Introduction to agreement

WEBVTT
Kind: captions
Language: en

00:00:00.368 --> 00:00:01.364
- [Voiceover] Hi grammarians.

00:00:01.364 --> 00:00:04.255
Today I want to talk about
this idea, in English,

00:00:04.255 --> 00:00:05.731
that we call agreement.

00:00:05.731 --> 00:00:08.078
So, I'm going to teach
you how to be agreeable,

00:00:08.078 --> 00:00:11.589
make it so that all of your
sentences get along really well.

00:00:11.589 --> 00:00:13.237
Let me give you an example.

00:00:13.237 --> 00:00:16.360
"The dog barks,"

00:00:16.360 --> 00:00:18.578
as opposed to,

00:00:18.578 --> 00:00:21.371
"The dogs bark."

00:00:21.371 --> 00:00:23.519
Do you see the difference?

00:00:23.519 --> 00:00:26.165
It's funny because the 's' seems

00:00:26.165 --> 00:00:30.640
to migrate, seems to move
from one part of the sentence

00:00:30.640 --> 00:00:32.173
to the other.

00:00:32.173 --> 00:00:33.990
So what you wouldn't say is,

00:00:35.920 --> 00:00:38.438
"The dog bark."

00:00:38.438 --> 00:00:39.667
That doesn't work.

00:00:39.667 --> 00:00:41.538
And you wouldn't say,

00:00:41.538 --> 00:00:46.298
"The dogs barks."

00:00:46.298 --> 00:00:48.249
I know that seems strange

00:00:48.249 --> 00:00:51.209
that you wouldn't want to have the 's'

00:00:51.209 --> 00:00:53.079
follow the other 's',

00:00:53.079 --> 00:00:57.409
but you actually only
want to have one of these

00:00:57.409 --> 00:00:58.315
in a sentence.

00:00:58.315 --> 00:00:59.847
So, "The dog barks."

00:00:59.847 --> 00:01:02.169
"The dogs bark."

00:01:02.169 --> 00:01:05.109
When there's only one
of something, strangely,

00:01:05.109 --> 00:01:08.604
in the present tense, it takes this 's'.

00:01:08.604 --> 00:01:12.900
So, "Jake bakes a cake."

00:01:12.900 --> 00:01:16.336
"We bake a cake."

00:01:16.336 --> 00:01:21.282
Not, "Jake bake a cake,"

00:01:21.282 --> 00:01:23.743
which, admittedly, is
a fun sentence to say.

00:01:23.743 --> 00:01:27.700
But this is not, in standard
English, this is not correct.

00:01:27.700 --> 00:01:31.394
It doesn't exhibit what we call agreement.

00:01:31.394 --> 00:01:33.833
Similarly, you also wouldn't say,

00:01:33.833 --> 00:01:38.047
"We bakes a cake."

00:01:38.047 --> 00:01:39.997
If you're familiar with
the Lord of the Rings books

00:01:39.997 --> 00:01:42.319
or movies, if you remember Gollum,

00:01:42.319 --> 00:01:47.066
the character Gollum is a
weird little bug-eyed critter

00:01:47.996 --> 00:01:51.631
whose just kind of
obsessed with the one ring,

00:01:51.631 --> 00:01:53.662
and he has this very
particular way of speaking

00:01:53.662 --> 00:01:57.507
that is not, strictly speaking, standard.

00:01:57.507 --> 00:01:59.661
He's just wild about that ring.

00:02:01.371 --> 00:02:03.891
And the way he refers to everything,

00:02:03.891 --> 00:02:08.465
he just throws on an 's' on the end

00:02:08.465 --> 00:02:11.077
of every word he uses, every verb.

00:02:11.077 --> 00:02:14.202
So, if he's talking about, you know,

00:02:14.202 --> 00:02:17.184
this magic ring that he's very fond of,

00:02:17.184 --> 00:02:18.970
which he calls the precious,

00:02:18.970 --> 00:02:21.155
he wouldn't just say, "I love it."

00:02:21.155 --> 00:02:23.810
He'd say, "I loves it."

00:02:25.160 --> 00:02:26.971
Like so, and that's not correct.

00:02:26.971 --> 00:02:29.537
That does not show agreement.

00:02:29.537 --> 00:02:32.463
So, the test here is really...

00:02:32.463 --> 00:02:34.982
You can figure out whether or not

00:02:34.982 --> 00:02:38.475
you are using agreement in your sentences

00:02:38.475 --> 00:02:41.703
if you sound like Gollum.

00:02:41.703 --> 00:02:43.980
Because if Gollum were being grammatical,

00:02:43.980 --> 00:02:47.009
he would say, "I love it."

00:02:49.959 --> 00:02:52.351
So this is the sense that
I want you to develop

00:02:52.351 --> 00:02:55.099
is you listen to yourself
speak and you ask,

00:02:55.099 --> 00:03:00.084
"Do I sound like Gollum,
the little ring monster?"

00:03:00.084 --> 00:03:01.473
Because if you do sound like Gollum,

00:03:01.473 --> 00:03:03.136
then you're probably not operating

00:03:03.136 --> 00:03:05.772
under the agreement rules

00:03:05.772 --> 00:03:08.465
of standard American English,

00:03:08.465 --> 00:03:10.962
or you're just doing a Gollum impression.

00:03:10.962 --> 00:03:14.212
That's the basic idea behind agreement

00:03:14.212 --> 00:03:15.397
Is you just want to make sure

00:03:15.397 --> 00:03:18.241
that the parts of the sentence match up.

00:03:18.241 --> 00:03:19.978
Sentence parts

00:03:24.058 --> 00:03:26.239
match.

00:03:26.239 --> 00:03:26.937
They agree.

00:03:26.937 --> 00:03:28.017
They get along.

00:03:28.017 --> 00:03:29.427
They work.

00:03:30.267 --> 00:03:31.929
You can learn anything.

00:03:31.929 --> 00:03:33.229
David out.


Confidence – Official SAT Practice on Khan Academy

WEBVTT
Kind: captions
Language: en

00:00:01.141 --> 00:00:03.457
(bell ringing)

00:00:11.365 --> 00:00:15.986
(upbeat music)

00:00:17.019 --> 00:00:21.723
(steady beat)

00:00:22.778 --> 00:00:27.626
(upbeat rock)

00:00:35.410 --> 00:00:37.245
- [Voiceover] Confidence.

00:00:37.245 --> 00:00:39.892
Nothing builds it like practice, and now,

00:00:39.892 --> 00:00:42.748
the best way to get
ready for the SAT is free

00:00:42.748 --> 00:00:44.622
for everyone.

00:00:52.344 --> 00:00:55.861
Get a plan built just for
you at satpractice.org.


2015 AP Calculus BC 5a

WEBVTT
Kind: captions
Language: en

00:00:00.574 --> 00:00:03.059
- [Voiceover] Consider the
function f of x is equal to one

00:00:03.059 --> 00:00:07.889
over x-squared minus k-x,
where k is a nonzero constant.

00:00:07.889 --> 00:00:10.420
The derivative of f is given by,

00:00:10.420 --> 00:00:12.335
and they give us this
expression right over here.

00:00:12.335 --> 00:00:15.435
That's nice that they took
the derivative for us.

00:00:15.435 --> 00:00:19.752
Now part a, let k equals
three so that f of x is equal

00:00:19.752 --> 00:00:22.422
to one over x-squared minus three-x.

00:00:22.422 --> 00:00:24.965
So they set k equal to three.

00:00:24.965 --> 00:00:27.832
Write an equation for the line tangent

00:00:27.832 --> 00:00:30.503
to the graph of f at the point

00:00:30.503 --> 00:00:34.053
whose x-coordinate is four.

00:00:34.053 --> 00:00:35.890
So to find an equation for a line,

00:00:35.890 --> 00:00:37.550
the equation of a line
is gonna be in the form

00:00:37.550 --> 00:00:40.721
y is equal to m-x plus b,

00:00:40.721 --> 00:00:42.251
where m is the slope of the line

00:00:42.251 --> 00:00:44.237
and b is the y-intercept.

00:00:44.237 --> 00:00:46.594
And the slope of the
line, right over here,

00:00:46.594 --> 00:00:50.321
this needs to be equal to
the derivative evaluated

00:00:50.321 --> 00:00:52.097
when x is equal to four.

00:00:52.097 --> 00:00:54.779
So we could say, y is equal to...

00:00:54.779 --> 00:00:56.486
Let me write it this way.

00:00:56.486 --> 00:01:01.292
We could say that m is
going to be equal to f-prime

00:01:01.292 --> 00:01:02.908
when x is equal to four.

00:01:02.908 --> 00:01:05.356
So, f-prime of four, which is equal to...

00:01:05.356 --> 00:01:07.097
Well, we know that k is equal to three.

00:01:07.097 --> 00:01:08.665
So, they gave us f-prime of x.

00:01:08.665 --> 00:01:12.577
So, it's going to be
three minus two times,

00:01:12.577 --> 00:01:13.738
we're taking f-prime of four,

00:01:13.738 --> 00:01:17.303
so minus two times four

00:01:17.303 --> 00:01:20.704
over x-squared.

00:01:20.704 --> 00:01:24.705
So that's going to be
four-squared, minus k,

00:01:24.705 --> 00:01:28.146
is three times four,

00:01:28.146 --> 00:01:31.873
three times four, and then
we square that whole thing.

00:01:31.873 --> 00:01:33.440
And so what is this going to be?

00:01:33.440 --> 00:01:34.673
This is an eight right over here,

00:01:34.673 --> 00:01:37.588
and all I did is f-prime of
x, when k is equal to three,

00:01:37.588 --> 00:01:40.244
is gonna be three minus
two-x over x-squared

00:01:40.244 --> 00:01:43.437
minus three-x and all of that squared.

00:01:43.437 --> 00:01:46.770
And I want to evaluate
what f-prime of four is.

00:01:46.770 --> 00:01:47.930
So, every place where I saw an x,

00:01:47.930 --> 00:01:49.392
I substituted with a four.

00:01:49.392 --> 00:01:51.504
Where I saw the k, k is three.

00:01:51.504 --> 00:01:53.282
And so this is going to be equal to...

00:01:53.282 --> 00:01:56.602
The numerator is three minus
eight, is negative five,

00:01:56.602 --> 00:02:01.098
over this is 16 minus 12,

00:02:01.098 --> 00:02:03.738
which is going to be four,

00:02:03.738 --> 00:02:06.575
16 minus 12 is four,
and then we square it.

00:02:06.575 --> 00:02:08.897
So it's gonna be negative five-sixteenths.

00:02:08.897 --> 00:02:09.883
And so let me write it this way,

00:02:09.883 --> 00:02:13.460
m is equal to negative five-sixteenths.

00:02:13.460 --> 00:02:16.409
So, how do we figure out b now?

00:02:16.409 --> 00:02:17.431
Well, what are the coordinates?

00:02:17.431 --> 00:02:21.602
When x is equal to four, what
is y going to be equal to?

00:02:21.602 --> 00:02:23.584
Well, y is equal to f of x.

00:02:23.584 --> 00:02:27.743
So we know that y, on
the curve, we know that y

00:02:27.743 --> 00:02:31.769
is going to be equal to f of four.

00:02:31.769 --> 00:02:33.754
So before we evaluated f-prime of four.

00:02:33.754 --> 00:02:37.259
Now we're going to evaluate
y as being f of four

00:02:37.259 --> 00:02:41.312
which is equal to one over four-squared,

00:02:41.312 --> 00:02:45.109
four-squared minus three times four,

00:02:45.109 --> 00:02:49.160
and so that is equal to one over

00:02:49.160 --> 00:02:52.980
16 minus 12, which is four.

00:02:52.980 --> 00:02:55.651
And so this point, right
here, when x is four,

00:02:55.651 --> 00:02:58.089
then y is equal to one-fourth.

00:02:58.089 --> 00:03:02.418
So, we can use that
information to solve for b.

00:03:02.418 --> 00:03:06.169
When y is one-fourth, x...

00:03:06.169 --> 00:03:08.352
So, we're gonna say if y is equal to m,

00:03:08.352 --> 00:03:11.011
which is negative five-sixteenths,

00:03:11.011 --> 00:03:13.541
negative five-sixteenths, times x.

00:03:13.541 --> 00:03:16.932
Well, when y is one-fourth, x is...

00:03:16.932 --> 00:03:20.450
Or when I say when x is equal to four,

00:03:20.450 --> 00:03:23.156
y is one-fourth,

00:03:23.156 --> 00:03:24.397
and then plus b.

00:03:24.397 --> 00:03:25.744
So I can now solve for b.

00:03:25.744 --> 00:03:29.006
So all I did is I used f-prime of x.

00:03:29.006 --> 00:03:31.618
I used f-prime of x to figure out m,

00:03:31.618 --> 00:03:33.279
when x is equal to four.

00:03:33.279 --> 00:03:35.601
And then, I said, "Okay,
well what is the value of y

00:03:35.601 --> 00:03:37.052
"when x is equal to four?"

00:03:37.052 --> 00:03:41.489
And so, if I know y, m, and
x, then I can solve for b.

00:03:41.489 --> 00:03:42.961
And so, let's just do that.

00:03:42.961 --> 00:03:46.235
We get one-fourth is equal to four times

00:03:46.235 --> 00:03:50.067
negative five-sixteenths,
is negative five over four,

00:03:50.067 --> 00:03:51.135
plus b.

00:03:51.135 --> 00:03:52.923
I can add five-fourths to both sides,

00:03:52.923 --> 00:03:57.381
and I get five-fourths plus
one-fourth is equal to b,

00:03:57.381 --> 00:04:01.934
or b is equal to six-fourths,

00:04:01.934 --> 00:04:04.788
six over four, which you can say...

