
very, very strong evidence for evolution.

00:09:50.224 --> 00:09:53.030
So this is a picture, I got this from,

00:09:53.030 --> 00:09:55.344
I got this from the site,
I should give proper

00:09:55.344 --> 00:09:57.011
credit, 23andme.com.

00:09:59.480 --> 00:10:01.060
But this and you'll see
other data like this

00:10:01.060 --> 00:10:03.412
that's very similar to
this, which is how much

00:10:03.412 --> 00:10:05.310
genetic similarity do we
have between different

00:10:05.310 --> 00:10:08.231
species, and these numbers
tell us how much genetic

00:10:08.231 --> 00:10:10.301
similarity at a high level do we have with

00:10:10.301 --> 00:10:14.472
chimpanzees, mice, fruit
flies, yeast, and plants.

00:10:14.472 --> 00:10:17.457
And the fact that we have 26% of our genes

00:10:17.457 --> 00:10:19.549
in common with yeast
is mind-blowing because

00:10:19.549 --> 00:10:21.446
at a macro level it
doesn't seem like there's

00:10:21.446 --> 00:10:23.299
a lot in common with
yeast, but when you get

00:10:23.299 --> 00:10:25.559
at a microbiological
level, there's a good bit

00:10:25.559 --> 00:10:27.188
that's in common with yeast.

00:10:27.188 --> 00:10:30.714
And chimpanzees, we do
relate to them, their facial

00:10:30.714 --> 00:10:33.767
expressions often feel
eerily human, their behaviors

00:10:33.767 --> 00:10:37.106
often feel eerily human,
but their genes, so just

00:10:37.106 --> 00:10:40.171
how close to human
beings they actually are.

00:10:40.171 --> 00:10:43.321
And this actually shows
that even mice are way

00:10:43.321 --> 00:10:46.183
closer, if you look at
the entire tree of life

00:10:46.183 --> 00:10:48.961
based on genetic
evidence, things like mice

00:10:48.961 --> 00:10:51.931
and even fruit flies are
awfully close to human

00:10:51.931 --> 00:10:54.526
beings, especially if
you were to compare it

00:10:54.526 --> 00:10:57.217
to bacteria or a plant.

00:10:57.217 --> 00:10:59.142
But once again, you
share all of these common

00:10:59.142 --> 00:11:01.777
processes and the fact
that we can now measure

00:11:01.777 --> 00:11:04.175
how far things are away
allows us to create

00:11:04.175 --> 00:11:08.110
a very accurate tree
of life, and especially

00:11:08.110 --> 00:11:10.650
thinking about how far in the past we had

00:11:10.650 --> 00:11:13.150
evolutionary common ancestors.

00:11:14.018 --> 00:11:16.363
Now the last thing that
I promised I would talk

00:11:16.363 --> 00:11:19.298
about is direct evidence, direct evidence

00:11:19.298 --> 00:11:22.912
of evolution, and I talk
about this in the first

00:11:22.912 --> 00:11:27.239
evolution video, but the
direct evidence we see

00:11:27.239 --> 00:11:30.364
all the time with things
like bacteria where

00:11:30.364 --> 00:11:33.696
you have bacteria, let's
say growing around,

00:11:33.696 --> 00:11:37.268
and we have antibiotics
that we use in our body

00:11:37.268 --> 00:11:40.909
to kill bacteria, but the reason why many

00:11:40.909 --> 00:11:42.925
physicians and scientists
will tell you "Don't

00:11:42.925 --> 00:11:45.612
"overuse antibiotics" is
because the more you use

00:11:45.612 --> 00:11:48.352
it, it causes a very
strong natural selection

00:11:48.352 --> 00:11:52.007
process for bacteria that
are going to be resistant

00:11:52.007 --> 00:11:53.299
to that antibiotic.

00:11:53.299 --> 00:11:56.167
So if you keep using an
antibiotic and the bacterias

00:11:56.167 --> 00:11:58.613
keep changing, there's
more and more variation,

00:11:58.613 --> 00:12:01.155
well you're gonna kill
a lot of the bacteria

00:12:01.155 --> 00:12:04.157
but if even one of them
is resistant to that

00:12:04.157 --> 00:12:06.895
antibiotic that you use,
well then all of its

00:12:06.895 --> 00:12:09.324
competition is gonna
get killed and so that

00:12:09.324 --> 00:12:12.763
drug-resistant superbug,
it's often called, is going

00:12:12.763 --> 00:12:15.309
to be able to go nuts
and that antibiotic isn't

00:12:15.309 --> 00:12:17.021
going to be able to do anything.

