Hypothesis testing for new cholesterol test

WEBVTT
Kind: captions
Language: en

00:00:00.633 --> 00:00:02.233
Voiceover:Karim has developed a new method

00:00:02.233 --> 00:00:04.233
for measuring cholesterol
level that doesn't

00:00:04.233 --> 00:00:06.967
involve any invasive
procedure, like drawing blood.

00:00:06.967 --> 00:00:08.266
Well, that would be very nice.

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Regulations demand that
the test should be correct

00:00:10.900 --> 00:00:13.534
99% of the time.

00:00:14.034 --> 00:00:16.333
Test should be correct 99% of the time.

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So he had to verify that his method meets

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the standards.

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He got a random sample
of 100 people, tested

00:00:23.634 --> 00:00:26.166
their cholesterol using his new method,

00:00:26.166 --> 00:00:28.033
compared the results to blood tests

00:00:28.033 --> 00:00:30.033
performed on the same people, and found

00:00:30.033 --> 00:00:33.266
they were the same 95% of the time.

00:00:33.766 --> 00:00:37.234
Since blood tests are
always correct, he concluded

00:00:37.234 --> 00:00:40.333
that his method was
correct 95% of the time.

00:00:40.700 --> 00:00:42.433
That seems reasonable,
at least for his sample

00:00:42.433 --> 00:00:43.733
right over here.

00:00:44.033 --> 00:00:46.300
In light of the results, Karim had to test

00:00:46.300 --> 00:00:48.900
the hypothesis that the test is, in fact,

00:00:48.900 --> 00:00:51.333
correct 99% of the time.

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So he wanted to test the hypothesis that

00:00:54.300 --> 00:00:57.266
the test is, in fact,
correct 99% of the time.

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So he performed 100 computer generated

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simulations of random
samples of 100 tests.

00:01:03.633 --> 00:01:04.767
So there's a lot of 100's here.

00:01:04.767 --> 00:01:07.100
So he did 100 simulations
and each of those

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samples essentially
simulated the test results

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of 100 people.

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So it's like using a computer that assumes

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this assumption.

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That assumes that the
test is, in fact, correct

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99% of the time, and to simulate 100

00:01:21.100 --> 00:01:23.800
of these samples.

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100 samples, each of them having 100

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people in them.

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Supposing the true percentage of correct

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result is 99%, to find how likely it is

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that a sample would have
95% correct results.

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The results of the
simulations are plotted below.

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How do the results of
the simulations affect

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the likelihood of the
hypothesis that Karim's

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test is correct 99% of the time?

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So I know this can be
a little bit confusing

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because we have all of these ...

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We're testing and 100's and all of this.

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But what he's doing, he said, "Okay."

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He got his test, on this
one sample, his test

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was right 95% of the time.

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But what he wants to really figure out is,

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he wants to get a sense
of what's the probability

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of that happening, assuming that his test

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is actually correct 99% of the time?

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So he has the hypothesis.

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So if we assumed the hypothesis ...

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If you assumed, if you assumed his test,

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his test, his test correct,
correct 99% of the time.

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So that's the hypothesis right over here.

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Then if we assume this,
he's trying to figure out

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what's the probability
that he gets these results?

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What is probability, probability of this

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right over here?

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Because if this probability is really low,

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if the probability of
getting this is really low,

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then he can say, "Well
then I can't make the ...

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"Then I really don't feel good about

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"assuming this hypothesis.
That maybe we can't

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"go with the hypothesis if
the probability of this ..."

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If the probability of
him seeing what he saw

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is very low, given this assumption.

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So let's look at that.

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So let's interpret what's happening

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in this chart here.

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So remember, this computer
is running 100 ...

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It's simulating 100 samples, each of those

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samples have 100 people.

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So for each sample ...

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So let me make it clear.

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So this is sample one.

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So sample, sample number
one and then there'd

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be sample number two
and there's going to be,

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there's going to be 100 of these samples.

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Each of these samples,
the computer is saying,

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"Okay, there's 100 people in each sample.

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"One, two, three ..."

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And it goes all the way to the hundredth,

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all the way to the hundredth person.

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For each of these the computer is saying,

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"Okay, I'm assuming there's a 99% chance

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"that it's correct."

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So the computer uses its random number

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generators, essentially kind of trying

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to roll the dice.

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And 99 out of 100 times, it'll say

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the sample is ...

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The new test matches up to the old test.

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But then, every now and then, there's

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that one percent chance that it says,

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"Okay, it doesn't match up."

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So let's say that this one had 99 checks

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and then one X.

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Then this would fall
into this category, right

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over here.

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Then this sample, this
sample right over here,

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would correspond ...

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Let me do this in a color.

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So this sample, right over ...

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I mean a new color.

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So this sample right over here would

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correlate to, say, this dot.

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And then if they did the test again and

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maybe it got 99 checks and one X,

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then it would be ...

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Say it's number two,
say it would be there.

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Let's say in one of the samples ...

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Let's say in one of the
samples of 100, I guess,

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virtual people right over here, it got ...

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You know, there's a 99% chance of getting

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a check here, so it got
all 100 people got a check.

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Then that would be ...

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Then it would stack it up right over here.

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And so we see using the
assumption that there's

00:04:57.834 --> 00:05:01.566
a 99% chance of the test matching up,

00:05:01.566 --> 00:05:05.433
or there's a 99% chance
that the test is correct,

00:05:05.433 --> 00:05:08.000
it got all of these results.

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And we see that out of
these 100 samples it took,

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none of them, none of
them, gave what Karim

00:05:15.233 --> 00:05:17.000
observed, right over here.

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So it seems like, if you
make this assumption,

00:05:19.966 --> 00:05:22.633
the probability of
observing what he observed

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right over here seems
very, very, very, very low.

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So this, to me or I
guess to any reasonable

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statistician or reasonable
person, these results,

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the results he got, the
results that he both observed

00:05:35.633 --> 00:05:38.633
and the results of the simulation tells us

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that the likelihood of
the hypothesis being

00:05:42.901 --> 00:05:45.467
correct is low.

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So this is a ...

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The likelihood of the
hypothesis that Karim's test

00:05:52.580 --> 00:05:55.233
is correct 99% of the time, the likelihood

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of this hypothesis is very low.

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Very, very, very low.

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Given this observation and the results

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of the simulation.

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And so you'd be likely
to reject the hypothesis.


Appropriate statistical study example

WEBVTT
Kind: captions
Language: en

00:00:00.771 --> 00:00:02.299
Voiceover:Alma has developed a new kind

00:00:02.299 --> 00:00:03.331
of antibiotic.

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For the antibiotic to be
sufficiently effective,

00:00:05.960 --> 00:00:08.458
it has to kill at least 90% of bacteria

00:00:08.458 --> 00:00:10.651
when applied to a
harmful bacteria culture.

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She applied her antibiotic to a petri dish

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full of harmful bacteria,

00:00:14.715 --> 00:00:16.442
waited for it to take effect,

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and then tried to estimate the percentage

00:00:18.443 --> 00:00:19.954
of dead bacteria in it.

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She took a random sample of 300 bacteria

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and found that 94% of them were dead.

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Then she calculated the margin of error

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and found that the true percentage of

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dead bacteria is most
likely to be above 90%.