00:04:04.788 --> 00:04:06.460
Well, there's a bunch of
ways you could write this.

00:04:06.460 --> 00:04:10.750
We could just say this
is just equal to 1.5.

00:04:10.750 --> 00:04:15.195
And so our equation is y
is equal to negative five

00:04:15.195 --> 00:04:19.359
over 16 x plus 1.5.

00:04:19.359 --> 00:04:21.170
Or if we wanted to write
everything as a fraction,

00:04:21.170 --> 00:04:24.656
we could say y is equal to negative five

00:04:24.656 --> 00:04:27.856
over 16 x plus three-halves.

00:04:27.856 --> 00:04:29.246
Six-fourths is the same thing

00:04:29.246 --> 00:04:33.096
as three-halves,

00:04:33.096 --> 00:04:34.880
and there you go.


2015 AP Calculus BC 5d

WEBVTT
Kind: captions
Language: en

00:00:00.383 --> 00:00:03.146
- [Voiceover] Let K equal six,
so that F of X is equal to

00:00:03.146 --> 00:00:05.165
one over X-squared minus six X.

00:00:05.165 --> 00:00:09.321
Find the partial fraction
decomposition for the function F.

00:00:09.321 --> 00:00:12.503
Find the integral of F of X D-X.

00:00:12.503 --> 00:00:14.222
And so let's first thinl about

00:00:14.222 --> 00:00:17.983
the partial fraction
decomposition for the function F.

00:00:17.983 --> 00:00:21.883
So, F of X, I could rewrite it

00:00:21.883 --> 00:00:23.997
where I factor the denominator.

00:00:23.997 --> 00:00:27.503
If I factor out an X, I
get X times X minus six.

00:00:27.503 --> 00:00:29.837
So, I can rewrite this as...

00:00:29.837 --> 00:00:31.462
And this is where I'm going

00:00:31.462 --> 00:00:33.099
to decompose it into partial fractions.

00:00:33.099 --> 00:00:36.153
A over X plus B over X minus six.

00:00:44.428 --> 00:00:46.136
If I actually had to add these two,

00:00:46.136 --> 00:00:48.110
I would try to find a common denominator.

00:00:48.110 --> 00:00:49.910
The best common denominator is

00:00:49.910 --> 00:00:51.488
just to take the product
of these two expressions.

00:00:51.488 --> 00:00:53.554
So I could multiply the
numerator and denomenator

00:00:53.554 --> 00:00:56.607
of this first time by X minus six.

00:00:56.607 --> 00:00:58.327
X minus six.

00:00:58.327 --> 00:01:00.416
And the numerator and
denominator of the second term,

00:01:00.416 --> 00:01:03.172
I can multiply it by the
denominator of the other one.

00:01:03.172 --> 00:01:07.166
So times X and times X.

00:01:07.166 --> 00:01:09.187
So that would give us,

00:01:09.187 --> 00:01:10.777
let's see if I distribute the A,

00:01:10.777 --> 00:01:14.266
it would give us A X minus six A plus B X

00:01:19.837 --> 00:01:23.845
over X times X minus six.

00:01:26.089 --> 00:01:28.375
If this looks completely foreign to you,

00:01:28.375 --> 00:01:30.697
I encourage you to watch
the videos on Kahn Academy

00:01:30.697 --> 00:01:34.855
on partial fraction decomposition.

00:01:34.855 --> 00:01:36.492
So let's see if we can,

00:01:36.492 --> 00:01:40.149
so what we wanna do is solve
for the A's and the B's.

00:01:43.849 --> 00:01:46.508
And so let's see,

00:01:46.508 --> 00:01:48.966
if we see that these have to add up to

00:01:48.966 --> 00:01:52.477
one over X times X minus six.

00:01:52.477 --> 00:01:54.161
So these have to add up.

00:01:54.161 --> 00:01:55.381
I'm just going back to
this, these have to add up.

00:01:55.381 --> 00:01:56.797
The numerator.

00:01:56.797 --> 00:01:58.143
So it has to add up to

00:01:58.143 --> 00:02:01.221
one over X times X minus six.

00:02:01.221 --> 00:02:04.181
So this numerator has to add up to one.

00:02:04.181 --> 00:02:05.842
So what we can see is

00:02:05.842 --> 00:02:07.978
is that the X terms right over here

00:02:07.978 --> 00:02:11.146
must cancel out since
we have no X terms here.

00:02:11.146 --> 00:02:16.146
We have A X plus B X
must be equal to zero.

00:02:16.244 --> 00:02:17.939
Or you could say,

00:02:17.939 --> 00:02:22.479
well that means A plus B is equal to zero.

00:02:22.479 --> 00:02:24.278
So we took care of that
term and that term.

00:02:24.278 --> 00:02:27.338
And then we know, that this
must be the constant term

00:02:27.338 --> 00:02:28.817
that adds up to one or
that is equal to one.

00:02:28.817 --> 00:02:30.617
And so we also know that
negative six A is equal to one.

00:02:30.617 --> 00:02:34.745
Or A is equal to, divide
both sides by negative six,

00:02:38.419 --> 00:02:40.508
negative one sixth.

00:02:40.508 --> 00:02:42.667
And if A is negative one sixth,

00:02:42.667 --> 00:02:45.012
well B is going to be
the negative of that.

00:02:45.012 --> 00:02:47.815
And negative one sixth plus B.

00:02:47.815 --> 00:02:51.101
I'm just substituting A
back into that equation.

00:02:51.101 --> 00:02:54.043
Need to be equal to zero.

00:02:54.043 --> 00:02:55.872
And so add one sixth to both sides,

00:02:55.872 --> 00:02:58.397
you get B is equal to one sixth.

00:02:58.397 --> 00:03:00.307
So I can decompose F of X.

00:03:01.183 --> 00:03:04.546
I can decompose F of X as
being equal to A over X

00:03:06.861 --> 00:03:10.982
So that's negative one sixth over X.

00:03:10.982 --> 00:03:12.231
I just write it that way.

00:03:12.231 --> 00:03:14.332
I could write it as
negative one over six X

00:03:14.332 --> 00:03:15.598
or something like that.

00:03:15.598 --> 00:03:17.165
But I'll just write it
like this, just be clear

00:03:17.165 --> 00:03:18.709
that this was our A.

00:03:18.709 --> 00:03:23.493
Plus B, which is one
sixth over X minus six.

00:03:27.713 --> 00:03:28.996
So that right over there,

00:03:28.996 --> 00:03:31.182
that's the partial fraction decomposition

00:03:31.182 --> 00:03:33.334
for our function F.

00:03:33.334 --> 00:03:35.401
And if I want to evaluate the integral.

00:03:35.401 --> 00:03:38.438
So the integral of F of X,
the indefinite integral.

00:03:39.417 --> 00:03:41.636
Well, that's where this
partial fractions decomposition

00:03:41.636 --> 00:03:43.046
is going to be valuable.

00:03:43.046 --> 00:03:44.719
That's going to be the indefinite integral

00:03:44.719 --> 00:03:48.086
of negative one sixth over
X plus positive one sixth

00:03:50.473 --> 00:03:54.246
over X minus six, and then we have D X.

00:03:56.250 --> 00:03:58.658
Well what's the anti-derivative
of this right over here?

00:03:58.658 --> 00:04:01.557
Well the anti-derivative of one over X

00:04:01.557 --> 00:04:03.954
is the natural log of
the absolute value of X.

00:04:03.954 --> 00:04:05.837
And so we can just this is going to be

00:04:05.837 --> 00:04:09.412
negative one sixth times
the natural log of the

00:04:09.412 --> 00:04:10.757
absolute value of X.

00:04:10.757 --> 00:04:12.860
Anti-derivative of this part.

00:04:12.860 --> 00:04:14.931
Then plus one sixth.

00:04:16.290 --> 00:04:18.543
You could do u-substitution
but you could just say

00:04:18.543 --> 00:04:21.050
"Hey, look the derivative of
this bottom part, X minus six."

00:04:21.050 --> 00:04:22.643
That's just one.

00:04:22.643 --> 00:04:25.169
So you could say that I have
that derivative laying around.

00:04:25.169 --> 00:04:29.185
So I can just take the
anti-derivative with respect to that.

00:04:29.185 --> 00:04:32.123
So that's going to be one
sixth times the natural log

00:04:32.123 --> 00:04:35.261
of the absolute value of X minus six,

00:04:35.261 --> 00:04:37.972
and then I have plus C.

00:04:39.185 --> 00:04:42.192
Don't forget this is an
indefinite integral over here.

00:04:42.192 --> 00:04:43.852
And then you're done.

00:04:43.852 --> 00:04:46.203
And you see that partial
fraction decomposition

00:04:46.203 --> 00:04:47.690
was actually quite useful.

00:04:47.690 --> 00:04:48.820
So they were helping us

00:04:48.820 --> 00:04:50.528
how to figure out this.

00:04:50.528 --> 00:04:52.362
You didn't just have to have that insight

00:04:52.362 --> 00:04:54.928
that, okay how do I evaluate
this anti-derivative.

00:04:54.928 --> 00:04:57.204
Well, the they're telling us to use

00:04:57.204 --> 00:04:59.277
partial fraction decomposition.


2015 AP Calculus BC 5b

WEBVTT
Kind: captions
Language: en

00:00:00.167 --> 00:00:02.245
- [Voiceover] Let k equal four,

00:00:02.245 --> 00:00:05.961
so that f of x is equal to one
over x squared minus four x.

00:00:05.961 --> 00:00:08.738
Determine whether f
has a relative minimum,

00:00:08.738 --> 00:00:12.449
a relative maximum, or
neither at x equals two.

00:00:12.449 --> 00:00:15.329
Justify your answer.

00:00:15.329 --> 00:00:16.051
All right.

00:00:16.051 --> 00:00:18.883
Well, if f of x is equal
to this, then f prime of x,

00:00:18.883 --> 00:00:21.655
they gave us f prime of x in terms of k.

00:00:21.655 --> 00:00:24.808
So f prime of x

00:00:24.808 --> 00:00:26.123
is going to be equal to,

00:00:26.123 --> 00:00:27.237
in this case k is four.

00:00:27.237 --> 00:00:29.756
So it's gonna be four minus two x.

00:00:29.756 --> 00:00:31.428
Four minus two x.

00:00:31.428 --> 00:00:35.138
Over x squared minus

00:00:35.138 --> 00:00:37.388
four x squared

00:00:37.388 --> 00:00:40.763
minus four x

00:00:40.763 --> 00:00:42.586
minus four x squared.

00:00:42.586 --> 00:00:45.595
So now we know f of x
and we know f prime of x.

00:00:45.595 --> 00:00:47.879
And if we were looking for relative minima

00:00:47.879 --> 00:00:49.329
or relative maxima,

00:00:49.329 --> 00:00:53.676
we would be interested in
points, and critical points.

00:00:53.676 --> 00:00:57.124
And especially where f
prime of x is equal to zero.

00:00:57.124 --> 00:01:01.591
And so if we said f prime
of x is equal to zero,

00:01:01.591 --> 00:01:04.982
well, we would say, when does
this numerator equal zero?

00:01:04.982 --> 00:01:09.326
And so you could say, when does
four minus two x equal zero?

00:01:09.326 --> 00:01:12.919
You add two x to both sides,
you get four is equal to two x.

00:01:12.919 --> 00:01:16.774
Or x is equal to two, x is equal to two.

00:01:16.774 --> 00:01:17.868
And they told us that.

00:01:17.868 --> 00:01:19.608
Or I guess we've confirmed that

00:01:19.608 --> 00:01:23.830
at f prime of two

00:01:23.830 --> 00:01:26.857
does indeed equal to zero.

00:01:26.857 --> 00:01:29.881
So this is definitely
an interesting point.

00:01:29.881 --> 00:01:32.207
So let's think about whether before,

00:01:32.207 --> 00:01:34.087
when x is less than two,

00:01:34.087 --> 00:01:37.187
is f prime of x increasing or decreasing?

00:01:37.187 --> 00:01:39.149
And then when x is greater than two,

00:01:39.149 --> 00:01:41.727
is f prime of x increasing or decreasing?

00:01:41.727 --> 00:01:44.792
And that will let us know if
this is a minima or a maxima.

00:01:44.792 --> 00:01:45.940
So let's say,

00:01:45.940 --> 00:01:50.715
so when x is less than two,

00:01:50.715 --> 00:01:52.508
we could test something out.

00:01:52.508 --> 00:01:54.704
We could say f prime of,

00:01:54.704 --> 00:01:57.770
so let's just say f prime of one.

00:01:57.770 --> 00:02:00.323
So if x is less than two.

00:02:00.323 --> 00:02:01.208
You know, you actually don't even

00:02:01.208 --> 00:02:02.822
have to test f prime of one.

00:02:02.822 --> 00:02:03.927
Ii mean, you could if you want.

00:02:03.927 --> 00:02:06.659
You would have four minus two.

00:02:06.659 --> 00:02:08.211
Four minus two times one.

00:02:08.211 --> 00:02:10.539
So that's going to be positive.

00:02:10.539 --> 00:02:11.704
And then this down over here

00:02:11.704 --> 00:02:14.095
is always going to be non-negative.

00:02:14.095 --> 00:02:16.527
Because you have a
squared right over here.

00:02:16.527 --> 00:02:18.330
So when x is less than two,

00:02:18.330 --> 00:02:22.369
f prime of x is greater than zero.

00:02:22.369 --> 00:02:25.494
And you could try this out with
different exes if you like.