00:12:17.021 --> 00:12:18.976
And if you look at
science today or you look

00:12:18.976 --> 00:12:21.283
at medicine today, this
is kind of an arms race.

00:12:21.283 --> 00:12:24.700
You have this constant need to create new

00:12:25.644 --> 00:12:28.555
antibiotics because more
and more bacteria are

00:12:28.555 --> 00:12:30.811
becoming drug-resistant,
they're becoming what's

00:12:30.811 --> 00:12:33.250
often called superbugs,
where they are resistant

00:12:33.250 --> 00:12:35.328
to the existing antibiotics.

00:12:35.328 --> 00:12:37.935
And this is evolution, natural selection,

00:12:37.935 --> 00:12:40.357
happening on a human scale.

00:12:40.357 --> 00:12:43.650
You could also think about
the flu virus where every

00:12:43.650 --> 00:12:46.052
year that vaccine for the
flu virus, you gotta get

00:12:46.052 --> 00:12:50.028
a new one every year because
the virus is changing.

00:12:50.028 --> 00:12:52.581
Your immune system's ability
to recognize it can't

00:12:52.581 --> 00:12:56.748
recognize the next year's
because it's changed so much.


Worked examples: finite geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:01.545
- [Voiceover] We're asked to find the sum

00:00:01.545 --> 00:00:05.167
of the first 50 terms of this series,

00:00:05.167 --> 00:00:06.537
and you might immediately recognize

00:00:06.537 --> 00:00:08.118
it is a geometric series.

00:00:08.118 --> 00:00:11.098
When we go from one term to
the next, what are we doing?

00:00:11.098 --> 00:00:15.108
Well, we're multiplying by
10/11, to go from one to 10/11,

00:00:15.108 --> 00:00:17.798
you multiply by 10 over 11,

00:00:17.798 --> 00:00:20.965
then you multiply by 10 over 11 again,

00:00:23.235 --> 00:00:24.265
and we keep doing this,

00:00:24.265 --> 00:00:26.908
and we wanna find the
first 50 terms of it.

00:00:26.908 --> 00:00:29.731
So we can apply the formula we derived

00:00:29.731 --> 00:00:33.289
for the sum of a finite geometric series

00:00:33.289 --> 00:00:35.540
and that tells us that the sum of,

00:00:35.540 --> 00:00:37.951
let's say in this case the first 50 terms,

00:00:37.951 --> 00:00:40.023
actually let me do it down here,

00:00:40.023 --> 00:00:42.690
so the sum of the first 50 terms

00:00:45.955 --> 00:00:50.623
is going to be equal to the
first term, which is one,

00:00:50.623 --> 00:00:53.706
so it's gonna be one times one minus,

00:00:55.049 --> 00:00:56.820
let me do that in a different color,

00:00:56.820 --> 00:00:59.903
one times one minus the common ratio,

00:01:01.621 --> 00:01:05.788
so the common ratio here is
10/11, 10/11 to the 50th power,

00:01:07.943 --> 00:01:10.902
to the power of how many terms we have,

00:01:10.902 --> 00:01:15.358
all of that over one
minus our common ratio.

00:01:15.358 --> 00:01:18.405
And so I'm not gonna solve it completely,

00:01:18.405 --> 00:01:20.214
but we can simplify this a little bit,

00:01:20.214 --> 00:01:22.286
this is gonna be one minus,

00:01:22.286 --> 00:01:24.178
let me put parenthesis
here just to make sure

00:01:24.178 --> 00:01:26.570
we're not just taking
10 to the 50th power.

00:01:26.570 --> 00:01:30.737
So one minus 10/11 to the 50th power over,

00:01:32.863 --> 00:01:37.292
this is 11/11 minus 10/11 is one over 11,

00:01:37.292 --> 00:01:38.251
and so this is the same thing

00:01:38.251 --> 00:01:40.470
as multiplying the numerator by 11,

00:01:40.470 --> 00:01:43.137
and so this is gonna be equal to

00:01:44.078 --> 00:01:48.245
11 times one minus 10/11 to the 50thpower,

00:01:50.008 --> 00:01:52.475
and you can try to
simplify this even more,

00:01:52.475 --> 00:01:53.337
but this gets up pretty far,

00:01:53.337 --> 00:01:55.717
at this point it is just arithmetic.