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So, what's happening over here,

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she's trying to figure out what percentage

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of the total population of bacteria died.

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Maybe there's something
about this bacteria,

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maybe when you look at it from, you know,

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the naked eye,

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you can't tell whether the bacteria

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died or not.

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So, she decides to estimate
the true percentage

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by sampling 300 individual bacterium,

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or by sampling ... I always forget the

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singular case.

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By sampling 300 bacteria,

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and then in her sample she found that 94%

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of them were dead.

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Then the margin of error tells us,

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because the margin of
error says it's unlikely,

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or that it's very likely that the true

00:01:13.226 --> 00:01:14.633
percentage is above 90%.

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That means that given that you sample

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300 bacteria,

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it's very unlikely that
the true percentage

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is below 90%.

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So, she could feel reasonably confident

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that in her petri dish,

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more than 90% of the population

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did indeed die.

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Now, let's answer these questions.

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What type of statistical
study did Alma use?

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Well, she used a ...

00:01:39.515 --> 00:01:42.138
she's trying to estimate a parameter

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for population, in this case,

00:01:43.271 --> 00:01:44.828
the parameter was the percentage of

00:01:44.828 --> 00:01:46.954
all of the bacteria that died.

00:01:46.954 --> 00:01:49.291
She couldn't observe that directly,

00:01:49.291 --> 00:01:51.529
so instead, she took a random sample

00:01:51.529 --> 00:01:53.437
of the bacteria in the petri dish,

00:01:53.437 --> 00:01:54.809
and she used ...

00:01:54.809 --> 00:01:57.204
she calculated the static for them,

00:01:57.204 --> 00:01:58.737
94% of them were dead,

00:01:58.737 --> 00:02:00.225
and that's her estimate for the

00:02:00.225 --> 00:02:01.661
population parameter.

00:02:01.661 --> 00:02:03.553
The percentage of the
population that died.

00:02:03.553 --> 00:02:05.160
So this is ... when you're using a ...

00:02:06.160 --> 00:02:07.686
when you're using a random sample

00:02:07.686 --> 00:02:09.693
to generate a statistic,

00:02:09.693 --> 00:02:12.022
which estimates a
parameter for a population,

00:02:12.022 --> 00:02:13.641
that's a sample study.

00:02:13.641 --> 00:02:16.398
So, she ran a sample study.

00:02:16.798 --> 00:02:18.195
Now the next question is.

00:02:18.195 --> 00:02:20.148
Is the study appropriate
for the statistical

00:02:20.148 --> 00:02:21.808
question it's supposed to answer?

00:02:22.469 --> 00:02:23.578
So what was the question that she's

00:02:23.578 --> 00:02:24.567
trying to answer?

00:02:24.567 --> 00:02:26.014
Well, at least the was it's written,

00:02:26.014 --> 00:02:27.721
it seems like she's trying to answer

00:02:27.721 --> 00:02:29.543
whether or not her antibiotic works,

00:02:29.543 --> 00:02:31.112
whether it's an effective antibiotic,

00:02:31.112 --> 00:02:33.517
whether it's capable of killing bacteria.

00:02:34.424 --> 00:02:35.837
You might be tempted to
say, "Okay, well look.

00:02:35.837 --> 00:02:37.108
It looks like it killed ...

00:02:37.108 --> 00:02:39.304
it killed more than 90% of the bacteria,

00:02:39.304 --> 00:02:41.143
or very likely it killed more than 90%

00:02:41.143 --> 00:02:44.192
of the bacteria given ...
given the sample size,

00:02:44.192 --> 00:02:45.858
and the margin error and all that."

00:02:45.858 --> 00:02:48.345
Even if it is indeed the case,

00:02:48.345 --> 00:02:51.313
that 95% of all the bacteria died,

00:02:51.313 --> 00:02:53.940
it doesn't necessarily
mean that it was caused

00:02:53.940 --> 00:02:54.924
by the antibiotic,

00:02:54.924 --> 00:02:56.356
maybe it was caused by the plastic in

00:02:56.356 --> 00:02:57.050
the petri dish.

00:02:57.050 --> 00:02:59.104
Maybe the air in the petri dish was

00:02:59.104 --> 00:03:00.439
too cold or went bad,

00:03:00.439 --> 00:03:02.093
or maybe it was handled in a weird way.

00:03:02.093 --> 00:03:03.684
Or, maybe that bacteria was just a bad ...

00:03:05.622 --> 00:03:06.651
a bad culture,

00:03:06.651 --> 00:03:08.750
and it somehow it just spontaneously

00:03:08.750 --> 00:03:09.784
died on its own.

00:03:10.322 --> 00:03:12.129
She can't say with confidence that it

00:03:12.129 --> 00:03:13.864
was definitely ... it was definitely

00:03:13.864 --> 00:03:15.101
the antibiotic.

00:03:15.101 --> 00:03:17.733
In order for her to make that statement,

00:03:17.733 --> 00:03:20.734
she would have to run a proper experiment.

00:03:20.734 --> 00:03:22.640
She would have to have a control and

00:03:22.640 --> 00:03:23.555
a treatment group,

00:03:23.555 --> 00:03:25.980
where everything is equal except for the

00:03:25.980 --> 00:03:27.423
treatment group has the treatment.

00:03:27.423 --> 00:03:28.900
So if she had 2 petri dishes

00:03:28.900 --> 00:03:31.762
that were kept in the same conditions

00:03:31.762 --> 00:03:34.118
with the same lightning, the same air,

00:03:34.118 --> 00:03:38.348
the same material that the bacteria

00:03:38.348 --> 00:03:40.394
is growing on, everything the same,

00:03:40.394 --> 00:03:42.678
except for the treatment group,

00:03:42.678 --> 00:03:45.562
has the antibiotic applied to it.

00:03:45.562 --> 00:03:48.429
Then she saw that in the treatment group

00:03:48.675 --> 00:03:51.037
that most of the bacteria died while in

00:03:51.037 --> 00:03:52.578
the control group, most of the bacteria

00:03:52.578 --> 00:03:53.289
didn't die,

00:03:53.289 --> 00:03:55.346
then she could say, "Okay. It looks

00:03:55.346 --> 00:03:57.702
like the antibiotic caused the bacteria

00:03:57.702 --> 00:03:58.575
to die.

00:03:58.575 --> 00:04:00.644
That there was actual causality here."

00:04:00.644 --> 00:04:03.109
So, she would have had
to run an experiment.

00:04:03.109 --> 00:04:06.754
The most appropriate statistical study,

00:04:06.754 --> 00:04:08.518
or the most appropriate study

00:04:08.518 --> 00:04:11.187
would have been a proper
controlled experiment.

00:04:11.187 --> 00:04:12.315
Where you have a control group,

00:04:12.315 --> 00:04:13.772
where they don't have the antibiotic,

00:04:13.772 --> 00:04:14.796
and a treatment group,

00:04:14.796 --> 00:04:16.211
where they do have the antibiotic.

00:04:17.364 --> 00:04:18.310
Let's see what are the choices here.

00:04:18.310 --> 00:04:20.388
Where they say is the study appropriate?

00:04:20.972 --> 00:04:23.569
So, yes because she is appropriate study.

00:04:23.569 --> 00:04:24.461
No, I don't like that answer.