00:02:25.494 --> 00:02:27.378
You could say, for example,

00:02:27.378 --> 00:02:29.799
for example,

00:02:30.843 --> 00:02:33.682
f prime of one is equal to

00:02:33.682 --> 00:02:36.017
four minus two, which is two

00:02:36.017 --> 00:02:39.704
over one minus four.

00:02:39.704 --> 00:02:41.931
Which is negative three, but then squared.

00:02:41.931 --> 00:02:44.965
Is equal to two ninths.

00:02:44.965 --> 00:02:46.387
And then we could say

00:02:47.227 --> 00:02:52.227
when x is greater than two,

00:02:52.760 --> 00:02:54.397
f prime of x,

00:02:55.407 --> 00:02:57.160
f prime of x.

00:02:57.160 --> 00:02:58.903
Well, when x is greater than two,

00:02:58.903 --> 00:03:02.026
you're gonna have four minus two times

00:03:02.026 --> 00:03:04.337
something larger than four.

00:03:04.337 --> 00:03:05.812
So this is going to be negative.

00:03:05.812 --> 00:03:08.097
So f prime of x is going
to be less than zero.

00:03:08.097 --> 00:03:09.855
Up here's gonna be negative.

00:03:09.855 --> 00:03:11.646
Down here's not going to be negative.

00:03:11.646 --> 00:03:14.606
And so f prime of x is
going to be less than zero.

00:03:14.606 --> 00:03:18.711
So if we are increasing
as we approach something,

00:03:18.711 --> 00:03:20.901
and then our slope is zero.

00:03:20.901 --> 00:03:22.929
And then we are decreasing,

00:03:22.929 --> 00:03:25.937
well then this is going
to be a maximum point.

00:03:25.937 --> 00:03:28.508
So that means that

00:03:28.508 --> 00:03:30.281
we have a maximum,

00:03:31.121 --> 00:03:32.703
maximum point

00:03:33.663 --> 00:03:34.585
point

00:03:34.585 --> 00:03:37.112
at x is equal

00:03:37.112 --> 00:03:40.002
at x is equal to,

00:03:40.002 --> 00:03:41.767
or we could say, yes.

00:03:41.767 --> 00:03:44.824
That we have, f has a relative maximum,

00:03:44.824 --> 00:03:47.220
relative maximum

00:03:47.220 --> 00:03:49.578
at x equals,

00:03:49.578 --> 00:03:51.042
at x equals two.

00:03:51.042 --> 00:03:52.409
Actually, let's make it right.

00:03:52.409 --> 00:03:55.611
X has a relative maximum.

00:03:55.611 --> 00:03:56.962
We're just going to use
the words they are using.

00:03:56.962 --> 00:03:58.627
Relative,

00:03:58.627 --> 00:04:01.705
relative maximum

00:04:01.705 --> 00:04:05.019
at x is equal to two.

00:04:05.019 --> 00:04:06.956
Now another way that you
could have tried to do it

00:04:06.956 --> 00:04:07.901
is you could have tried to take

00:04:07.901 --> 00:04:09.315
the second derivative of this

00:04:09.315 --> 00:04:12.197
and then saw whether that
was positive or negative,

00:04:12.197 --> 00:04:14.043
and whether it was concave
upwards or downwards.

00:04:14.043 --> 00:04:15.928
But taking the second derivative of this,

00:04:15.928 --> 00:04:17.141
it gets quite hairy.

00:04:17.141 --> 00:04:18.849
And any time you're taking the AP test

00:04:18.849 --> 00:04:20.016
and you find yourself going down

00:04:20.016 --> 00:04:21.599
a really, really hairy path.

00:04:21.599 --> 00:04:23.583
Like taking the derivative of,

00:04:23.583 --> 00:04:25.650
or like taking the second derivative of f,

00:04:25.650 --> 00:04:27.860
which would be the first
derivative of f prime.

00:04:27.860 --> 00:04:30.598
Any time you see yourself going
down a hairy path like that,

00:04:30.598 --> 00:04:34.468
it might work, but it's
probably not the optimal path.

00:04:34.468 --> 00:04:36.186
So an easier way is to just think,

00:04:36.186 --> 00:04:39.345
okay, well what is f prime
doing as we approach this?

00:04:39.345 --> 00:04:40.264
Or another way is thinking,

00:04:40.264 --> 00:04:43.650
is the function increasing
as we approach from below?

00:04:43.650 --> 00:04:46.242
And is it increasing or decreasing

00:04:46.242 --> 00:04:50.742
as we get beyond that
point, beyond x equals two?


2015 AP Calculus BC 6c

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:02.720
- [Voiceover] Write
the first nonzero terms

00:00:02.720 --> 00:00:05.844
of the Maclaurin series for e to the x.

00:00:05.844 --> 00:00:08.363
Use the Maclaurin series for e to the x

00:00:08.363 --> 00:00:11.068
to write the third-degree
Taylor polynomial

00:00:11.068 --> 00:00:13.733
for g to the x is equal to e to the x

00:00:13.733 --> 00:00:16.914
times f of x about x equals zero.

00:00:16.914 --> 00:00:19.596
So Maclaurin series, if
that looks familiar to you,

00:00:19.596 --> 00:00:21.860
watch the videos on Khan
Academy on Maclaurin series.

00:00:21.860 --> 00:00:23.393
I'll give a little bit of primary here.

00:00:23.393 --> 00:00:26.017
For taking the Maclaurin series of f of x,

00:00:26.017 --> 00:00:29.941
that's going to be equal to f of zero.

00:00:29.941 --> 00:00:32.820
We could view this f of
zero times x to the zero

00:00:32.820 --> 00:00:34.399
over zero factorial,

00:00:34.399 --> 00:00:36.535
if you assume zero
factorial is equal to one,

00:00:36.535 --> 00:00:40.761
and then plus f prime of zero

00:00:40.761 --> 00:00:44.441
times x to the first over one factorial

00:00:45.411 --> 00:00:50.258
plus f prime prime, the second
derivative evaluated at zero,

00:00:50.258 --> 00:00:53.988
times x squared over two factorial.

00:00:53.988 --> 00:00:54.975
I think you see the pattern here.

00:00:54.975 --> 00:00:58.829
Plus f prime prime prime,
the third derivative

00:00:58.829 --> 00:01:01.255
evaluated at zero of x to the third power

00:01:01.255 --> 00:01:03.019
over three factorial.

00:01:03.019 --> 00:01:05.399
I think you see where this is going.

00:01:05.399 --> 00:01:07.199
And of course, this
first one, x to the zero

00:01:07.199 --> 00:01:08.662
over zero factorial, that's just one.

00:01:08.662 --> 00:01:11.460
So oftentimes it will
just be written f of zero.

00:01:11.460 --> 00:01:13.735
And this term one factorial is just one.

00:01:13.735 --> 00:01:15.689
So oftentimes it's just
a zero, and this f prime

00:01:15.689 --> 00:01:18.614
of zero times x, so on and so forth.

00:01:18.614 --> 00:01:20.931
So, actually, let me just write

00:01:20.931 --> 00:01:23.288
approximately right over there.

00:01:23.288 --> 00:01:26.248
And so let's do it for e to the x.

00:01:26.248 --> 00:01:30.405
So e to the x is approximately equal to,

00:01:30.405 --> 00:01:33.855
well it's going to be e
to the zero which is one.

00:01:33.855 --> 00:01:36.055
Plus, you might already know

00:01:36.055 --> 00:01:40.128
that f of x.

00:01:40.128 --> 00:01:42.729
If f of x is equal to e to the x,

00:01:42.729 --> 00:01:46.142
then f prime of x is
also equal to e to the x.

00:01:46.142 --> 00:01:48.476
That's one of the magical
things about e to the x.

00:01:48.476 --> 00:01:52.319
The slope of the tangent
line at any point is,

00:01:52.319 --> 00:01:55.314
well, equal to the value is
equal to the x value there.

00:01:55.314 --> 00:01:58.437
And if you take the third
derivative or second derivative,

00:01:58.437 --> 00:02:00.190
it's also, you're gonna get as
many derivatives as you want.

00:02:00.190 --> 00:02:01.537
You'll still get e to the x.

00:02:01.537 --> 00:02:03.429
That's one of the special things about e.

00:02:03.429 --> 00:02:05.473
So the first derivative evaluated zero,

00:02:05.473 --> 00:02:09.014
well that's still e to the zero
power times x to the first.

00:02:09.014 --> 00:02:11.556
So plus x.

00:02:11.556 --> 00:02:14.935
And then we have the second
derivative evaluated at zero.

00:02:14.935 --> 00:02:15.771
Well, that's still one.

00:02:15.771 --> 00:02:18.778
So it's going to be times,
or so it's one times

00:02:18.778 --> 00:02:20.357
x squared over two factorial.

00:02:20.357 --> 00:02:23.898
So we just say plus x squared over two,

00:02:23.898 --> 00:02:26.150
plus x squared over two.

00:02:26.150 --> 00:02:27.950
Two factorials is two times one

00:02:27.950 --> 00:02:29.238
and then plus.

00:02:29.238 --> 00:02:30.991
Once again, the third
derivative evaluated at zero,

00:02:30.991 --> 00:02:32.868
that's just e to the x evaluated at zero

00:02:32.868 --> 00:02:34.388
which is e to the zero which is one.

00:02:34.388 --> 00:02:36.966
So x to the third over three factorial.

00:02:36.966 --> 00:02:40.611
So plus x to the third
over three factorial.

00:02:40.611 --> 00:02:42.063
We could write it as three factorial

00:02:42.063 --> 00:02:43.607
or that's three times two times one.

00:02:43.607 --> 00:02:44.965
That's equal to six.

00:02:44.965 --> 00:02:46.683
And then we keep going.

00:02:46.683 --> 00:02:49.702
So we just wrote the
first four nonzero terms

00:02:49.702 --> 00:02:51.327
of the Maclaurin series for e to the x.

00:02:51.327 --> 00:02:54.265
That's one, two, three,
four nonzero terms.

00:02:54.265 --> 00:02:57.469
Now we wanna use, so let
me just underline that.

00:02:57.469 --> 00:02:59.965
That's part of what
they're asking us to do.

00:02:59.965 --> 00:03:03.367
Then they say use the
Maclaurin series for e to the x

00:03:03.367 --> 00:03:05.608
to write the third-degree
Taylor polynomial

00:03:05.608 --> 00:03:07.953
for g of x which is equal to the product

00:03:07.953 --> 00:03:09.903
of e to the x and f of x.

00:03:09.903 --> 00:03:11.889
So what I'm gonna do is I'm gonna write

00:03:11.889 --> 00:03:13.491
our original f of x.

00:03:13.491 --> 00:03:16.289
Let's write down the
first few terms of it.

00:03:16.289 --> 00:03:17.573
And then what we could
do this thing about,

00:03:17.573 --> 00:03:21.139
well, how can we multiply
those two polynomials?

00:03:21.139 --> 00:03:23.414
And we just have to know
enough about the multiplication

00:03:23.414 --> 00:03:24.750
of those two polynomials

00:03:24.750 --> 00:03:27.931
to just get us our

00:03:27.931 --> 00:03:30.566
terms that are no higher
than third degree.

00:03:30.566 --> 00:03:34.211
So f of x is approximately equal to,

00:03:34.211 --> 00:03:36.429
let's see, it's x, I have a bad memory,

00:03:36.429 --> 00:03:40.167
it is x minus 3/2 x squared.

00:03:41.157 --> 00:03:46.157
X minus 3/2

00:03:46.214 --> 00:03:49.659
x squared and then,

00:03:49.659 --> 00:03:52.882
plus three x to the third power,

00:03:52.882 --> 00:03:55.741
plus three x to the third power.

00:03:55.741 --> 00:03:58.439
Plus three x to the third power.

00:03:58.439 --> 00:04:01.446
Plus, and actually you
could say minus if you like

00:04:01.446 --> 00:04:02.723
because that is gonna be,

00:04:02.723 --> 00:04:05.788
you are gonna have, it's
plus minus plus minus,

00:04:05.788 --> 00:04:07.123
however you want us to do it.

00:04:07.123 --> 00:04:08.041
And that's enough.

00:04:08.041 --> 00:04:09.921
And, why do I feel confident
that that's enough?

00:04:09.921 --> 00:04:12.861
Well, we only want we only want to write

00:04:12.861 --> 00:04:15.959
the third degree Taylor polynomial.

00:04:15.959 --> 00:04:18.752
So if we multiply this
and we involve terms

00:04:18.752 --> 00:04:20.337
that are higher than third degree,

00:04:20.337 --> 00:04:23.268
well, we're gonna, that's going to give us

00:04:23.268 --> 00:04:25.039
the terms that are polynomial

00:04:25.039 --> 00:04:26.853
that are higher than third degree.

00:04:26.853 --> 00:04:28.024
So let's just think about it.

00:04:28.024 --> 00:04:29.336
What's gonna be the product?

00:04:29.336 --> 00:04:33.516
So e to the x times f of x.

00:04:33.516 --> 00:04:36.058
That's going to be approximately equal to,

00:04:36.058 --> 00:04:37.196
well let's see.

00:04:37.196 --> 00:04:40.760
We are going to multiply
this infinite polynomial

00:04:40.760 --> 00:04:42.676
times this infinite polynomial

00:04:42.676 --> 00:04:44.917
and that might seem
intimidating to you at first.

00:04:44.917 --> 00:04:48.214
What you could do is you could
go for each of these terms.

00:04:48.214 --> 00:04:49.386
Must start multiplying it

00:04:49.386 --> 00:04:50.884
times each of these terms at here.