00:01:55.717 --> 00:01:58.516
Let's do another one of
these, this is kinda fun.

00:01:58.516 --> 00:02:02.099
So this is more clearly
a geometric series,

00:02:03.005 --> 00:02:05.199
and let's just first
think about how many terms

00:02:05.199 --> 00:02:06.180
we're gonna take the sum of.

00:02:06.180 --> 00:02:07.013
You might be tempted to say

00:02:07.013 --> 00:02:09.176
"Okay I'm gonna take it to the 79th power,

00:02:09.176 --> 00:02:12.087
"there must be 79 terms
here", but be very careful,

00:02:12.087 --> 00:02:13.983
because the first term is
when we're taking things

00:02:13.983 --> 00:02:17.619
to the zeroth power, we're
taking 0.99 to the zeroth power.

00:02:17.619 --> 00:02:21.146
The second term is where we're
taking it to the first power,

00:02:21.146 --> 00:02:23.325
the third term is where we're
taking it to the second power,

00:02:23.325 --> 00:02:27.271
the fourth term is where we're
taking it to the third power

00:02:27.271 --> 00:02:31.963
so on and so forth, so this
right over here is the 80th,

00:02:31.963 --> 00:02:34.046
the 80th term, 80th term.

00:02:35.795 --> 00:02:39.962
So we wanna find S sub 80, and
so this is gonna be equal to

00:02:41.393 --> 00:02:44.393
our first term is gonna be one times

00:02:47.384 --> 00:02:51.134
one minus our common
ratio to the 80th power,

00:02:52.687 --> 00:02:56.266
to the 80th power, all over,
and I'm leaving a blank

00:02:56.266 --> 00:02:58.278
because we still need to
figure out our common ratio,

00:02:58.278 --> 00:03:00.845
all over one minus our common ratio.

00:03:00.845 --> 00:03:02.102
So at first you might say

00:03:02.102 --> 00:03:05.071
well maybe the common ratio here is 0.99,

00:03:05.071 --> 00:03:08.320
but notice we have a change in sign here,

00:03:08.320 --> 00:03:10.722
and the key thing is to say
well to go from term to the next

00:03:10.722 --> 00:03:12.383
what are we multiplying by?

00:03:12.383 --> 00:03:15.982
Well to go from the first
term to the second term,

00:03:15.982 --> 00:03:18.399
we multiply by negative 0.99.

00:03:19.604 --> 00:03:24.420
And then, so we're
multiplying by negative 0.99.

00:03:24.420 --> 00:03:25.863
Now to go to the next term,

00:03:25.863 --> 00:03:29.280
we're again multiplying by negative 0.99,

00:03:30.665 --> 00:03:34.082
so the common ratio is not positive 0.99,

00:03:36.244 --> 00:03:38.994
but negative zero, negative 0.99,

00:03:39.960 --> 00:03:42.960
so let me write that, negative 0.99,

00:03:45.734 --> 00:03:49.524
and of course that is going
to be to the 80th power,

00:03:49.524 --> 00:03:52.274
all over one minus negative 0.99.

00:03:56.046 --> 00:03:59.126
And so we could simplify
this a little bit,

00:03:59.126 --> 00:04:01.886
this is all going to be equal to,

00:04:01.886 --> 00:04:04.449
oh that one we don't have to
worry too much about that,

00:04:04.449 --> 00:04:07.279
and so this is going to be one minus,

00:04:07.279 --> 00:04:10.890
so negative 0.99 to the 80th power,

00:04:10.890 --> 00:04:12.376
I should put parenthesis
there to make sure

00:04:12.376 --> 00:04:16.543
we are taking the negative
0.99 to the 80th power.