00:04:24.461 --> 00:04:26.079
No, because she can't know for certain

00:04:26.079 --> 00:04:27.530
that the true percentage of dead bacteria

00:04:27.530 --> 00:04:28.908
is above 90%.

00:04:29.308 --> 00:04:30.577
Well, I'm not going to click on that,

00:04:30.577 --> 00:04:32.335
because even if she knew for certain

00:04:32.335 --> 00:04:33.964
that the true percentage of the dead

00:04:33.964 --> 00:04:35.834
bacteria were 95%,

00:04:35.834 --> 00:04:37.953
she can't feel confident that it was due

00:04:37.953 --> 00:04:39.288
to the antibiotic.

00:04:39.288 --> 00:04:40.694
Once again, it could be caused by the

00:04:40.694 --> 00:04:41.446
air conditioner.

00:04:41.446 --> 00:04:43.418
It could have been
caused by the petri dish.

00:04:43.418 --> 00:04:44.898
It could have been caused by the lighting

00:04:44.898 --> 00:04:45.665
in the room.

00:04:45.665 --> 00:04:47.564
So, no, because the study didn't have a

00:04:47.564 --> 00:04:48.983
treatment and a control group.

00:04:48.983 --> 00:04:50.820
Yeah, I would go with
that one right over there.

00:04:50.820 --> 00:04:52.223
Yes, because she found that the antibiotic

00:04:52.223 --> 00:04:54.297
killed more than 90% of harmful bacteria.

00:04:54.297 --> 00:04:57.120
Once again, even if she knew for sure

00:04:57.120 --> 00:04:59.195
that more than 90% of the population

00:04:59.195 --> 00:05:00.356
had been killed,

00:05:00.356 --> 00:05:02.722
she doesn't know that it was caused by

00:05:02.722 --> 00:05:03.305
the antibiotic.

00:05:03.305 --> 00:05:04.481
It could have been caused by a whole

00:05:04.481 --> 00:05:05.314
bunch of things.

00:05:05.314 --> 00:05:06.920
If she had a controlled group

00:05:06.920 --> 00:05:08.570
that had the exact same conditions

00:05:08.570 --> 00:05:11.274
and the bacteria didn't die,

00:05:11.274 --> 00:05:12.980
then she could feel better

00:05:12.980 --> 00:05:15.365
that it was the bacteria death

00:05:15.365 --> 00:05:17.047
was due to the antibiotic.


Types of statistical studies

WEBVTT
Kind: captions
Language: en

00:00:00.456 --> 00:00:02.393
Voiceover: Let's say that you have a hunch that

00:00:02.393 --> 00:00:06.977
sugar is somehow causing heart disease.

00:00:06.977 --> 00:00:10.159
Is somehow causing heart disease.

00:00:10.159 --> 00:00:11.840
So you want to research this further.

00:00:11.840 --> 00:00:14.531
You want to see what kind of statistical studies

00:00:14.531 --> 00:00:17.937
can you perform to better understand sugar intake in

00:00:17.937 --> 00:00:21.120
the population generally and whether that seems

00:00:21.120 --> 00:00:24.405
to be causing heart disease in some way.

00:00:24.405 --> 00:00:26.312
Well, the first that you might want to do is just to try

00:00:26.312 --> 00:00:30.414
to get a sense of sugar intake in the population as a whole.

00:00:30.414 --> 00:00:33.060
Now clearly, you don't know know, there's no way

00:00:33.060 --> 00:00:36.889
of measuring exactly how much sugar every member, let's say

00:00:36.889 --> 00:00:38.440
you're talking about the United States, every member

00:00:38.440 --> 00:00:42.579
of that 300 million population is consuming everyday.

00:00:42.579 --> 00:00:45.188
The way that we try to get a sense of how much sugar

00:00:45.188 --> 00:00:48.995
is being consumed is by conducting a sample study.

00:00:48.995 --> 00:00:53.566
So you take your population, your 300 million people.

00:00:53.566 --> 00:00:57.844
So you take your population right over here.

00:00:57.844 --> 00:00:59.832
Population.

00:00:59.832 --> 00:01:01.826
And you sample it.

00:01:01.826 --> 00:01:04.849
You sample your population and not only

00:01:04.849 --> 00:01:06.678
do you sample it, but you randomly sample it.

00:01:06.678 --> 00:01:09.825
Obviously you don't want to just survey people who are

00:01:09.825 --> 00:01:13.229
exiting a cupcake store or people who are exiting a gym.

00:01:13.229 --> 00:01:16.335
You want it to be a random sample of people,

00:01:16.335 --> 00:01:20.218
where where you're sampling them shouldn't somehow

00:01:20.218 --> 00:01:23.294
affect whether or not their answer or how much sugar

00:01:23.294 --> 00:01:25.673
they might say that they are consuming.

00:01:25.673 --> 00:01:27.441
But they are gonna tell you how much sugar

00:01:27.441 --> 00:01:30.067
they consume, let's say, on an average day maybe

00:01:30.067 --> 00:01:32.880
by filling out a survey or through some other way.

00:01:32.880 --> 00:01:36.303
And through that you would take this data and obviously,

00:01:36.303 --> 00:01:38.266
the more samples you have the better and we talk

00:01:38.266 --> 00:01:40.345
about that in depth in other statistics videos.

00:01:40.345 --> 00:01:44.913
About how you get a better predictor of the actual true

00:01:44.913 --> 00:01:48.922
population parameter, the more samples that you might take.

00:01:48.922 --> 00:01:52.345
But you might do that to get a gauge of how much sugar

00:01:52.345 --> 00:01:55.847
the average American consumes in a given day.

00:01:55.847 --> 00:01:59.276
So this right over here where you are taking random samples

00:01:59.276 --> 00:02:03.129
of the population to, essentially, generate a statistic

00:02:03.129 --> 00:02:07.389
that is estimating a true parameter, which is the actual

00:02:07.389 --> 00:02:11.181
amount of sugar Americans are consuming each day.

00:02:11.181 --> 00:02:13.396
We call this a sample study.

00:02:14.027 --> 00:02:17.338
Sample study.

00:02:17.338 --> 00:02:19.746
It's a way, once again, of just estimating what the

00:02:19.746 --> 00:02:23.407
actual amount of sugar people are having each day.

00:02:23.407 --> 00:02:24.806
Let's say you want to go further.

00:02:24.806 --> 00:02:27.952
This will give you a sense of what is likely the amount

00:02:27.952 --> 00:02:30.702
of sugar that people are consuming each day, but you

00:02:30.702 --> 00:02:33.726
really want to see how that's related to heart disease.

00:02:33.726 --> 00:02:37.501
So instead, what you do is you go survey people.

00:02:37.501 --> 00:02:43.499
You go survey people and you say, "How much sugar have

00:02:43.499 --> 00:02:46.703
"you consumed over", and once again, when you pick

00:02:46.703 --> 00:02:49.706
these people, you should be doing it randomly.

00:02:49.706 --> 00:02:54.477
So let's say you go survey a random sample of 60 year olds.

00:02:54.477 --> 00:02:56.530
So once again, you wouldn't want to sample people

00:02:56.530 --> 00:02:58.832
who are in the hospital, you wouldn't want to

00:02:58.832 --> 00:03:03.066
sample people who are at or just at the gym.