00:04:50.884 --> 00:04:52.778
Essentially, you know, when
you're multiplying polynomials,

00:04:52.778 --> 00:04:54.763
you're just repeatedly doing
the distributive property

00:04:54.763 --> 00:04:57.526
so you take this and
distribute it on that.

00:04:57.526 --> 00:04:59.140
But we should only worry about the terms

00:04:59.140 --> 00:05:01.624
up to third degree because
anything beyond that,

00:05:01.624 --> 00:05:03.540
well, that's going to add up

00:05:03.540 --> 00:05:05.618
to a higher than third degree term.

00:05:05.618 --> 00:05:08.416
So x times one is x.

00:05:08.416 --> 00:05:10.854
X times x squared,

00:05:10.854 --> 00:05:13.269
or sorry, x times x is x squared

00:05:13.269 --> 00:05:14.988
so plus x squared.

00:05:14.988 --> 00:05:17.135
X times x squared over two is

00:05:17.135 --> 00:05:20.735
x to the third over two.

00:05:20.735 --> 00:05:23.475
So I'll just write plus 1/2

00:05:23.475 --> 00:05:26.516
x to the third power.

00:05:26.516 --> 00:05:27.376
And I'm gonna stop there

00:05:27.376 --> 00:05:28.571
because if I do x times that,

00:05:28.571 --> 00:05:29.814
that's gonna be fourth degree term

00:05:29.814 --> 00:05:30.870
and I don't wanna worry about that.

00:05:30.870 --> 00:05:33.134
We are writing the
third-degree Taylor polynomial.

00:05:33.134 --> 00:05:37.685
So that's x, x squared,
x to the third over two.

00:05:37.685 --> 00:05:42.685
So now let's distribute
the negative 3/2 x squared.

00:05:43.646 --> 00:05:44.953
I'll use another color here

00:05:44.953 --> 00:05:47.441
just to help explain it a little bit.

00:05:47.441 --> 00:05:49.984
So if you multiply that times one,

00:05:49.984 --> 00:05:54.767
that's going to be minus 3/2 x squared.

00:05:54.767 --> 00:05:55.986
I'm just going to a second line here

00:05:55.986 --> 00:05:58.575
so I can line things
up and add them nicely.

00:05:58.575 --> 00:06:01.478
And then this times this is

00:06:01.478 --> 00:06:04.276
negative 3/2 x to the third.

00:06:04.276 --> 00:06:07.538
Negative 3/2 x to the third.

00:06:07.538 --> 00:06:08.384
And once again, I'm gonna start there.

00:06:08.384 --> 00:06:10.963
If I multiply these two, I'm
gonna get a fourth degree term.

00:06:10.963 --> 00:06:13.111
I don't care about the
fourth degree terms.

00:06:13.111 --> 00:06:16.745
And then, and then,

00:06:16.745 --> 00:06:19.061
let's do, let's worry about this guy.

00:06:19.061 --> 00:06:20.292
And so let's start distributing.

00:06:20.292 --> 00:06:22.207
So if I multiply it times this one,

00:06:22.207 --> 00:06:24.251
I'm gonna get three x to the third power.

00:06:24.251 --> 00:06:25.470
Three x to the third power.

00:06:25.470 --> 00:06:26.248
And I'm gonna stop there.

00:06:26.248 --> 00:06:28.105
Because then if I start
multiplying it times that guy,

00:06:28.105 --> 00:06:29.626
it's getting me a fourth degree term

00:06:29.626 --> 00:06:30.787
and then a fifth degree term

00:06:30.787 --> 00:06:31.762
and then a sixth degree term

00:06:31.762 --> 00:06:33.167
which I don't need to worry about.

00:06:33.167 --> 00:06:35.710
So these are all the pieces
that are going to make up

00:06:35.710 --> 00:06:37.532
that third degree polynomial.

00:06:37.532 --> 00:06:40.616
And so, what is that going to be equal to?

00:06:40.616 --> 00:06:41.830
What is that going to be equal to?

00:06:41.830 --> 00:06:44.983
This is a little bit of,
you know, getting your math

00:06:44.983 --> 00:06:48.234
intuition for multiplying
infinite polynomials.

00:06:48.234 --> 00:06:50.335
Let's see, you're gonna have x.

00:06:50.335 --> 00:06:51.682
And then you're going to have,

00:06:51.682 --> 00:06:55.757
this is one x squared
minus 1 1/2 x squared.

00:06:55.757 --> 00:06:59.902
So that's going to be
negative 1/2 x squared.

00:06:59.902 --> 00:07:00.993
Now let's see.

00:07:00.993 --> 00:07:04.696
Here, you have plus 1/2 minus 1 1/2

00:07:04.696 --> 00:07:07.494
which would give you negative two

00:07:07.494 --> 00:07:10.037
and then plus three,

00:07:10.037 --> 00:07:11.189
plus, no, sorry,

00:07:11.189 --> 00:07:14.207
1/2 minus 1 1/2

00:07:14.207 --> 00:07:16.220
which would be negative one

00:07:16.220 --> 00:07:18.240
plus three is positive two

00:07:18.240 --> 00:07:21.119
plus two x to the third power.

00:07:21.119 --> 00:07:24.707
So we could say e to the x times f of x,

00:07:24.707 --> 00:07:28.979
the third degree Taylor
polynomial for this is

00:07:28.979 --> 00:07:32.422
x minus 1/2 x squared

00:07:32.422 --> 00:07:35.185
plus two x to the third power.

00:07:35.185 --> 00:07:37.983
And we are done.

00:07:37.983 --> 00:07:40.503
So this was a little bit tricky.

00:07:40.503 --> 00:07:42.662
You have to appreciate how
to, and besides calculus,

00:07:42.662 --> 00:07:44.079
it's a little bit of just algebra.

00:07:44.079 --> 00:07:45.681
I just appreciate, okay, I
only need the third degree.

00:07:45.681 --> 00:07:46.545
I don't have to distribute this

00:07:46.545 --> 00:07:48.444
times the infinite number of terms.

00:07:48.444 --> 00:07:49.923
Because, at first you might
say, "That's super hard.

00:07:49.923 --> 00:07:51.827
"How do I multiply two
infinite polynomials?"

00:07:51.827 --> 00:07:54.109
The key is we only worry
about the third degree,

00:07:54.109 --> 00:07:55.924
up to the third degree.


2015 AP Calculus BC 5c

WEBVTT
Kind: captions
Language: en

00:00:00.100 --> 00:00:01.344
- [Voiceover] Find the value of k

00:00:01.344 --> 00:00:05.347
for which f has a critical
point at x equals negative five.

00:00:05.347 --> 00:00:07.082
All right, so let's just remind ourselves

00:00:07.082 --> 00:00:09.807
what f of x and f prime of x were.

00:00:09.807 --> 00:00:11.646
They gave it to us at the top,

00:00:11.646 --> 00:00:14.979
f of x is equal to one
over x squared minus kx,

00:00:14.979 --> 00:00:17.840
and then f prime of x is
equal to all of this business.

00:00:17.840 --> 00:00:18.862
Let me rewrite it,

00:00:18.862 --> 00:00:21.230
let me rewrite it down here,

00:00:21.230 --> 00:00:23.787
so f,

00:00:23.787 --> 00:00:27.503
f of x is equal to one over

00:00:27.503 --> 00:00:30.113
x squared minus kx

00:00:30.113 --> 00:00:34.153
and f prime of x is equal to

00:00:34.153 --> 00:00:37.000
k minus two x

00:00:37.000 --> 00:00:41.394
over x squared minus kx,

00:00:41.394 --> 00:00:43.631
whole quantity squared.

00:00:43.631 --> 00:00:45.690
And we wanna find the value of k

00:00:45.690 --> 00:00:50.441
for which f has a critical
point at x equals negative five,

00:00:50.441 --> 00:00:54.535
so f has a critical point
at x equals negative five.

00:00:54.535 --> 00:00:58.426
This means that x,

00:00:58.426 --> 00:01:03.426
that means that x equals
negative five is in domain,

00:01:03.507 --> 00:01:06.646
so this is in domain,

00:01:06.646 --> 00:01:08.345
which means our function itself

00:01:08.345 --> 00:01:12.488
is defined at x equals negative five,

00:01:12.488 --> 00:01:17.488
and it means that f prime of negative five

00:01:17.764 --> 00:01:21.897
is equal to zero or undefined,

00:01:21.897 --> 00:01:23.977
or undefined.

00:01:23.977 --> 00:01:27.368
A critical point is a member
of the functions domain

00:01:27.368 --> 00:01:29.893
where the derivative is equal to zero,

00:01:29.893 --> 00:01:31.947
or it's undefined.

00:01:31.947 --> 00:01:34.841
So let's evaluate f
prime of negative five.

00:01:34.841 --> 00:01:38.373
F prime of negative five in terms of k

00:01:38.373 --> 00:01:42.869
is going to be equal to k
minus two times negative five,

00:01:42.869 --> 00:01:45.255
minus two times negative five,

00:01:45.255 --> 00:01:46.501
all of that over

00:01:46.501 --> 00:01:51.501
negative five squared minus k

00:01:51.698 --> 00:01:54.799
times negative five,

00:01:54.799 --> 00:01:57.646
and then we want to square all of this,

00:01:57.646 --> 00:02:01.346
and so this is going to
be equal to k plus 10,

00:02:01.346 --> 00:02:04.649
k plus 10 over,

00:02:04.649 --> 00:02:08.076
we have 25, what is this?

00:02:08.076 --> 00:02:13.018
25 plus five k,

00:02:13.018 --> 00:02:17.152
plus five k squared.

00:02:17.152 --> 00:02:20.366
So what value of k makes f prime

00:02:20.366 --> 00:02:23.202
of negative five equal to zero?

00:02:23.202 --> 00:02:26.329
Well, f prime of negative five

00:02:26.329 --> 00:02:28.388
is equal to zero

00:02:28.388 --> 00:02:32.502
if k is equal to negative ten.

00:02:32.502 --> 00:02:34.610
So that's a value of k for which f has

00:02:34.610 --> 00:02:36.979
a critical point at negative five.

00:02:36.979 --> 00:02:38.372
Now, you might be saying,

00:02:38.372 --> 00:02:39.700
well, what values of k?

00:02:39.700 --> 00:02:40.866
That's the value of k that makes

00:02:40.866 --> 00:02:42.102
the function equal zero,

00:02:42.102 --> 00:02:43.225
it makes the numerator equal zero,

00:02:43.225 --> 00:02:45.274
and therefore makes the
whole function equal zero.

00:02:45.274 --> 00:02:49.470
But why can't I pick a value
of k that makes the derivative,

00:02:49.470 --> 00:02:53.256
or it makes the numerator
of the derivative zero,

00:02:53.256 --> 00:02:55.211
and so therefore makes
the derivative equal zero,

00:02:55.211 --> 00:02:56.154
I think I said function,

00:02:56.154 --> 00:02:57.891
not the derivative of the function.

00:02:57.891 --> 00:02:58.727
But you might be saying,

00:02:58.727 --> 00:03:00.662
well, why can't I pick a value of k

00:03:00.662 --> 00:03:03.981
that makes the derivative undefined?

00:03:03.981 --> 00:03:05.914
And so, you can think of what that is,

00:03:05.914 --> 00:03:07.271
what would make undefined,

00:03:07.271 --> 00:03:09.900
well, if 25 plus five k is equal to zero,

00:03:09.900 --> 00:03:11.448
well, then this is going to be,

00:03:11.448 --> 00:03:13.170
you're gonna have zero squared,

00:03:13.170 --> 00:03:14.051
and divide it by zero,

00:03:14.051 --> 00:03:15.296
it's going to be undefined,

00:03:15.296 --> 00:03:19.817
so you could say f prime of negative five

00:03:19.817 --> 00:03:21.733
undefined,

00:03:21.733 --> 00:03:26.098
undefined if k is equal to what?

00:03:26.098 --> 00:03:28.413
Negative five, right?

00:03:28.413 --> 00:03:31.207
Five times negative five plus 25 is zero,

00:03:31.207 --> 00:03:32.306
but this,

00:03:32.306 --> 00:03:34.825
if k is equal to negative five,

00:03:34.825 --> 00:03:37.593
then this can't be a
critical point anymore,

00:03:37.593 --> 00:03:39.641
x equals negative five
can't be a critical point

00:03:39.641 --> 00:03:42.070
because it won't be in
it's domain anymore.

00:03:42.070 --> 00:03:43.779
If k is equal to negative five,

00:03:43.779 --> 00:03:45.980
then f of x would be equal to one

00:03:45.980 --> 00:03:50.120
over x squared plus five x,

00:03:50.120 --> 00:03:51.215
plus five x,

00:03:51.215 --> 00:03:54.368
and then f will not be
defined at negative five,

00:03:54.368 --> 00:03:56.123
and so the negative five
couldn't be a critical point

00:03:56.123 --> 00:03:58.808
because it's not even in the domain.

00:03:58.808 --> 00:04:00.238
So the important thing is,

00:04:00.238 --> 00:04:01.585
in order to be a critical point,

00:04:01.585 --> 00:04:02.987
it has to be in the domain,

00:04:02.987 --> 00:04:05.086
and the derivative at that point

00:04:05.086 --> 00:04:07.130
has to be equal to zero undefined.

00:04:07.130 --> 00:04:09.712
I can get the derivative to
be undefined at negative five,

00:04:09.712 --> 00:04:11.505
if we said k equals negative five,

00:04:11.505 --> 00:04:13.402
but with we said k equals negative five,

00:04:13.402 --> 00:04:16.928
then x equals negative five
will no longer be in the domain,

00:04:16.928 --> 00:04:20.068
so we wanna go with the k
that just makes the numerator

00:04:20.068 --> 00:04:21.582
of our derivative equal to zero,

00:04:21.582 --> 00:04:24.628
or sets our entire
derivative equal to zero.