00:04:17.417 --> 00:04:19.429
Well, we're taking it to an even power,

00:04:19.429 --> 00:04:21.217
so it's going to be positive,
so that's going to be

00:04:21.217 --> 00:04:24.634
the same thing as 0.99 to the 80th power,

00:04:26.978 --> 00:04:30.679
and all of that over, well
subtracting a negative

00:04:30.679 --> 00:04:33.181
that's just gonna be adding the positive,

00:04:33.181 --> 00:04:35.264
so all of that over 1.99,

00:04:36.870 --> 00:04:39.471
and we could attempt to
simplify it more but,

00:04:39.471 --> 00:04:41.009
if we had a calculator we could

00:04:41.009 --> 00:04:44.327
actually find this exact
value or close value actually,

00:04:44.327 --> 00:04:47.032
most calculators don't
give you the exact value

00:04:47.032 --> 00:04:49.153
when you take something to the 80th power,

00:04:49.153 --> 00:04:52.715
but this is what that sum is going to be.

00:04:52.715 --> 00:04:54.984
Let's do one more of these.

00:04:54.984 --> 00:04:58.511
Alright, so here we have a
series defined recursively

00:04:58.511 --> 00:05:01.056
and so it's useful to just think about

00:05:01.056 --> 00:05:02.435
what it would actually look like.

00:05:02.435 --> 00:05:06.435
So the first term is 10,
and then the next term,

00:05:07.575 --> 00:05:11.076
so the second term A sub two is equal to

00:05:11.076 --> 00:05:14.231
A sub one times 9/10, alright.

00:05:14.231 --> 00:05:16.890
So the next term is gonna be
the previous term times 9/10,

00:05:16.890 --> 00:05:19.919
so it's gonna be 10 times nine over 10,

00:05:19.919 --> 00:05:22.018
and then the next term
is gonna be that times,

00:05:22.018 --> 00:05:23.237
is gonna be the second term,

00:05:23.237 --> 00:05:25.879
the third term is the
second term times 9/10,

00:05:25.879 --> 00:05:30.618
so 10 times nine over
10, nine over 10 squared.

00:05:30.618 --> 00:05:32.417
And the way it's written right now,

00:05:32.417 --> 00:05:36.838
we don't have it written as
a finite geometric series,

00:05:36.838 --> 00:05:38.549
so let's say we wanna take the sum,

00:05:38.549 --> 00:05:41.466
let's say we want the sum of first,

00:05:43.698 --> 00:05:46.031
first I don't know 30 terms,

00:05:46.882 --> 00:05:48.715
sum of first 30 terms.

00:05:49.892 --> 00:05:51.320
So what will this be?

00:05:51.320 --> 00:05:55.487
Well we're gonna take S sub
one, S sub 30, oh I wrote ten,

00:05:57.778 --> 00:06:00.042
S sub 30, the sum of the first 30 terms,

00:06:00.042 --> 00:06:03.436
is gonna be equal to the first
term, we've done this before,

00:06:03.436 --> 00:06:07.436
the first term times one
minus the common ratio,

00:06:08.503 --> 00:06:12.253
one minus the common
ratio to the 30th power,

00:06:13.795 --> 00:06:17.462
all of that over one
minus the common ratio.

00:06:18.511 --> 00:06:20.853
And let's see we could, one minus 9/10,

00:06:20.853 --> 00:06:24.026
this is 1/10 right over
here, you divide by 1/10,

00:06:24.026 --> 00:06:25.991
this is the same thing
as multiplying by 100,

00:06:25.991 --> 00:06:30.158
so this is gonna be 100
times one minus 9/10 to the,

00:06:32.769 --> 00:06:36.936
oh let me write it this
way, 9/10 to the 30th power.

00:06:37.819 --> 00:06:40.402
And, actually these
parenthesis you always wanna

00:06:40.402 --> 00:06:41.960
put parenthesis there to make sure we see

00:06:41.960 --> 00:06:43.991
we're taking both the nine and the 10

00:06:43.991 --> 00:06:46.143
or the 9/10, the whole
thing to the 30th power,

00:06:46.143 --> 00:06:49.726
not just the nine, so
there you go, did I..

00:06:51.732 --> 00:06:53.869
yep there you go we're done.


Worked example: finite geometric series (sigma notation)

WEBVTT
Kind: captions
Language: en

00:00:00.497 --> 00:00:01.827
- [Voiceover] Let's do some examples

00:00:01.827 --> 00:00:06.444
where we're finding sums
of finite geometric series.

00:00:06.444 --> 00:00:08.438
Now let's just remind
ourselves in a previous video

00:00:08.438 --> 00:00:12.376
we derived the formula where
the sum of the first n terms

00:00:12.376 --> 00:00:14.543
is equal to our first term

00:00:15.802 --> 00:00:19.269
times one minus our common ratio

00:00:19.269 --> 00:00:24.199
to the nth power all over
one minus our common ratio.