00:03:03.066 --> 00:03:04.702
You would want to find a random sample

00:03:04.702 --> 00:03:06.796
or sample them in places where it shouldn't

00:03:06.796 --> 00:03:09.289
affect their answer, which way they are going to go.

00:03:09.289 --> 00:03:14.644
Let's say you surveyed 300 60 year olds and you asked them

00:03:14.644 --> 00:03:18.594
how much sugar have they consumed over the last 30 years.

00:03:18.594 --> 00:03:21.810
And you also asked them their condition of their heart.

00:03:21.810 --> 00:03:24.027
And what you get is something like this.

00:03:24.027 --> 00:03:28.652
So on the horizontal axis you plot sugar consumption

00:03:28.652 --> 00:03:32.780
and then on the vertical axis you plot heart disease risk

00:03:32.780 --> 00:03:35.512
or their level of heart disease.

00:03:35.512 --> 00:03:40.908
Heart disease risk, let's say at 60.

00:03:41.862 --> 00:03:44.938
And you find a plot that looks something like this.

00:03:44.938 --> 00:03:46.474
So each of these points.

00:03:46.474 --> 00:03:51.736
This is someone who consumed 200 grams of sugar per day

00:03:51.736 --> 00:03:55.997
and they're now at high heart disease risk now at age 60.

00:03:55.997 --> 00:03:58.826
But maybe this is someone who is at low heart disease risk

00:03:58.826 --> 00:04:02.237
at age 60 even though they consumed a lot of sugar everyday.

00:04:02.806 --> 00:04:06.584
And so we just keep plotting all of these points.

00:04:07.476 --> 00:04:10.077
And you say, "Well, you know what, it does look"

00:04:10.077 --> 00:04:12.374
and obviously I'm not going to all 300.

00:04:12.374 --> 00:04:14.871
You say, "Well, actually, it looks like there is

00:04:14.871 --> 00:04:17.410
"a rough correlation right over here."

00:04:17.410 --> 00:04:19.589
That if you tried to plot a line, there's

00:04:19.589 --> 00:04:22.338
definitely some outliers here, but it looks

00:04:22.338 --> 00:04:27.112
like there is a line that you could fit.

00:04:27.112 --> 00:04:30.237
And then you might say, "It looks like sugar and heart

00:04:30.237 --> 00:04:33.088
"disease risk at age 60, that they are correlated,

00:04:33.088 --> 00:04:35.066
"that they're related in this way, that they move together,

00:04:35.066 --> 00:04:36.910
"that if someone consumed a lot of sugar over

00:04:36.910 --> 00:04:39.719
"the last 30 years they seem to have a worse

00:04:39.719 --> 00:04:43.094
"heart situation and if they consumed a lot lower sugar,

00:04:43.094 --> 00:04:47.072
"they seem to have a lower or better heart situation."

00:04:47.626 --> 00:04:51.653
This often happens in medical science, when people see

00:04:51.653 --> 00:04:53.989
something like this they often jump to the conclusion,

00:04:53.989 --> 00:04:57.486
oh therefore consuming more sugar

00:04:57.486 --> 00:05:00.199
must drive up heart disease risk.

00:05:00.199 --> 00:05:02.455
That's very dangerous because just seeing this data

00:05:02.455 --> 00:05:06.185
doesn't tell you that sugar is causing heart disease risk.

00:05:06.185 --> 00:05:08.351
It could go the other way around.

00:05:08.351 --> 00:05:10.853
It could be that people who end up having a high

00:05:10.853 --> 00:05:14.491
heart disease risk, maybe they crave sugar, but actually

00:05:14.491 --> 00:05:18.502
there's some other underlying cause that's making it happen.

00:05:18.502 --> 00:05:19.937
Maybe they have some other deficiency

00:05:19.937 --> 00:05:21.593
that's making them crave sugar.

00:05:21.593 --> 00:05:24.416
So it's not clear which way the causality is happening.

00:05:24.416 --> 00:05:27.568
Is the sugar consumption driving the unhealthy heart

00:05:27.568 --> 00:05:29.723
or is somehow the unhealthy heart driving the sugar

00:05:29.723 --> 00:05:32.289
consumption or maybe there's some other factor.

00:05:32.289 --> 00:05:35.251
Maybe fat consumption is driving the heart disease

00:05:35.251 --> 00:05:36.845
and maybe people who have more fat also

00:05:36.845 --> 00:05:39.388
will have more sugar or vice versa, who knows.

00:05:39.697 --> 00:05:42.427
All this is telling you is that there is a correlation.

00:05:42.427 --> 00:05:44.082
So this right over here, you would

00:05:44.082 --> 00:05:46.131
call this an observational study.

00:05:46.131 --> 00:05:48.020
You've observed a relationship, but you

00:05:48.020 --> 00:05:50.399
really can't say what is causing what.

00:05:50.399 --> 00:05:51.807
So let me write this down.

00:05:51.807 --> 00:05:57.534
This is an observational study.

00:05:57.887 --> 00:06:01.304
You're probably saying, "Alright, then how could you prove

00:06:01.304 --> 00:06:05.564
"or feel better about the idea that sugar is actually a

00:06:05.564 --> 00:06:07.736
"cause, that there's actually causality there?"

00:06:08.520 --> 00:06:12.924
To do that, you would actually have to run an experiment.

00:06:12.924 --> 00:06:15.115
An experiment.

00:06:15.115 --> 00:06:18.387
To do an experimental study here, what you

00:06:18.387 --> 00:06:20.944
would do is try to take two groups of folks.

00:06:20.944 --> 00:06:22.946
You would have your experimental group.

00:06:22.946 --> 00:06:24.893
So that's your experimental group.

00:06:24.893 --> 00:06:26.998
Actually, let me make it a circle

00:06:26.998 --> 00:06:29.768
here so it's a pool of people.

00:06:29.768 --> 00:06:33.827
Let's say you have 100 people that are experimental.

00:06:33.827 --> 00:06:35.589
That's your experimental pool.

00:06:35.589 --> 00:06:37.716
Then you have your control.

00:06:37.716 --> 00:06:39.880
You have your control.

00:06:39.880 --> 00:06:43.364
What you would do, if you wanted to run this type of

00:06:43.364 --> 00:06:45.839
experiment and as we'll see, this type of experiment

00:06:45.839 --> 00:06:48.443
probably wouldn't be run because some would consider it

00:06:48.443 --> 00:06:51.541
unethical or actually I would consider it unethical as well.

00:06:51.541 --> 00:06:53.389
But what you would do is you would randomly, let's say

00:06:53.389 --> 00:06:56.380
take 30 year olds, you would randomly take 30 year olds

00:06:56.380 --> 00:06:58.217
and put them in one of these two groups.

00:06:58.217 --> 00:07:00.506
Once again, when we say randomly, you don't want to put

00:07:00.506 --> 00:07:02.600
all the healthy people in one group and all the unhealthy

00:07:02.600 --> 00:07:04.539
people in the other group or vice versa.

00:07:04.539 --> 00:07:06.283
You want it to be random, you don't want to put all of the

00:07:06.283 --> 00:07:09.049
people of one type, one demographic, one economic status

00:07:09.049 --> 00:07:11.243
in one group or the other, you want it to be random.