2015 AP Calculus BC 6a

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.020
- [Voiceover] The Maclaurin
series for a function f

00:00:02.020 --> 00:00:04.544
is given by, they give it
to us in sigma notation

00:00:04.544 --> 00:00:06.883
and then they expand it out for us,

00:00:06.883 --> 00:00:08.551
and converges to f of x

00:00:08.551 --> 00:00:11.525
for the absolute value
of x being less than R,

00:00:11.525 --> 00:00:13.917
where R is the radius of convergence

00:00:13.917 --> 00:00:16.108
of the Maclaurin series.

00:00:16.108 --> 00:00:17.275
Part a.

00:00:17.275 --> 00:00:20.128
Use the ratio test to find R.

00:00:20.128 --> 00:00:22.645
So first of all, if terms
like Maclaurin series

00:00:22.645 --> 00:00:23.961
and radius of convergence

00:00:23.961 --> 00:00:25.985
or even convergence or ratio test

00:00:25.985 --> 00:00:28.992
seem foreign to you or you
have some foggy memories of it,

00:00:28.992 --> 00:00:30.752
you might want to review
all of those concepts

00:00:30.752 --> 00:00:31.659
on Khan Academy.

00:00:31.659 --> 00:00:34.155
We actually have multiple
videos and exercises

00:00:34.155 --> 00:00:37.026
on each of these concepts on Khan Academy.

00:00:37.026 --> 00:00:38.281
But if you kind of know what it is,

00:00:38.281 --> 00:00:40.086
I will give you a little bit of a reminder

00:00:40.086 --> 00:00:42.228
for the ratio test.

00:00:42.228 --> 00:00:47.139
So the ratio test tells us if
we have an infinite series,

00:00:47.139 --> 00:00:50.022
so we go from n equals one to infinity,

00:00:50.022 --> 00:00:51.901
and each term is a sub n,

00:00:51.901 --> 00:00:53.924
the ratio test,

00:00:53.924 --> 00:00:56.386
ratio test,

00:00:56.386 --> 00:00:59.782
says alright, let's consider the ratio

00:00:59.782 --> 00:01:01.307
between successive terms.

00:01:01.307 --> 00:01:05.002
So we could say the
ratio of a sub n plus one

00:01:05.002 --> 00:01:07.262
over a sub n

00:01:07.262 --> 00:01:08.592
and in particular we want to focus on

00:01:08.592 --> 00:01:11.938
the absolute value of this ratio.

00:01:11.938 --> 00:01:14.128
And this by itself, it
might not be a constant

00:01:14.128 --> 00:01:15.384
like we would see in a geometric series,

00:01:15.384 --> 00:01:18.222
it actually might be a
function of n itself.

00:01:18.222 --> 00:01:20.671
And so we want to see the
behavior of this ratio

00:01:20.671 --> 00:01:22.395
as n gets really, really, really large,

00:01:22.395 --> 00:01:23.980
as we're kind of adding those terms

00:01:23.980 --> 00:01:25.739
as we're getting close to infinity.

00:01:25.739 --> 00:01:30.559
So we want to take the limit
as n approaches infinity here.

00:01:30.559 --> 00:01:33.867
And if this limit
exists, let's say it's L,

00:01:33.867 --> 00:01:38.700
and if L is less than one,

00:01:38.700 --> 00:01:41.813
then the series,

00:01:41.813 --> 00:01:45.923
the series converges.

00:01:45.923 --> 00:01:47.035
So what we're gonna want to do,

00:01:47.035 --> 00:01:49.203
and if L is greater than one, it diverges,

00:01:49.203 --> 00:01:52.380
if it's equal to one, it's inconclusive.

00:01:52.380 --> 00:01:54.083
And so what we want to do here is

00:01:54.083 --> 00:01:56.422
we want to figure out the
absolute value of the ratio here,

00:01:56.422 --> 00:01:59.184
take the limit and then see

00:01:59.184 --> 00:02:02.023
for what x values does that limit,

00:02:02.023 --> 00:02:05.113
will that limit be less than one?

00:02:05.113 --> 00:02:06.061
So let's do that.

00:02:06.061 --> 00:02:08.540
So let's first think about this ratio.

00:02:08.540 --> 00:02:11.483
So a sub n plus one

00:02:11.483 --> 00:02:13.713
over a sub n

00:02:13.713 --> 00:02:15.372
is going to be equal to...

00:02:15.372 --> 00:02:18.298
So if we put n plus one
into this expression here,

00:02:18.298 --> 00:02:19.505
we're gonna have...

00:02:19.505 --> 00:02:21.084
so let me make this clear,

00:02:21.084 --> 00:02:22.594
So I'm gonna do a sub n plus one up here,

00:02:22.594 --> 00:02:25.622
so we're gonna have negative three,

00:02:25.622 --> 00:02:28.352
and I could write to the n
plus one instead of an n,

00:02:28.352 --> 00:02:29.861
and then minus one, so it would be

00:02:29.861 --> 00:02:33.134
n plus one minus one.

00:02:33.134 --> 00:02:34.947
Well plus one minus one is just zero

00:02:34.947 --> 00:02:35.933
so that's just the same thing as

00:02:35.933 --> 00:02:40.059
negative three to the n,
times x to the n plus one,

00:02:40.059 --> 00:02:42.726
x to the n plus one,

00:02:42.726 --> 00:02:45.960
over n plus one.

00:02:45.960 --> 00:02:47.893
So that's a sub n plus one there.

00:02:47.893 --> 00:02:49.956
And a sub n, well that's just

00:02:49.956 --> 00:02:52.854
negative three to the n minus one

00:02:52.854 --> 00:02:57.135
times x to the n over n.

00:02:57.135 --> 00:03:00.470
So what does this right
over here simplify to?

00:03:00.470 --> 00:03:02.787
This is going to be equal to,

00:03:02.787 --> 00:03:05.415
well we could just say
this divided by that's

00:03:05.415 --> 00:03:06.971
the same thing as multiplying by

00:03:06.971 --> 00:03:08.922
the reciprocal of all
of this stuff down here.

00:03:08.922 --> 00:03:13.519
So it's negative three to the n,

00:03:13.519 --> 00:03:16.007
x to the n plus one

00:03:16.007 --> 00:03:18.657
over n plus one,

00:03:18.657 --> 00:03:21.553
times the reciprocal of this business.

00:03:21.553 --> 00:03:26.399
So times n over

00:03:26.399 --> 00:03:29.809
negative three to the n minus one,

00:03:29.809 --> 00:03:31.775
x to the n.

00:03:31.775 --> 00:03:33.672
And so can we simplify this?

00:03:33.672 --> 00:03:35.715
Well we can divide both the
numerator and the denominator

00:03:35.715 --> 00:03:37.273
here by x to the n.

00:03:37.273 --> 00:03:39.494
So this divided by x to the n is just one.

00:03:39.494 --> 00:03:43.032
This divided by x to
the n is going to be x,

00:03:43.032 --> 00:03:44.771
or x to the first power.

00:03:44.771 --> 00:03:46.197
And we can divide the
numerator and the denominator

00:03:46.197 --> 00:03:48.476
by negative three to the n minus one.

00:03:48.476 --> 00:03:49.915
Well this is just going to be one.

00:03:49.915 --> 00:03:51.624
And if you divide negative three to the n

00:03:51.624 --> 00:03:54.104
by negative three to the n
minus one, well that's just

00:03:54.104 --> 00:03:56.276
going to be negative
three to the first power.

00:03:56.276 --> 00:03:58.084
So this is all going to be

00:03:58.084 --> 00:04:02.304
negative three x n

00:04:02.304 --> 00:04:07.244
over n plus one.

00:04:07.244 --> 00:04:08.923
So now let's think about what the limit

00:04:08.923 --> 00:04:10.859
of the absolute value of this

00:04:10.859 --> 00:04:12.885
as n approaches infinity is.

00:04:12.885 --> 00:04:14.950
So the limit

00:04:14.950 --> 00:04:18.550
the limit as n approaches infinity

00:04:18.550 --> 00:04:20.356
of the absolute value

00:04:20.356 --> 00:04:24.016
of negative three x n

00:04:24.016 --> 00:04:26.716
over n plus one.

00:04:26.716 --> 00:04:28.858
Now some of you might
recognize if we focus on n,

00:04:28.858 --> 00:04:31.607
we have the same degree up
here, same degree down here,

00:04:31.607 --> 00:04:34.373
it's both n to the first,
so these are going to,

00:04:34.373 --> 00:04:37.144
the n over n plus one is
going to approach one.

00:04:37.144 --> 00:04:38.365
And so you might say "Okay,
well this is going to be

00:04:38.365 --> 00:04:40.858
the absolute value of negative three x",

00:04:40.858 --> 00:04:42.589
but if you want to make
that a little bit clearer

00:04:42.589 --> 00:04:44.470
this is equal to the limit

00:04:44.470 --> 00:04:48.460
as n approaches infinity of,

00:04:48.460 --> 00:04:51.745
and I'll write it this way,

00:04:51.745 --> 00:04:55.098
so I could write the absolute
value of negative three x

00:04:55.098 --> 00:04:58.057
or the absolute value of negative
three x is the same thing

00:04:58.057 --> 00:05:01.827
as three times the absolute value of x

00:05:01.827 --> 00:05:06.015
times the absolute value of,

00:05:06.015 --> 00:05:08.816
if I divide the numerator
here by n, I would get one

00:05:08.816 --> 00:05:11.656
and if I divide the
denominator by n, I could do...

00:05:11.656 --> 00:05:14.619
if I multiply or divide the
numerator and the denominator

00:05:14.619 --> 00:05:16.902
by the same thing, I'm
not changing the value.

00:05:16.902 --> 00:05:19.151
So if I divide both of them by n,

00:05:19.151 --> 00:05:20.334
in the numerator I just get one,

00:05:20.334 --> 00:05:24.122
in the denominator, n divided by n is one,

00:05:24.122 --> 00:05:27.979
one divided by n is plus one over n.

00:05:27.979 --> 00:05:30.191
And so this might be a
little bit clearer that

00:05:30.191 --> 00:05:32.509
as n approaches infinity,
well we don't know,

00:05:32.509 --> 00:05:34.103
this doesn't deal with n.

00:05:34.103 --> 00:05:37.379
But this over here, one over
n is going to approach zero,

00:05:37.379 --> 00:05:39.353
and so this whole thing
is going to approach one

00:05:39.353 --> 00:05:40.898
and so the limit is going to be

00:05:40.898 --> 00:05:44.246
three times the absolute value of x.

00:05:44.246 --> 00:05:47.529
And so remember, this series converges

00:05:47.529 --> 00:05:51.757
if this limit is less than one.

00:05:51.757 --> 00:05:53.944
So converges,

00:05:53.944 --> 00:05:55.954
converges,

00:05:55.954 --> 00:06:00.294
converges if three times
the absolute value of x

00:06:00.294 --> 00:06:01.758
is less than one.

00:06:01.758 --> 00:06:04.126
Or we could say the absolute value of x,

00:06:04.126 --> 00:06:05.995
just divide both sides by three,

00:06:05.995 --> 00:06:08.352
is less than one third.

00:06:08.352 --> 00:06:12.703
And so we have just found
our radius of convergence.

00:06:12.703 --> 00:06:16.066
So we could say R is equal to one third.

00:06:16.066 --> 00:06:18.263
This Maclaurin series is going to converge

00:06:18.263 --> 00:06:20.689
as long as the absolute value of x

00:06:20.689 --> 00:06:23.209
is less than one third.

00:06:23.209 --> 00:06:28.021
Or we could say our radius
of convergence is equal to,

00:06:28.021 --> 00:06:31.520
our radius of convergence
is equal to one third.

00:06:31.520 --> 00:06:33.473
So there you go.


2015 AP Calculus BC 6b

WEBVTT
Kind: captions
Language: en

00:00:00.193 --> 00:00:03.008
- [Voiceover] Part b,
write the first four,

00:00:03.008 --> 00:00:05.394
write the first nonzero terms

00:00:05.394 --> 00:00:09.741
of the Maclaurin series for
f prime, the derivative of f.

00:00:09.741 --> 00:00:12.369
Express f prime as a rational function

00:00:12.369 --> 00:00:15.714
for the absolute value
of x being less than R,

00:00:15.714 --> 00:00:17.756
our radius of convergence.

00:00:17.756 --> 00:00:19.837
So, if we wanna find f prime,

00:00:19.837 --> 00:00:21.211
we could just take each of,

00:00:21.211 --> 00:00:22.327
we could just take the derivative

00:00:22.327 --> 00:00:25.044
of each of these terms with respect to x.

00:00:25.044 --> 00:00:27.119
And so, we could just say,

00:00:27.119 --> 00:00:30.407
if this is f, then f prime,

00:00:30.407 --> 00:00:31.489
and I'm not gonna scroll down

00:00:31.489 --> 00:00:33.044
just so I can see this up here.

00:00:33.044 --> 00:00:35.534
We could just say f prime,

00:00:35.534 --> 00:00:39.258
f prime of x is going to be,

00:00:39.258 --> 00:00:41.301
or the Maclaurin series for f prime of x,

00:00:41.301 --> 00:00:43.131
maybe I should write it that way,

00:00:43.131 --> 00:00:45.035
so let me write it this way.