00:00:24.199 --> 00:00:28.091
So let's apply that to this
finite geometric series

00:00:28.091 --> 00:00:29.553
right over here.

00:00:29.553 --> 00:00:32.231
So what is our first term
and what is our common ratio?

00:00:32.231 --> 00:00:33.870
And what is our n?

00:00:33.870 --> 00:00:35.416
Well, some of you might
just be able to pick it out

00:00:35.416 --> 00:00:38.112
by inspecting this here, but
for the sake of this example,

00:00:38.112 --> 00:00:39.860
let's expand this out a little bit.

00:00:39.860 --> 00:00:43.225
This is going to be equal to
two times three to the zero,

00:00:43.225 --> 00:00:44.498
which is just two,

00:00:44.498 --> 00:00:47.709
plus two times three to the first power,

00:00:47.709 --> 00:00:51.222
plus two times three to the second power,

00:00:51.222 --> 00:00:52.620
I can write first power there,

00:00:52.620 --> 00:00:55.595
plus two times three to the third power,

00:00:55.595 --> 00:00:59.881
and we're gonna go all
the way to two times three

00:00:59.881 --> 00:01:01.985
to the 99th power.

00:01:01.985 --> 00:01:03.617
So what is our first term?

00:01:03.617 --> 00:01:05.084
What is our a?

00:01:05.084 --> 00:01:07.082
Well, a is going to be two.

00:01:07.082 --> 00:01:10.189
And we see that in all
of these terms here.

00:01:10.189 --> 00:01:12.533
So a is going to be two.

00:01:12.533 --> 00:01:13.824
What is r?

00:01:13.824 --> 00:01:17.023
Well, each successive term,
as k increases by one,

00:01:17.023 --> 00:01:18.901
we're multiplying by three again.

00:01:18.901 --> 00:01:22.121
So, three is our common ratio.

00:01:22.121 --> 00:01:24.750
So that right over there, that is r.

00:01:24.750 --> 00:01:28.272
Let me make sure that we, that is a.

00:01:28.272 --> 00:01:30.686
And now what is n going to be?

00:01:30.686 --> 00:01:32.029
Well, you might be tempted to say,

00:01:32.029 --> 00:01:35.220
well, we're going up to k
equals 99, maybe n is 99,

00:01:35.220 --> 00:01:37.850
but we have to realize that
we're starting at k equals zero.

00:01:37.850 --> 00:01:39.675
So there is actually 100 terms here.

00:01:39.675 --> 00:01:42.121
Notice, when k equals zero,
that's our first term,

00:01:42.121 --> 00:01:43.766
when k equals one, that's our second term,

00:01:43.766 --> 00:01:45.775
when k equals two, that's our third term,

00:01:45.775 --> 00:01:47.752
when k equals three,
that's our fourth term,

00:01:47.752 --> 00:01:51.919
when k equals 99, this is
our 100th term, 100th term.

00:01:53.658 --> 00:01:56.451
So what we really want
to find is S sub 100.

00:01:56.451 --> 00:01:59.665
So let's write that down, S sub 100,

00:01:59.665 --> 00:02:03.748
for this geometric series
is going to be equal to

00:02:06.348 --> 00:02:10.098
two times one minus
three to the 100th power,

00:02:12.277 --> 00:02:14.944
to the 100th power, all of that,

00:02:16.467 --> 00:02:20.634
all of that over, all of
that over one minus three.

00:02:23.565 --> 00:02:25.820
And we could simplify
this, I mean at this point

00:02:25.820 --> 00:02:28.384
it is arithmetic that
you'd be dealing with,

00:02:28.384 --> 00:02:30.803
but down here you would
have a negative two,

00:02:30.803 --> 00:02:34.023
and so you'd have two
divided by negative two

00:02:34.023 --> 00:02:35.355
so that is just a negative.

00:02:35.355 --> 00:02:38.348
And so negative of one
minus three to the 100th,

00:02:38.348 --> 00:02:41.844
that's the same thing, this is
equal to three to the 100th,

00:02:41.844 --> 00:02:44.761
three to the 100th power minus one.

00:02:45.675 --> 00:02:46.878
And we're done.