00:07:11.243 --> 00:07:16.520
So you randomly put people into these two groups and then

00:07:16.520 --> 00:07:19.515
the experimental group, you would change one variable.

00:07:19.515 --> 00:07:21.565
The variable you care about is sugar.

00:07:21.565 --> 00:07:24.376
So what you might say is, "Okay, all of the people in this

00:07:24.376 --> 00:07:27.068
"group right over here, whatever sugar that they would have

00:07:27.068 --> 00:07:30.485
"consumed, on top of that they have to drink, I don't know,

00:07:30.485 --> 00:07:32.961
"they have to drink a cup of syrup every night or they

00:07:32.961 --> 00:07:36.058
"have to have a minimum sugar intake."

00:07:36.058 --> 00:07:39.189
So that you essentially force sugar into this group

00:07:40.203 --> 00:07:42.766
that you're not forcing into this group.

00:07:42.766 --> 00:07:46.149
And then 30 years later, so one, this is probably unethical

00:07:46.149 --> 00:07:49.488
to force people to have something that is very likely

00:07:49.488 --> 00:07:52.299
not good for their health and then you would have to run

00:07:52.299 --> 00:07:55.813
it for a long period of time, you would wait 30 years.

00:07:55.813 --> 00:07:59.135
You would wait 30 years when they're 60 years old and

00:07:59.135 --> 00:08:02.372
you would see what was the heart condition of these folks.

00:08:02.372 --> 00:08:04.361
How many people maybe had heart attacks?

00:08:04.361 --> 00:08:09.815
At age 60, what is their health condition?

00:08:09.815 --> 00:08:14.518
And then statistically, is it unlikely that the

00:08:14.518 --> 00:08:20.499
difference would be due purely to chance alone?

00:08:20.499 --> 00:08:23.257
For example, if you did this, and let's say that yeah these

00:08:23.257 --> 00:08:27.071
people had a slightly higher chance for heart disease

00:08:27.071 --> 00:08:29.512
or heart attacks than these folks after 30 years.

00:08:29.512 --> 00:08:34.719
It would be a good experiment, but it wouldn't allow you

00:08:34.719 --> 00:08:36.784
to conclude that sugar is causing it, cause that might

00:08:36.784 --> 00:08:38.890
have happened through chance alone.

00:08:38.890 --> 00:08:41.989
But if, for example, after 30 years, let's say this group

00:08:41.989 --> 00:08:45.214
right over here has 10 times the risk of a heart attack or

00:08:45.214 --> 00:08:49.753
10 times of whatever the risk factors for heart disease.

00:08:49.753 --> 00:08:52.779
You would statistically say the odds of that happening

00:08:52.779 --> 00:08:55.963
by chance alone, that the 100 people in this group right

00:08:55.963 --> 00:08:59.666
over here having 10 times the chance of heart attack as this

00:08:59.666 --> 00:09:02.892
group right over here, that's unlikely due to chance alone.

00:09:02.892 --> 00:09:04.201
So you would say, "Okay!"

00:09:04.201 --> 00:09:05.923
We would feel good about our conclusion that

00:09:05.923 --> 00:09:09.159
this forced sugar is what is causing that.

00:09:09.621 --> 00:09:12.685
Anyway, we'll dig deeper into each of these three types,

00:09:12.685 --> 00:09:14.753
but the whole point of this video is to just give you an

00:09:14.753 --> 00:09:17.148
appreciation that, you know, we use statistics a lot, but

00:09:17.148 --> 00:09:19.505
this gives you a context for how we're using it in different

00:09:19.505 --> 00:09:22.613
situations when we're performing statistical studies.

00:09:22.613 --> 00:09:26.261
This is to estimate the true parameter for a population.

00:09:26.261 --> 00:09:28.833
What is the actual sugar intake for the population?

00:09:29.064 --> 00:09:32.126
You randomly sample and then you use that sample data

00:09:32.126 --> 00:09:35.521
to create a statistic that estimates the true parameter.

00:09:35.521 --> 00:09:37.985
Observational study, you observe what's going on.

00:09:37.985 --> 00:09:40.182
Sugar intake versus heart risk.

00:09:40.182 --> 00:09:42.694
You say, "Hey, there's a relationship maybe

00:09:42.694 --> 00:09:44.852
"this is worth doing an experiment on."

00:09:44.852 --> 00:09:46.448
Because only through an experiment could

00:09:46.448 --> 00:09:50.369
you attempt to find some type of causality.


Day length in Alaska

WEBVTT
Kind: captions
Language: en

00:00:00.489 --> 00:00:01.967
Sal:The longest day of the year in

00:00:01.967 --> 00:00:08.366
Juneau, Alaska is June 21st.
It's 1,096.5 minutes long.

00:00:08.366 --> 00:00:09.966
Half a year later when
the days are at their

00:00:09.966 --> 00:00:14.866
shortest the days are
about 382.5 minutes long.

00:00:14.966 --> 00:00:19.225
If it's not a leap year
the year is 365 days long

00:00:19.234 --> 00:00:23.753
and June 21st is the
172nd day of the year.

00:00:23.766 --> 00:00:26.227
Write a trigonometric function that models

00:00:26.242 --> 00:00:29.535
the length L of the T-th day of the year.

00:00:29.535 --> 00:00:32.446
It's going to be L as a function of T,

00:00:32.500 --> 00:00:34.902
assuming it's not a leap year.

00:00:35.300 --> 00:00:36.967
I encourage you to pause
this video and try to do

00:00:36.967 --> 00:00:40.216
this on your own before
I try to work through it.

00:00:41.109 --> 00:00:42.966
Let me give a go at it.

00:00:42.966 --> 00:00:45.167
Instead of first starting
at L of T I'm going

00:00:45.167 --> 00:00:48.694
to actually start with
L as a function of U,

00:00:48.694 --> 00:00:53.588
where U is another variable
I'll use as an intermediary

00:00:53.588 --> 00:00:57.419
variable that will help
set it up in a simpler way.

00:00:57.834 --> 00:01:09.534
Where U is days after June 21st.

00:01:10.042 --> 00:01:12.098
Let's just think about this a little bit.

00:01:13.499 --> 00:01:15.366
June 21st.

00:01:15.366 --> 00:01:18.154
If we're thinking about in
terms of U, U is going to be

00:01:18.154 --> 00:01:21.830
equal to zero because it's
zero days after June 21st.

00:01:21.834 --> 00:01:24.367
But if we're thinking
about it in terms of T,

00:01:24.367 --> 00:01:27.671
June 21st is the 172nd day of the year.

00:01:27.671 --> 00:01:29.895
172.

00:01:29.895 --> 00:01:33.366
What's the relationship between U and T?

00:01:33.366 --> 00:01:35.900
It's shifted by 172 days.

00:01:35.900 --> 00:01:42.865
U is going to be equal to T minus 172.

00:01:43.330 --> 00:01:48.225
Notice when T is 172, U is equal to zero.

00:01:48.233 --> 00:01:50.433
Let's figure out L of
U first and then later

00:01:50.433 --> 00:01:54.656
we can just substitute U with T minus 172.

00:01:55.475 --> 00:01:58.367
First of all what's happening
when U is equal to zero?

00:01:58.367 --> 00:01:59.600
Let me write all of this down.