00:00:45.035 --> 00:00:46.204
Maclaurin,

00:00:47.338 --> 00:00:49.009
Maclaurin,

00:00:49.864 --> 00:00:51.276
Maclaurin,

00:00:51.276 --> 00:00:52.716
that a looked funny,

00:00:52.716 --> 00:00:56.856
Maclaurin series

00:00:56.856 --> 00:01:00.414
for f prime,

00:01:00.414 --> 00:01:02.913
f prime of x,

00:01:02.913 --> 00:01:04.844
well, it's going to be the sum

00:01:06.192 --> 00:01:10.093
from n equals one to infinity,

00:01:10.093 --> 00:01:12.693
and we would just take the derivative

00:01:12.693 --> 00:01:15.517
of this right over here with respect to x.

00:01:15.517 --> 00:01:18.545
And so, this is just
application of the power rule.

00:01:18.545 --> 00:01:20.560
Take the exponent right over here,

00:01:20.560 --> 00:01:22.334
multiply it by the coefficient,

00:01:22.334 --> 00:01:23.997
so if you take n times this,

00:01:23.997 --> 00:01:25.678
it cancels out with this n,

00:01:25.678 --> 00:01:28.446
so it's going to be negative three

00:01:28.446 --> 00:01:30.591
to the n minus one,

00:01:30.591 --> 00:01:32.254
and then decrement your exponent,

00:01:32.254 --> 00:01:35.969
times x to the n minus one.

00:01:35.969 --> 00:01:37.957
And so, they want us
to write the first four

00:01:37.957 --> 00:01:40.483
nonzero terms of the Maclaurin series,

00:01:40.483 --> 00:01:43.465
so that is going to be equal to,

00:01:43.465 --> 00:01:45.193
so, I'll write approximately equal to,

00:01:45.193 --> 00:01:46.791
because we're one going
to write the first four

00:01:46.791 --> 00:01:49.188
of terms of this infinite series,

00:01:49.188 --> 00:01:50.310
and just to be clear what I did,

00:01:50.310 --> 00:01:51.908
I just did the power rule here,

00:01:51.908 --> 00:01:53.737
I looked at this exponent, which is n,

00:01:53.737 --> 00:01:55.576
multiplied by this coefficient,

00:01:55.576 --> 00:01:56.756
which had a n in the denominator,

00:01:56.756 --> 00:01:59.105
so that n and this n cancel out,

00:01:59.105 --> 00:02:01.649
so I'm just left with negative
three to the n minus one,

00:02:01.649 --> 00:02:03.507
and then I decrement that exponent.

00:02:03.507 --> 00:02:04.673
That's straight out of the power rule,

00:02:04.673 --> 00:02:06.096
it's one of the first things you'll learn

00:02:06.096 --> 00:02:07.632
about taking derivatives.

00:02:07.632 --> 00:02:10.559
And so, if we want the
first four nonzero terms,

00:02:10.559 --> 00:02:13.233
when n equals one this is going to be

00:02:13.233 --> 00:02:15.657
negative three to the one minus one power,

00:02:15.657 --> 00:02:16.446
let me just write it,

00:02:16.446 --> 00:02:19.121
negative three to the one minus one

00:02:19.121 --> 00:02:22.344
times x to the one minus one,

00:02:22.344 --> 00:02:24.137
that's when n equals one,

00:02:24.137 --> 00:02:27.137
plus negative three,

00:02:27.137 --> 00:02:30.459
negative three to the two minus one,

00:02:30.459 --> 00:02:31.879
two minus one,

00:02:31.879 --> 00:02:35.823
times x to the two minus one,

00:02:35.823 --> 00:02:36.733
and then,

00:02:36.733 --> 00:02:38.136
and actually I could've reassured

00:02:38.136 --> 00:02:40.123
this is negative three
x to the n minus one,

00:02:40.123 --> 00:02:42.157
actually let me me do that just for,

00:02:42.157 --> 00:02:44.967
so this is, this I could just write,

00:02:44.967 --> 00:02:46.365
because they have the same exponent,

00:02:46.365 --> 00:02:47.739
this will simplify a little bit,

00:02:47.739 --> 00:02:51.047
as negative three x

00:02:51.047 --> 00:02:53.787
to the n minus one,

00:02:53.787 --> 00:02:56.026
and so this going to be approximately,

00:02:56.026 --> 00:02:58.245
when n is equal to one,

00:02:58.245 --> 00:03:00.056
this is gonna be equal to zero,

00:03:00.056 --> 00:03:03.892
so negative three x to the zero
power is just gonna be one,

00:03:03.892 --> 00:03:08.536
when n is equal to two,

00:03:08.536 --> 00:03:10.264
this is going to be two minus one,

00:03:10.264 --> 00:03:11.907
so it's gonna be the first power,

00:03:11.907 --> 00:03:14.202
so negative three x to the first power,

00:03:14.202 --> 00:03:17.998
so I could just write
this as negative three x,

00:03:17.998 --> 00:03:20.190
and then when n is three,

00:03:20.190 --> 00:03:23.552
well, this is going to be
negative three x squared,

00:03:23.552 --> 00:03:26.403
so negative three x
squared is going to be,

00:03:26.403 --> 00:03:30.676
negative three squared is nine x squared,

00:03:30.676 --> 00:03:32.691
and then the fourth term

00:03:32.691 --> 00:03:35.812
is gonna be negative three x
to the four minus one power,

00:03:35.812 --> 00:03:37.317
so to the third power,

00:03:37.317 --> 00:03:41.394
so negative three to the
third power is negative 27

00:03:41.394 --> 00:03:45.462
times x to the third power.

00:03:45.462 --> 00:03:46.345
So there you have it,

00:03:46.345 --> 00:03:49.874
that's the first four nonzero
terms of the Maclaurin series.

00:03:49.874 --> 00:03:51.462
You could've also just looked over here,

00:03:51.462 --> 00:03:53.301
and said okay, the derivative of x,

00:03:53.301 --> 00:03:55.447
with respect to x is one,

00:03:55.447 --> 00:03:57.685
derivative of negative
three have x squared,

00:03:57.685 --> 00:04:01.371
with respect to x is negative
three, you could've said,

00:04:01.371 --> 00:04:05.144
the derivative of this is nine
x squared, right over there,

00:04:05.144 --> 00:04:07.252
and then you'd have to
write out the fourth term

00:04:07.252 --> 00:04:09.109
and take out the
derivative in the same way,

00:04:09.109 --> 00:04:10.744
and you'd have gotten
this right over here.

00:04:10.744 --> 00:04:11.896
So we did the first part,

00:04:11.896 --> 00:04:14.274
we wrote the first four nonzero terms

00:04:14.274 --> 00:04:15.973
of the Maclaurin series for f prime,

00:04:15.973 --> 00:04:17.097
the derivative of f,

00:04:17.097 --> 00:04:17.930
and then they said,

00:04:17.930 --> 00:04:20.369
express f prime as a rational function

00:04:20.369 --> 00:04:24.054
for the absolute value
of x being less than R.

00:04:24.054 --> 00:04:26.313
So this sum,

00:04:26.313 --> 00:04:27.424
if we assume it converges,

00:04:27.424 --> 00:04:31.040
and we know the radius of
convergence already, so,

00:04:31.040 --> 00:04:32.772
or assuming that we're dealing

00:04:32.772 --> 00:04:35.874
with x that are withing
the radius of convergence.

00:04:35.874 --> 00:04:37.406
So this right over here,

00:04:37.406 --> 00:04:39.041
you might recognize this,

00:04:39.041 --> 00:04:40.369
so I could write it like that,

00:04:40.369 --> 00:04:43.308
or I could also write it,

00:04:43.308 --> 00:04:47.270
if I start at n equals zero,

00:04:47.270 --> 00:04:48.513
so I could also write this as

00:04:48.513 --> 00:04:51.819
from n equals one to infinity of,

00:04:51.819 --> 00:04:55.831
or actually, from n
equals zero to infinity

00:04:55.831 --> 00:05:00.810
of negative three x to the n minus one,

00:05:00.810 --> 00:05:01.845
either, I'm sorry,

00:05:01.845 --> 00:05:03.085
to the n,

00:05:03.085 --> 00:05:05.500
because now the first term
was to the zero power,

00:05:05.500 --> 00:05:07.005
that first term was to zero power,

00:05:07.005 --> 00:05:09.243
so whether you do one minus one,

00:05:09.243 --> 00:05:11.565
is where you start, or you just at zero,

00:05:11.565 --> 00:05:13.432
these two things are equivalent.

00:05:13.432 --> 00:05:15.958
You might recognize these
as a geometrix series

00:05:15.958 --> 00:05:19.256
with common ration of negative,

00:05:19.256 --> 00:05:21.681
of negative three x.

00:05:21.681 --> 00:05:24.448
And so, what's the sum
of a geometric series

00:05:24.448 --> 00:05:27.076
with common ration, with
a certain common ration?

00:05:27.076 --> 00:05:30.336
Well, it's going to be
equal to the first term,

00:05:30.336 --> 00:05:31.860
and regardless of how you view this,

00:05:31.860 --> 00:05:34.181
the first term is going to be one

00:05:34.181 --> 00:05:37.441
divided by one minus the common ratio.

00:05:37.441 --> 00:05:39.721
So our common ratio is negative three x,

00:05:39.721 --> 00:05:41.463
one minus negative three x,

00:05:41.463 --> 00:05:45.540
well, that's just going
to be one plus three x.

00:05:45.540 --> 00:05:48.660
If what I just did here
looks unfamiliar to you,

00:05:48.660 --> 00:05:50.257
I encourage you to watch the sum

00:05:50.257 --> 00:05:52.579
of infinite geometric series,

00:05:52.579 --> 00:05:53.796
and not only do we show you

00:05:53.796 --> 00:05:55.106
this formula and how to apply it,

00:05:55.106 --> 00:05:57.018
but we show how you
can prove this formula,

00:05:57.018 --> 00:05:59.048
it's actually a pretty fun prove.

00:05:59.048 --> 00:05:59.818
But anyway,

00:05:59.818 --> 00:06:02.100
regardless of how you view
this Maclaurin series,

00:06:02.100 --> 00:06:04.069
it is an infinite geometric series,

00:06:04.069 --> 00:06:06.202
and this assuming that our x

00:06:06.202 --> 00:06:07.771
is in our radius of convergence.

00:06:07.771 --> 00:06:09.916
This is what our sum is,

00:06:09.916 --> 00:06:13.148
this is what we are going to converge to.


Sal interviews the AP Calculus Lead at College Board

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:02.660
- So this is Sal Khan,
founder of the Khan Academy.

00:00:02.660 --> 00:00:05.680
And this is a very exciting
Skype call that we're on.

00:00:05.680 --> 00:00:07.260
I'm with Ben Hedrick,

00:00:07.260 --> 00:00:10.080
who's the lead for AP Calculus.

00:00:10.080 --> 00:00:11.980
What do you at the College Board?

00:00:12.830 --> 00:00:16.570
- Really, anything with AP
Calculus and AP Statistics

00:00:16.570 --> 00:00:18.900
is something that I have a
little bit of control over.

00:00:18.900 --> 00:00:20.800
So for example, for AP Calculus,

00:00:20.800 --> 00:00:23.640
anything with the curriculum,
anything with the assessment

00:00:23.640 --> 00:00:25.837
is something that I work with.

00:00:25.837 --> 00:00:29.020
- And I'm sure that you're as excited

00:00:29.020 --> 00:00:30.650
as I am and, you know,

00:00:30.650 --> 00:00:33.040
several hundred thousands of
students are around the country

00:00:33.040 --> 00:00:35.720
for the AP exams that are coming up

00:00:35.720 --> 00:00:38.600
in roughly a little less than a week.

00:00:38.600 --> 00:00:42.294
And I thought a fun
place to start would be

00:00:42.294 --> 00:00:44.460
when you guys write the test,

00:00:44.460 --> 00:00:46.020
what are you trying to assess?

00:00:46.020 --> 00:00:48.290
What's the spirit of the questions?

00:00:48.290 --> 00:00:51.622
Both on the multiple choice
and the free response section.

00:00:51.622 --> 00:00:53.620
- So what we're looking for from students

00:00:53.620 --> 00:00:57.520
is evidence of good, conceptual
understanding in calculus.

00:00:57.520 --> 00:00:59.890
Now, we have some things that
are gonna test procedures,

00:00:59.890 --> 00:01:02.740
but really, we want students
to understand what's going on

00:01:02.740 --> 00:01:05.560
with the calculus rather
than just calculating things

00:01:05.560 --> 00:01:07.110
or putting numbers on paper.

00:01:07.110 --> 00:01:09.780
A lot of thinking
questions, a lot of insight

00:01:09.780 --> 00:01:12.240
to make sure that whatever
good calculus those students

00:01:12.240 --> 00:01:13.510
have learned in their courses,

00:01:13.510 --> 00:01:15.920
we're assessing it on the AP exam.

00:01:15.920 --> 00:01:20.360
- And I've done a bunch of
example problems for the AP exam,

00:01:20.360 --> 00:01:22.724
and yeah, my sense of it is

00:01:22.724 --> 00:01:26.760
you sometimes will do
some of the minutiae work

00:01:26.760 --> 00:01:28.140
or you'll give the formula,

00:01:28.140 --> 00:01:29.380
and you're really trying to understand

00:01:29.380 --> 00:01:30.880
whether students understand the essence

00:01:30.880 --> 00:01:31.800
of what an integral is,

00:01:31.800 --> 00:01:34.680
or the essence of derivative
as a rate of change,

00:01:34.680 --> 00:01:36.240
or the slope of a tangent line.

00:01:36.240 --> 00:01:39.320
So, if you were preparing for this exam,

00:01:39.320 --> 00:01:40.420
if you were the students,

00:01:40.420 --> 00:01:42.280
what you suggest to them?

00:01:42.280 --> 00:01:44.170
You know, obviously we have
a little less than a week

00:01:44.170 --> 00:01:45.100
for this current test,

00:01:45.100 --> 00:01:46.960
so what advice would you
have for those students?

00:01:46.960 --> 00:01:49.040
And then just generally, as
students go through the year,

00:01:49.040 --> 00:01:51.189
how should they think about it?

00:01:51.189 --> 00:01:52.930
- Yeah, the advice I would give

00:01:52.930 --> 00:01:54.980
is dependent on how much
time the students have

00:01:54.980 --> 00:01:57.080
to prepare, so the advice that I give

00:01:57.080 --> 00:01:58.900
at the beginning of the year
is different from the advice

00:01:58.900 --> 00:02:01.662
that I give now, several
days before the exam.

00:02:01.662 --> 00:02:03.970
Getting ready for it, first, relax.

00:02:03.970 --> 00:02:05.660
You know, this is a calculus exam.

00:02:05.660 --> 00:02:07.990
We're not gonna be throwing
anything at you from left field.

00:02:07.990 --> 00:02:09.140
It's calculus.

00:02:09.140 --> 00:02:10.360
You need to know limits.

00:02:10.360 --> 00:02:11.560
You need to know derivatives.

00:02:11.560 --> 00:02:12.920
You need to know integrals.

00:02:12.920 --> 00:02:15.730
In terms of preparation, make
sure you're taking the time

00:02:15.730 --> 00:02:17.254
to review the basics.

00:02:17.254 --> 00:02:18.500
Every question that they see

00:02:18.500 --> 00:02:20.500
is gonna hit one of those big topics,

00:02:20.500 --> 00:02:22.570
so make sure you know
your derivative rules,

00:02:22.570 --> 00:02:24.546
make sure you know your integral rules.

00:02:24.546 --> 00:02:26.100
Take a look at the old exams

00:02:26.100 --> 00:02:28.100
and see how we're asking questions.

00:02:28.100 --> 00:02:30.940
And you'll at the patterns and
see that we're always asking

00:02:30.940 --> 00:02:33.420
about understanding
what a derivative means,

00:02:33.420 --> 00:02:36.920
understanding integration
or the idea of accumulation.

00:02:36.920 --> 00:02:38.730
There are no secrets on the exams,

00:02:38.730 --> 00:02:41.130
so really stick with what we've done.

00:02:41.130 --> 00:02:42.650
Make sure that you're thinking of ways

00:02:42.650 --> 00:02:44.220
to apply to your good knowledge,

00:02:44.220 --> 00:02:46.611
and just review your basics.

00:02:46.611 --> 00:02:49.060
- And my experience with the
exam is there's a lot of,

00:02:49.060 --> 00:02:52.020
as you just said, really the
mainstream, meaty topics.

00:02:52.020 --> 00:02:54.720
You know, chain rule, fundamental theorem,

00:02:54.720 --> 00:02:57.260
or theorems of calculus, things like that.

00:02:57.260 --> 00:02:58.950
What's your sense of the more,

00:02:58.950 --> 00:03:00.920
what I would concern maybe a
little bit more, the minutiae,

00:03:00.920 --> 00:03:02.770
some of the more special case derivatives,

00:03:02.770 --> 00:03:06.284
or special case integrals,
you know, arc tangent and,

00:03:06.284 --> 00:03:10.355
how much does that play into the exam?

00:03:10.355 --> 00:03:12.250
- You know, I think, Sal,
you've actually answered

00:03:12.250 --> 00:03:13.570
your own question in asking that,

00:03:13.570 --> 00:03:15.470
is it is the minutiae.

00:03:15.470 --> 00:03:18.020
So really, at this point in
time, if you're getting ready

00:03:18.020 --> 00:03:20.320
for the exam, spending time on minutiae

00:03:20.320 --> 00:03:22.836
is not where you're gonna be
best served with your time.

00:03:22.836 --> 00:03:24.250
You should be hitting the major things,

00:03:24.250 --> 00:03:26.040
and you said it right at
the beginning, chain rule.

00:03:26.040 --> 00:03:28.660
I mean, any kind of question
we do with derivatives,

00:03:28.660 --> 00:03:31.000
you know, if you're doing maxes or mins,

00:03:31.000 --> 00:03:32.440
if you're doing tangent lines,

00:03:32.440 --> 00:03:34.350
if you're doing a related rates problem,

00:03:34.350 --> 00:03:36.870
they're all derivatives,
derivatives, derivatives.

00:03:36.870 --> 00:03:39.090
And if you're having
trouble with chain rule,

00:03:39.090 --> 00:03:41.940
it doesn't matter how much
minutiae you spend time with,

00:03:41.940 --> 00:03:43.480
the chain rule's gonna come up.

00:03:43.480 --> 00:03:46.680
So I would say don't sweat the minutiae.

00:03:46.680 --> 00:03:48.210
There might be something on the exam

00:03:48.210 --> 00:03:50.430
that you haven't reviewed
as much as you want to,

00:03:50.430 --> 00:03:53.649
but it's gonna be one
single, solitary question.

00:03:53.649 --> 00:03:56.050
Basic derivatives are going to form, well,

00:03:56.050 --> 00:03:57.860
the basis for all those problems.

00:03:57.860 --> 00:03:59.580
So that's where your time is best spent.

00:03:59.580 --> 00:04:00.770
- Yeah, I definitely get that sense.

00:04:00.770 --> 00:04:03.210
Especially, you know, how
do you interpret the first,

00:04:03.210 --> 00:04:06.910
second, you know, derivatives, concavity.

00:04:07.830 --> 00:04:10.140
Things like that, those
seem to show up a lot.

00:04:10.140 --> 00:04:13.930
But once again, that's a
conceptual idea of derivatives.

00:04:13.930 --> 00:04:15.390
And on the integration side,

00:04:15.390 --> 00:04:17.280
I've been doing a bunch
of the free response.

00:04:17.280 --> 00:04:18.840
It just seems like, I mean, you know,

00:04:18.840 --> 00:04:21.300
what happens if you swap
the bounds of integration,

00:04:21.300 --> 00:04:22.620
or if you add integrals.

00:04:22.620 --> 00:04:25.300
So a lot of the properties and
the conceptual understanding

00:04:25.300 --> 00:04:26.734
of what an integral is,

00:04:26.734 --> 00:04:29.740
but not necessarily the fancy tricks.

00:04:29.740 --> 00:04:31.840
At least at this point,
if you're studying.

00:04:31.840 --> 00:04:34.080
- Absolutely, there's
nothing that we go after

00:04:34.080 --> 00:04:35.010
the fancy tricks.

00:04:35.010 --> 00:04:37.200
We're not trying to ask trick questions

00:04:37.200 --> 00:04:38.800
or play any gotcha moments on it.

00:04:38.800 --> 00:04:40.880
It's really assessment of calculus.

00:04:40.880 --> 00:04:42.330
And for the students who
have been going through

00:04:42.330 --> 00:04:43.960
these great courses all year,

00:04:43.960 --> 00:04:46.280
this is a wonderful opportunity to show us

00:04:46.280 --> 00:04:47.590
everything that you know

00:04:47.590 --> 00:04:49.731
and get a really great score on the exam.

00:04:49.731 --> 00:04:51.440
- And what about calculator?

00:04:51.440 --> 00:04:52.960
I mean, I was doing some
of the free response.

00:04:52.960 --> 00:04:54.440
That first part of the free response,

00:04:54.440 --> 00:04:56.940
knowing your calculator well helps.

00:04:56.940 --> 00:04:57.890
- Oh, absolutely.

00:04:57.890 --> 00:04:59.250
The calculator's a tool,

00:04:59.250 --> 00:05:00.930
and like any tool, you want to make sure

00:05:00.930 --> 00:05:03.390
that you're applying it
appropriately and strategically,

00:05:03.390 --> 00:05:05.780
which sometimes means
using your calculator,

00:05:05.780 --> 00:05:07.737
and using it well and correctly,

00:05:07.737 --> 00:05:09.500
but other times not using your calculator.

00:05:09.500 --> 00:05:11.250
I mean, even on the free response section

00:05:11.250 --> 00:05:13.220
where the calculator is being used,

00:05:13.220 --> 00:05:15.220
we expect students to show their work.

00:05:15.220 --> 00:05:16.800
If you're taking an integral,

00:05:16.800 --> 00:05:19.290
we wanna see that you've
actually taken that integral.

00:05:19.290 --> 00:05:21.470
Now, the calculator
will do the work of it,

00:05:21.470 --> 00:05:23.080
but we wanna see the notation that says

00:05:23.080 --> 00:05:24.680
this is the integral you calculated,

00:05:24.680 --> 00:05:26.930
and this is the answer you obtained.

00:05:26.930 --> 00:05:29.250
And then do whatever you
will with that answer

00:05:29.250 --> 00:05:30.560
as required by the question,

00:05:30.560 --> 00:05:33.180
but good communication through writing,

00:05:33.180 --> 00:05:35.942
even on the calculator
section, is necessary.

00:05:35.942 --> 00:05:39.485
- And for students taking the BC exam,

00:05:39.485 --> 00:05:42.240
above and beyond the core differentiation,

00:05:42.240 --> 00:05:44.530
integration, you know,
a lot of what you learn

00:05:44.530 --> 00:05:46.850
about parametric equations
is actually just an extension

00:05:46.850 --> 00:05:48.980
of what you learn in
differentiation, integration.

00:05:48.980 --> 00:05:52.420
But probably some of the convergence tests

00:05:52.420 --> 00:05:55.320
are something to become
pretty familiar with.

00:05:55.320 --> 00:05:57.166
- That would be correct, yes.

00:05:57.166 --> 00:06:01.060
- And in terms of grading of the exam,

00:06:01.060 --> 00:06:02.882
you know, some people,

00:06:02.882 --> 00:06:06.081
to get a five or a high,

00:06:06.081 --> 00:06:07.820
my understanding is you
don't need to answer

00:06:07.820 --> 00:06:10.493
absolutely every question perfectly.

00:06:10.493 --> 00:06:12.570
- Well, like any test,
if you want to do well,

00:06:12.570 --> 00:06:14.140
you don't have to have perfection on it,

00:06:14.140 --> 00:06:17.400
and this is a test where
students are coming in

00:06:17.400 --> 00:06:20.200
and it's a three hour timed test,

00:06:20.200 --> 00:06:22.340
well, a little over three hour timed test,

00:06:22.340 --> 00:06:25.450
and you know, you can do
a lot of really good work

00:06:25.450 --> 00:06:27.830
and earn a five pretty easily.

00:06:27.830 --> 00:06:30.380
So I wouldn't tell anyone
to go in there worrying

00:06:30.380 --> 00:06:33.570
about trying to figure out
what magical numbers they need

00:06:33.570 --> 00:06:36.100
to get in order to get a
perfect score on the test

00:06:36.100 --> 00:06:37.210
or to get a five on the test

00:06:37.210 --> 00:06:39.720
or whatever it is that they're going for.

00:06:39.720 --> 00:06:41.760
Just go in like you would on any test,

00:06:41.760 --> 00:06:43.300
do your best on every question,

00:06:43.300 --> 00:06:45.046
get every point that you can,

00:06:45.046 --> 00:06:47.110
and hopefully it'll work out fine for you.

00:06:47.110 --> 00:06:48.530
- What advice would you have for students,

00:06:48.530 --> 00:06:49.680
especially on the free response

00:06:49.680 --> 00:06:50.920
where there's multiple sections,

00:06:50.920 --> 00:06:52.330
and if, you know, on part A they say,

00:06:52.330 --> 00:06:53.300
what, that, that's,

00:06:53.300 --> 00:06:55.070
I don't get what that, what,

00:06:55.070 --> 00:06:57.010
you know, should they
skip to the next free,

00:06:57.010 --> 00:06:58.718
what should they do?

00:06:58.718 --> 00:07:02.660
- I actually have very
common advice on that one

00:07:02.660 --> 00:07:04.480
and that's to make sure
that students are trying

00:07:04.480 --> 00:07:06.500
every part of a free response question.

00:07:06.500 --> 00:07:08.500
One of the misconceptions students have

00:07:08.500 --> 00:07:09.570
is exactly what you said.

00:07:09.570 --> 00:07:11.920
If you hit part A and you're
freaking out a little bit

00:07:11.920 --> 00:07:14.195
because you don't know
what part A is about,

00:07:14.195 --> 00:07:15.950
that's okay, move on to part B.

00:07:15.950 --> 00:07:18.460
There are some questions
where the answer for part b

00:07:18.460 --> 00:07:21.340
depends on part A, but for
the majority of the questions,

00:07:21.340 --> 00:07:22.590
they're separate pieces.

00:07:22.590 --> 00:07:24.560
So part A, part B, part C,

00:07:24.560 --> 00:07:25.720
if there's a part D,

00:07:25.720 --> 00:07:28.140
they might all be asking
very different things.

00:07:28.140 --> 00:07:31.020
And if you have trouble even with part C,

00:07:31.020 --> 00:07:32.520
that doesn't mean the question

00:07:32.520 --> 00:07:34.400
gets progressively more difficult.

00:07:34.400 --> 00:07:35.920
Give part D a try.

00:07:35.920 --> 00:07:38.380
And a lot of the things
that we do give points for

00:07:38.380 --> 00:07:40.940
are good, solid calculus work.

00:07:40.940 --> 00:07:42.660
So if you can set up a problem,

00:07:42.660 --> 00:07:44.350
maybe you don't have time to finish it,

00:07:44.350 --> 00:07:46.520
maybe you make some
mistakes along the way,

00:07:46.520 --> 00:07:48.650
the answer point is only one little point

00:07:48.650 --> 00:07:49.720
out of the nine.