00:01:59.600 --> 00:02:02.832
What is happening when U is equal to zero.

00:02:02.832 --> 00:02:07.976
U equals zero is June 21st
and that's the maximum point.

00:02:07.976 --> 00:02:11.151
What trig function hits its maximum point

00:02:11.151 --> 00:02:14.800
when the input into the
trig function is zero?

00:02:15.100 --> 00:02:18.330
Well sine of zero is zero
while cosine of zero is one.

00:02:18.330 --> 00:02:20.500
Cosine hits its maximum point.

00:02:21.730 --> 00:02:24.430
It seems a little bit easier
to model this with a cosine.

00:02:24.433 --> 00:02:28.309
It's going to be some amplitude

00:02:28.309 --> 00:02:34.623
times cosine of, let's say,

00:02:34.623 --> 00:02:38.566
I'll write some coefficient
C right over here.

00:02:38.566 --> 00:02:41.273
Actually just let me use a
B since I already used an A.

00:02:41.273 --> 00:02:47.683
Some coefficient over
here times our U plus some

00:02:47.683 --> 00:02:52.381
constant that will shift the
entire function up or down.

00:02:52.381 --> 00:02:55.433
This is the form that our
function of U is going to take.

00:02:55.433 --> 00:02:59.191
Now we just have to figure out
what each of these parts are.

00:02:59.234 --> 00:03:01.100
First let's think about what the amplitude

00:03:01.100 --> 00:03:02.700
and what the mid line is going to be.

00:03:02.700 --> 00:03:03.894
The mid line is essentially how

00:03:03.894 --> 00:03:07.400
much we're shifting the function up.

00:03:07.966 --> 00:03:09.933
Let's get our calculator out.

00:03:10.700 --> 00:03:11.827
The mid line is going to be half

00:03:11.827 --> 00:03:13.466
way between these two numbers.

00:03:13.700 --> 00:03:24.116
We could say 1,096.5 plus 382.5

00:03:24.161 --> 00:03:30.266
divided by two gets us to 739.5.

00:03:30.966 --> 00:03:36.400
That's what C is equal
to. C is equal to 739.5.

00:03:38.500 --> 00:03:42.670
Now the amplitude is how much
do we vary from that mid line.

00:03:42.670 --> 00:03:44.500
We could take 1,096 minus this or

00:03:44.500 --> 00:03:47.666
we could take this minus 382.5.

00:03:47.866 --> 00:03:49.110
Let's do that.

00:03:49.110 --> 00:03:55.398
Let's take 1,096.5 minus

00:03:55.398 --> 00:04:03.564
what we just got, 739.5

00:04:05.933 --> 00:04:12.415
and we get 357. This is how
much we vary from that mid line.

00:04:12.415 --> 00:04:15.075
A is equal to 357.

00:04:15.522 --> 00:04:18.469
This right over here is equal to 357.

00:04:18.700 --> 00:04:21.016
What's B going to be equal to?

00:04:21.167 --> 00:04:22.981
For that I always think,
"What's the behavior of the

00:04:22.981 --> 00:04:25.954
"function, what's the period
of the function going to be?"

00:04:26.233 --> 00:04:27.841
Let me make a little table here.

00:04:32.550 --> 00:04:34.766
Let's put some different inputs from U.

00:04:34.766 --> 00:04:40.300
When U is zero we're zero
days after June 21st,

00:04:40.300 --> 00:04:41.901
we're at our maximum point.

00:04:41.901 --> 00:04:45.540
And we already said that what
we want the cosine function

00:04:45.540 --> 00:04:48.290
to evaluate to at that
point is essentially we want

00:04:48.290 --> 00:04:59.904
it to evaluate as 357 times
cosine of zero plus 739.5.

00:05:01.367 --> 00:05:05.900
Now what's a full period?
A full period is a year.

00:05:05.900 --> 00:05:08.233
At a year we get to the
same point in the year,

00:05:08.233 --> 00:05:10.301
which is I guess a little
bit of common sense.

00:05:10.301 --> 00:05:13.300
You go all the way to 365,

00:05:13.853 --> 00:05:17.163
when U is 365 we should
have completed a period.

00:05:17.233 --> 00:05:19.666
We should be back to that maximum point.

00:05:20.300 --> 00:05:28.400
This should essentially be
357 times cosine of two pi.

00:05:28.433 --> 00:05:31.330
If we were just thinking in terms of a

00:05:31.330 --> 00:05:35.500
traditional trig function, if
we just had a theta in here,

00:05:35.500 --> 00:05:37.666
you complete a period every two pi.

00:05:37.669 --> 00:05:41.134
This should be equivalent to
what I'm writing out over here.

00:05:41.700 --> 00:05:45.800
Plus 739.5.

00:05:46.634 --> 00:05:48.817
One way to think about it is B

00:05:48.817 --> 00:05:52.515
times 365 should be equal to two pi.

00:05:52.515 --> 00:05:55.392
Notice this is going to be B times 365,

00:05:55.392 --> 00:05:56.867
so let's write that down.

00:05:57.233 --> 00:06:01.133
B times 365.

00:06:03.563 --> 00:06:05.354
That's the input into the cosine

00:06:05.354 --> 00:06:08.569
function needs to be equal to two pi.

00:06:09.300 --> 00:06:13.134
Or B is equal to two pi over 365.

00:06:14.049 --> 00:06:20.692
B is equal to two pi over 365.

00:06:21.166 --> 00:06:23.767
We are almost done. We
figured out what A, B and C

00:06:23.767 --> 00:06:28.619
are and now we just have to
substitute U with T minus 172

00:06:28.633 --> 00:06:29.966
to get our function of T.

00:06:29.966 --> 00:06:31.266
Let's just do that.

00:06:31.433 --> 00:06:34.100
We get -- we deserve a little
bit of a drum roll now.

00:06:34.101 --> 00:06:39.412
L of T is equal to A,

00:06:39.412 --> 00:06:50.563
which is 357 times cosine
of B two pi over 365.

00:06:50.907 --> 00:06:55.761
Two pi over 365 times

00:06:55.761 --> 00:06:58.300
not U, but now we're going
to write it in terms of T.

00:06:58.300 --> 00:06:59.433
We want to think about day of the year,

00:06:59.433 --> 00:07:01.966
not days after June 21st.

00:07:01.966 --> 00:07:11.660
So times T minus 172 and then
finally plus our mid line.

00:07:11.660 --> 00:07:15.292
Plus 739.5.

00:07:16.972 --> 00:07:19.866
We are done.

00:07:20.166 --> 00:07:21.900
It seems like a very
complicated expression but

00:07:21.900 --> 00:07:24.894
if you just break it
down and think about it,

00:07:24.894 --> 00:07:27.700
make the point that we're
talking about the extreme point,

00:07:27.700 --> 00:07:29.166
either the minimum or the maximum.

00:07:29.166 --> 00:07:33.966
Make that when the input into
our function is zero or two pi.

00:07:33.966 --> 00:07:36.119
Zero is actually the easiest one.

00:07:36.119 --> 00:07:39.366
Then later you can worry about the shift.

00:07:39.366 --> 00:07:41.882
Hopefully you found that helpful.


Three way tug of war vectors

WEBVTT
Kind: captions
Language: en

00:00:00.399 --> 00:00:02.452
Voiceover: Teams A, B, and C

00:00:02.452 --> 00:00:04.222
are playing a game of strength.

00:00:04.222 --> 00:00:07.418
Each team has attached
a rope to a metal ring

00:00:07.418 --> 00:00:10.418
and is trying to pull the
ring into their own area.

00:00:10.418 --> 00:00:12.525
Team areas shown below.

00:00:12.525 --> 00:00:14.145
Let's see what that's ...

00:00:14.145 --> 00:00:15.628
We see that over here team A

00:00:15.628 --> 00:00:17.267
is pulling on the ring in that direction.

00:00:17.267 --> 00:00:18.850
Team B is pulling on the
ring in that direction.

00:00:18.850 --> 00:00:21.516
Team C is pulling on the
ring in that direction.

00:00:21.516 --> 00:00:22.944
They're each trying to get the ring

00:00:22.944 --> 00:00:26.455
into their respective areas,

00:00:26.455 --> 00:00:28.282
it's a three-way tug of war.

00:00:28.282 --> 00:00:32.132
They tell us team A pulls
with a force vector a

00:00:32.132 --> 00:00:35.462
of four times a unit vector i plus zero j.

00:00:35.462 --> 00:00:38.730
This is four times, there's
no vertical component here

00:00:38.730 --> 00:00:41.103
so it's going straight in the positive,

00:00:41.103 --> 00:00:42.931
I guess you can say x direction

00:00:42.931 --> 00:00:44.595
and we can see it right over there.

00:00:44.595 --> 00:00:46.711
A is going straight in
the positive x direction

00:00:46.711 --> 00:00:49.146
so just a multiple of the unit vector i.

00:00:50.546 --> 00:00:53.621
Team B pulls with force
vector negative two i

00:00:53.621 --> 00:00:57.211
plus four j, so it's to the left and up.

00:00:57.211 --> 00:01:00.083
We see that negative two i plus four j

00:01:01.206 --> 00:01:04.206
then we have team C
pulls with a force vector

00:01:04.206 --> 00:01:06.311
negative three i minus three j.

00:01:06.311 --> 00:01:10.722
Let's see, negative three i minus three j.

00:01:10.722 --> 00:01:13.500
Yup that seems consistent
with the diagram.

00:01:13.500 --> 00:01:17.258
Forces are given in
kilo newtons, all right.

00:01:17.258 --> 00:01:19.642
Which team will win the game?

00:01:19.642 --> 00:01:21.730
Think about which team will win the game

00:01:21.730 --> 00:01:23.134
if you think about which one is pulling

00:01:23.134 --> 00:01:25.518
with the most magnitude
in their directions.

00:01:25.518 --> 00:01:27.229
Let me get my little scratch pad out

00:01:27.229 --> 00:01:28.829
and let's think about the magnitudes

00:01:28.829 --> 00:01:30.325
of each of these vectors.

00:01:30.325 --> 00:01:34.517
Let's first think about
the magnitude of vector a,

00:01:34.517 --> 00:01:36.355
what is that going to be?

00:01:36.355 --> 00:01:39.798
Well, it's four times the unit vector i,

00:01:39.798 --> 00:01:41.995
so it's just going to
have a magnitude of four.

00:01:41.995 --> 00:01:43.694
The unit vector i has a magnitude of one

00:01:43.694 --> 00:01:45.854
so it's magnitude is just four times that,

00:01:45.854 --> 00:01:47.700
it's magnitude is equal to four.

00:01:47.700 --> 00:01:50.984
Let's put the units there,
it's four kilo newtons.

00:01:50.984 --> 00:01:53.646
What about the magnitude of vector b?

00:01:56.215 --> 00:01:58.498
We can use a Pythagorean theorem.

00:01:58.498 --> 00:02:03.989
It's negative two i plus four j

00:02:03.989 --> 00:02:06.561
so the magnitude squared is going to be

00:02:06.561 --> 00:02:09.629
negative two squared plus four squared

00:02:09.629 --> 00:02:11.961
which is really just coming
out of the Pythagorean theorem.

00:02:11.961 --> 00:02:16.614
The magnitude of vector b is going to be

00:02:16.614 --> 00:02:22.660
equal to the square root of
negative two squared plus four

00:02:22.660 --> 00:02:25.179
or one way to think about this
is the Pythagorean theorem

00:02:25.179 --> 00:02:27.285
or the distance formula between the origin

00:02:27.285 --> 00:02:28.864
and that point right over there.

00:02:28.864 --> 00:02:31.226
Plus four squared, that's four plus 16

00:02:31.226 --> 00:02:33.670
is the square root of 20.

00:02:33.670 --> 00:02:36.172
Let's see, this is
definitely greater than four.

00:02:36.172 --> 00:02:37.617
Four squared is just 16,

00:02:37.617 --> 00:02:39.692
this is some place between four and five.

00:02:39.692 --> 00:02:45.838
Let's see, team C, the
magnitude of vector c same logic

00:02:45.838 --> 00:02:49.239
is going to be the square
root of negative three squared

00:02:49.239 --> 00:02:53.394
plus negative three squared
which is equal to ...

00:02:53.394 --> 00:02:57.643
Let's see this is square root
of 18, right nine plus nine.

00:02:57.643 --> 00:02:59.126
Square root of 18.

00:02:59.126 --> 00:03:01.844
Square root of 20 is clearly larger than

00:03:01.844 --> 00:03:02.780
square root of 18.

00:03:02.780 --> 00:03:06.553
We already said that square
root of 20 is larger than four.

00:03:07.506 --> 00:03:09.398
Team B is going to win.

00:03:09.398 --> 00:03:11.019
Which team will win the game?

00:03:11.019 --> 00:03:16.211
Team B, and we can even, let's
just fill it out on our ...

00:03:16.211 --> 00:03:18.415
Team B is going to win the game.

00:03:18.415 --> 00:03:21.087
Now the next thing they ask us is

00:03:21.087 --> 00:03:23.894
what is the magnitude of
the teams combined forces

00:03:23.894 --> 00:03:25.405
acting on the ring?

00:03:25.405 --> 00:03:28.934
To figure that out we can just
add up all of these vectors.

00:03:28.934 --> 00:03:32.379
The combined force, so let's write this.

00:03:32.379 --> 00:03:36.636
The force, let's say force
combined, this is a vector

00:03:36.636 --> 00:03:44.747
it's going to be the sum of
vectors a plus vector b plus c

00:03:47.054 --> 00:03:53.489
which is going to be equal
to four i plus zero j,

00:03:55.335 --> 00:04:07.594
minus two i plus four j,
minus three i minus three j.

00:04:07.594 --> 00:04:11.988
Now we can just add up the
i's and add up the j's.

00:04:11.988 --> 00:04:20.228
We have four i minus two i minus three i,

00:04:20.228 --> 00:04:21.652
so four of something minus two

00:04:21.652 --> 00:04:22.759
you're going to have two left.

00:04:22.759 --> 00:04:23.828
Then you subtract another three

00:04:23.828 --> 00:04:25.352
you're going to have negative one of that,

00:04:25.352 --> 00:04:27.365
so that's going to be equal to negative,

00:04:27.365 --> 00:04:30.650
I could write negative
one i or just negative i

00:04:30.650 --> 00:04:37.427
and then we have zero j
plus four j minus three j

00:04:37.427 --> 00:04:40.951
so that's was zero plus four minus three

00:04:40.951 --> 00:04:42.440
is just going to be one j.