00:07:49.720 --> 00:07:51.320
There are other points for the setup

00:07:51.320 --> 00:07:53.470
and the conceptual
understanding of the problem,

00:07:53.470 --> 00:07:56.280
and a lot of students have
good conceptual understanding

00:07:56.280 --> 00:07:57.558
that they could set up,

00:07:57.558 --> 00:07:59.560
pick up a point here,
pick up a point there,

00:07:59.560 --> 00:08:01.340
but they panic a little bit,

00:08:01.340 --> 00:08:02.800
and they move on to another question

00:08:02.800 --> 00:08:04.214
and never come back.

00:08:04.214 --> 00:08:06.730
And that's okay if you want to
move on to another question,

00:08:06.730 --> 00:08:08.440
but absolutely I would say,

00:08:08.440 --> 00:08:10.338
if you're having trouble with part A,

00:08:10.338 --> 00:08:12.769
that's all right, take a look at part B.

00:08:12.769 --> 00:08:15.176
- And one thing that I've observed is

00:08:15.176 --> 00:08:16.770
a lot of these questions, at
first when you look at 'em,

00:08:16.770 --> 00:08:18.120
like oh wow, this looks like some really,

00:08:18.120 --> 00:08:19.660
you know, super deep thing,

00:08:19.660 --> 00:08:23.060
but it really is some
core basic calculus ideas,

00:08:23.060 --> 00:08:24.780
and that if you're on the right track,

00:08:24.780 --> 00:08:26.260
it's actually quite simple.

00:08:27.150 --> 00:08:29.450
Would you say it's fair
if a student finds himself

00:08:29.450 --> 00:08:31.160
doing a very hairy calculation

00:08:31.160 --> 00:08:34.349
that they might question
whether they need to?

00:08:34.349 --> 00:08:36.100
- I would go beyond might.

00:08:36.100 --> 00:08:37.660
They should question what they're doing.

00:08:37.660 --> 00:08:39.420
If you've done something
where you feel like

00:08:39.420 --> 00:08:41.090
you need to prove a new calculus theorem

00:08:41.090 --> 00:08:42.100
in order to answer it,

00:08:42.100 --> 00:08:44.915
something has gone
terribly, terribly wrong.

00:08:44.915 --> 00:08:46.250
And also, if you feel like you need

00:08:46.250 --> 00:08:47.840
to bring in some sort of mathematics

00:08:47.840 --> 00:08:49.660
you've never seen before in calculus,

00:08:50.980 --> 00:08:53.700
as much as I hate to say it,
you know, this is calculus.

00:08:53.700 --> 00:08:56.580
It's limits, it's
derivatives, it's integrals.

00:08:56.580 --> 00:08:58.460
Even though there's a lot
of wealth in that one,

00:08:58.460 --> 00:09:00.630
if you're doing something beyond that,

00:09:00.630 --> 00:09:02.859
probably something has gone wrong.

00:09:02.859 --> 00:09:04.040
- Yeah, yeah.

00:09:04.040 --> 00:09:04.770
Well, thanks.

00:09:04.770 --> 00:09:06.240
I think, you know,

00:09:06.240 --> 00:09:09.180
thousands of AP students are
gonna really appreciate that.

00:09:09.180 --> 00:09:12.749
Any other parting words
for the coming test?

00:09:12.749 --> 00:09:14.400
- You know, it's the same thing

00:09:14.400 --> 00:09:15.760
that I used to tell my students

00:09:15.760 --> 00:09:17.160
when I was teaching AP Calculus,

00:09:17.160 --> 00:09:18.520
this is just another test,

00:09:18.520 --> 00:09:21.090
and the beauty of AP is
the teachers have been

00:09:21.090 --> 00:09:23.570
prepping their students since
the beginning of the year.

00:09:23.570 --> 00:09:24.920
It's one of the few tests

00:09:24.920 --> 00:09:27.248
where everyone knows what's coming,

00:09:27.248 --> 00:09:30.210
so when you sit down and you've
got free response questions,

00:09:30.210 --> 00:09:32.030
you've got multiple choice questions,

00:09:32.030 --> 00:09:34.320
you know, there really
shouldn't be any surprises.

00:09:34.320 --> 00:09:36.693
This is the stuff they've seen all year.

00:09:36.693 --> 00:09:38.361
They know what to do.

00:09:38.361 --> 00:09:40.615
Take a moment, take a breath,

00:09:40.615 --> 00:09:42.350
take a look at the problem,

00:09:42.350 --> 00:09:43.580
get all the points that you can.

00:09:43.580 --> 00:09:44.940
Show us that good calculus.

00:09:44.940 --> 00:09:46.320
- Awesome, and I'll throw in a plug.

00:09:46.320 --> 00:09:48.720
We have tons of resources
for the students as well,

00:09:48.720 --> 00:09:51.520
and we're gonna have our
office hours on Monday.

00:09:51.520 --> 00:09:52.920
So, super exciting.

00:09:52.920 --> 00:09:54.230
Thanks a bunch, Ben.

00:09:54.230 --> 00:09:55.670
- No, thanks for having me.


2015 AP Calculus BC 2d

WEBVTT
Kind: captions
Language: en

00:00:00.218 --> 00:00:02.189
- [Voiceover] Find the total distance

00:00:02.189 --> 00:00:03.895
traveled by the particle from

00:00:03.895 --> 00:00:05.826
time t equals zero to

00:00:05.826 --> 00:00:08.548
t equals one.

00:00:08.548 --> 00:00:10.987
Always remember they didn't say
find the total displacement,

00:00:10.987 --> 00:00:13.201
they said find the total distance

00:00:13.201 --> 00:00:15.395
traveled by the particle.

00:00:15.395 --> 00:00:18.788
So if something goes to the right by one,

00:00:18.788 --> 00:00:21.328
and then goes up by one,

00:00:21.328 --> 00:00:23.644
their distance is two.

00:00:23.644 --> 00:00:25.635
And actually, and then if they go back,

00:00:25.635 --> 00:00:28.656
if they go back, over here,

00:00:28.656 --> 00:00:30.288
just let me draw a straighter line,

00:00:30.288 --> 00:00:32.056
they then go back to

00:00:32.056 --> 00:00:33.579
the original starting point.

00:00:33.579 --> 00:00:36.079
And then this distance
right here would be what?

00:00:36.079 --> 00:00:39.187
The square root of,
the square root of two,

00:00:39.187 --> 00:00:41.138
the displacement would be zero.

00:00:41.138 --> 00:00:42.661
They got back to where they started.

00:00:42.661 --> 00:00:44.978
For the distance would be one plus one

00:00:44.978 --> 00:00:47.416
plus the square root of two.

00:00:47.416 --> 00:00:49.265
So how do we figure out the total distance

00:00:49.265 --> 00:00:51.417
in this scenario right over here?

00:00:51.417 --> 00:00:53.653
Let me erase this since it's
not relevant to the problem,

00:00:53.653 --> 00:00:55.116
it's just a, for remind ourselves

00:00:55.116 --> 00:00:56.782
we're talking about distance.

00:00:56.782 --> 00:00:59.058
Well distance is equal to rate times time.

00:00:59.058 --> 00:01:01.374
Or it's equal to speed times time.

00:01:01.374 --> 00:01:03.141
So in a very small amount of time,

00:01:03.141 --> 00:01:04.747
if you want to figure out the distance,

00:01:04.747 --> 00:01:07.652
well, you could take your speed,

00:01:07.652 --> 00:01:11.573
which is the magnus magnitude
of your velocity function.

00:01:11.573 --> 00:01:13.564
If you took your speed,

00:01:13.564 --> 00:01:15.941
that's your speed right over there,

00:01:15.941 --> 00:01:19.110
and if you multiply it by a
little small change in time,

00:01:19.110 --> 00:01:20.066
that's gonna give you

00:01:20.066 --> 00:01:23.828
your infinitesimally
small change in distance

00:01:23.828 --> 00:01:26.527
over that infinitesimally
small change in time.

00:01:26.527 --> 00:01:27.746
And if you wanted to change your,

00:01:27.746 --> 00:01:29.963
if you wanted to find the total distance

00:01:29.963 --> 00:01:33.211
over a non-infinitesimal change in time,

00:01:33.211 --> 00:01:34.999
well, then you could integrate.

00:01:34.999 --> 00:01:37.072
And you can integrate those
little changes in time

00:01:37.072 --> 00:01:41.196
in this case from t equals
zero to t equals one.

00:01:41.196 --> 00:01:44.711
This is going to be the
expression for the total distance.

00:01:44.711 --> 00:01:46.357
Well what is this going to be?

00:01:46.357 --> 00:01:47.941
Well this is going to
be equal to the interval

00:01:47.941 --> 00:01:49.628
from zero to one.

00:01:49.628 --> 00:01:50.664
In the last part of the problem,

00:01:50.664 --> 00:01:54.117
we already have an
expression for our speed.

00:01:54.117 --> 00:01:58.137
Our speed we saw was this
business right over here.

00:01:58.137 --> 00:02:01.922
So it's going to be
equal to the square root,

00:02:01.922 --> 00:02:03.647
give myself a little bit more

00:02:03.647 --> 00:02:05.679
space to work with, is
going to be equal to

00:02:05.679 --> 00:02:10.559
the square root of the
x-component of the velocity,

00:02:10.559 --> 00:02:14.828
cosign of t squared squared,

00:02:14.828 --> 00:02:17.117
let me put the little
outer parenthesis there,

00:02:17.117 --> 00:02:21.404
squared, plus the y-component
of our velocity squared,

00:02:21.404 --> 00:02:24.452
the rate of change of y with
respect to time squared.

00:02:24.452 --> 00:02:29.002
So plus e to the 0.5 t,

00:02:29.002 --> 00:02:30.052
that's the rate of change of y

00:02:30.052 --> 00:02:33.148
with respect to time, or
the y-component of velocity,

00:02:33.148 --> 00:02:34.407
we're gonna square that.

00:02:34.407 --> 00:02:38.207
This is our expression for our
speed as a function of time.

00:02:38.207 --> 00:02:40.523
And then you multiply that times dt

00:02:40.523 --> 00:02:41.580
and we're gonna integrate all this

00:02:41.580 --> 00:02:43.611
and this is gonna give us our total,

00:02:43.611 --> 00:02:45.887
this is gonna give us our total distance.

00:02:45.887 --> 00:02:48.020
And lucky for us we can use our calculator

00:02:48.020 --> 00:02:52.023
in this part of the AP exam,
so let's evaluate this.

00:02:52.023 --> 00:02:56.005
So let me turn it on.

00:02:56.005 --> 00:02:57.462
Let me clear out.

00:02:57.462 --> 00:02:59.840
And so we're just going to evaluate,

00:02:59.840 --> 00:03:01.526
let's go to math,

00:03:01.526 --> 00:03:04.615
and function integral right over there,

00:03:04.615 --> 00:03:06.775
that's for evaluating definite integrals,

00:03:06.775 --> 00:03:10.596
and we want to evaluate the square root,

00:03:10.596 --> 00:03:12.904
an open parenthesis,

00:03:12.904 --> 00:03:15.240
of, I think I'm gonna have
a lot of parenthesis here,

00:03:15.240 --> 00:03:16.643
but we'll see if we can do it.

00:03:16.643 --> 00:03:20.096
So the square root of, cosign of,

00:03:20.096 --> 00:03:22.127
and I'm gonna us x as
my variable integration

00:03:22.127 --> 00:03:24.465
cosign of x squared,

00:03:24.465 --> 00:03:27.452
so that's that parenthesis,
I need another parenthesis,

00:03:27.452 --> 00:03:30.966
squared, squared,

00:03:30.966 --> 00:03:34.197
plus e to this business squared.

00:03:34.197 --> 00:03:35.802
Well this we already solved before.

00:03:35.802 --> 00:03:39.662
This is the same thing
as e to the t, right?

00:03:39.662 --> 00:03:41.821
If we raise something to the 0.5 t,

00:03:41.821 --> 00:03:43.156
and then we raise that to the second power

00:03:43.156 --> 00:03:45.859
two times 0.5 t is just t.

00:03:45.859 --> 00:03:47.761
So I can do that if I like.

00:03:47.761 --> 00:03:50.186
I could just type in all of
this business if you like.

00:03:50.186 --> 00:03:53.538
So plus second e to the,

00:03:53.538 --> 00:03:56.323
my variable of integration here is x,

00:03:56.323 --> 00:03:58.840
e to the x,

00:03:58.840 --> 00:04:02.180
so close that, and then I close

00:04:02.180 --> 00:04:04.815
my square root.

00:04:04.815 --> 00:04:05.891
Did I do that right?

00:04:05.891 --> 00:04:07.518
This closes around the x squared,

00:04:07.518 --> 00:04:10.870
that closes with that,
yep ok that looks right.

00:04:10.870 --> 00:04:13.369
And then my variable of integration is x,

00:04:13.369 --> 00:04:17.960
and I'm integrating from zero to one.

00:04:17.960 --> 00:04:21.577
And now let's let the calculator
munch on it a little bit.

00:04:21.577 --> 00:04:25.965
And I get approximately 1.595.

00:04:25.965 --> 00:04:28.356
So this is approximately

00:04:28.356 --> 00:04:32.206
1.595, the total distance

00:04:32.206 --> 00:04:33.341
traveled by the particle

00:04:33.341 --> 00:04:35.271
in time equals zero to t equals one

00:04:35.271 --> 00:04:37.039
which is kind of neat that
we can do these things,

00:04:37.039 --> 00:04:39.722
just from, you know, the
information that they gave us

00:04:39.722 --> 00:04:43.154
at the beginning of the problem.