00:04:42.440 --> 00:04:45.004
Or you can say just plus j.

00:04:45.004 --> 00:04:49.090
That's the resulting vector
here is negative i plus j.

00:04:49.090 --> 00:04:50.771
It might look something like this.

00:04:50.771 --> 00:05:00.248
Negative i plus a j, gets us
a vector that looks like that.

00:05:00.248 --> 00:05:02.298
Let me actually draw it
a little bit bigger here.

00:05:02.298 --> 00:05:06.688
If I were to draw our coordinate axis

00:05:06.688 --> 00:05:11.620
that's my y-axis, this is my x-axis.

00:05:14.143 --> 00:05:19.009
It's negative one times i, so
let's say it looks like that

00:05:19.009 --> 00:05:20.162
that's negative i.

00:05:20.162 --> 00:05:22.627
That's negative i right over there.

00:05:22.627 --> 00:05:26.119
Negative i, let me write that

00:05:26.119 --> 00:05:28.892
that is negative i plus j.

00:05:28.892 --> 00:05:33.910
Plus the j unit vector which
might look something like this

00:05:33.910 --> 00:05:36.465
both of them have a magnitude of one,

00:05:36.465 --> 00:05:38.301
so that is the j unit vector.

00:05:38.301 --> 00:05:41.216
The sum of those two
is our combined force.

00:05:41.216 --> 00:05:46.777
Our combined force is going to
be going into this direction.

00:05:47.746 --> 00:05:49.319
Hence this is its magnitude.

00:05:49.319 --> 00:05:51.742
The length of it is this
magnitude, combined force

00:05:51.742 --> 00:05:53.242
and that goes in line with

00:05:53.242 --> 00:05:54.920
we're going to be going in that direction

00:05:54.920 --> 00:05:56.998
which takes us to the zone of B.

00:05:56.998 --> 00:05:58.980
They ask us what is the magnitude

00:05:58.980 --> 00:06:00.555
of the teams' combined force?

00:06:00.555 --> 00:06:04.666
Well the magnitude of this side is one,

00:06:04.666 --> 00:06:06.530
it's a unit vector where it's
going to the left direction.

00:06:06.530 --> 00:06:09.016
The magnitude of this side is one

00:06:09.016 --> 00:06:10.760
so the magnitude of this side

00:06:10.760 --> 00:06:12.127
we just use a Pythagorean theorem,

00:06:12.127 --> 00:06:15.357
it's a square root of one
squared plus one squared

00:06:15.357 --> 00:06:19.619
which is just the square root of two.

00:06:20.388 --> 00:06:21.804
The magnitude here,

00:06:21.804 --> 00:06:25.907
what is the magnitude of
the teams combined forces

00:06:25.907 --> 00:06:27.330
is square root of two.

00:06:27.330 --> 00:06:29.750
In what direction is
the ring getting pulled?

00:06:29.750 --> 00:06:30.774
Actually let me just fill out

00:06:30.774 --> 00:06:33.034
what they've already
asked us so we can ...

00:06:33.034 --> 00:06:35.788
What is the magnitude of
the teams' combined force

00:06:35.788 --> 00:06:36.643
acting on the ring?

00:06:36.643 --> 00:06:37.847
Round to the nearest tenth.

00:06:37.847 --> 00:06:39.647
Let's just write square root of two.

00:06:39.647 --> 00:06:43.175
Square root of two is equal to

00:06:43.175 --> 00:06:46.647
and round to the nearest
tenth 1.4 kilo newtons.

00:06:49.154 --> 00:06:51.873
In what direction is
the ring getting pulled

00:06:51.873 --> 00:06:53.178
and they want this is in radians.

00:06:53.178 --> 00:06:54.988
Round it to the nearest tenth,

00:06:54.988 --> 00:06:57.363
your answer should be between
zero and two pi radians.

00:06:57.363 --> 00:07:00.946
That, go back to our diagram,

00:07:00.946 --> 00:07:05.067
we need to figure out what
this angle is right over here.

00:07:05.067 --> 00:07:07.537
We need to figure out what
that angle is in radians

00:07:07.537 --> 00:07:10.771
and we know that this whole thing

00:07:10.771 --> 00:07:12.514
if we went from the positive x-axis

00:07:12.514 --> 00:07:15.913
all the way to the negative
x-axis that is pi radians.

00:07:15.913 --> 00:07:19.801
We also know what this
angle right over here is.

00:07:19.801 --> 00:07:20.938
How do we know that?

00:07:20.938 --> 00:07:22.449
Well, I've just constructed a ...

00:07:22.449 --> 00:07:23.724
Notice this is a right angle

00:07:23.724 --> 00:07:25.690
and this is an isosceles triangle,

00:07:25.690 --> 00:07:27.570
isosceles right triangle.

00:07:27.570 --> 00:07:31.467
It's going to be a 45, 45, 90 triangle

00:07:31.467 --> 00:07:33.392
but if you're thinking in terms of radians

00:07:33.392 --> 00:07:35.750
45 degrees is pi over four.

00:07:35.750 --> 00:07:39.457
This is pi over four, pi over four,

00:07:39.457 --> 00:07:41.959
pi over two triangle.

00:07:41.959 --> 00:07:44.001
Same thing as 45, 45, 90.

00:07:44.001 --> 00:07:46.296
If this whole purple angle is pi

00:07:46.296 --> 00:07:47.494
and this is pi over four,

00:07:47.494 --> 00:07:49.227
you just subtract the
pi over four from the pi

00:07:49.227 --> 00:07:50.685
to get what theta is.

00:07:50.685 --> 00:07:54.168
Theta is going to be equal to ...

00:07:54.168 --> 00:07:58.227
Pi is the same thing as four pi over four

00:07:58.227 --> 00:08:00.441
minus pi over four.

00:08:02.393 --> 00:08:04.120
I just did that to get
a common denominator

00:08:04.120 --> 00:08:06.973
which is equal to three pi over four.

00:08:06.973 --> 00:08:10.473
Three pi over four radians.

00:08:10.473 --> 00:08:12.731
Let's go back over here.

00:08:12.731 --> 00:08:15.170
Now they want us to round
it to the nearest tenth,

00:08:15.170 --> 00:08:21.062
so let's take three
times pi divided by ...

00:08:21.062 --> 00:08:23.576
Just to make sure you can
see what I'm doing here

00:08:23.576 --> 00:08:26.075
so three times pi divided by four

00:08:26.075 --> 00:08:32.877
is equal to 2.356, round
it to the nearest tenth

00:08:32.877 --> 00:08:35.532
would be 2.4 radians.

00:08:35.532 --> 00:08:42.407
Approximately 2.4 radians
and we got it right.


Force contribution while pushing box

WEBVTT
Kind: captions
Language: en

00:00:00.591 --> 00:00:03.022
Voiceover: Let's say that
you have two folks that are

00:00:03.022 --> 00:00:07.377
trying to collectively
push a box across the snow

00:00:07.377 --> 00:00:10.428
towards a target, so
this is where the box is,

00:00:10.428 --> 00:00:14.840
right over here and this is
the target, right over here.

00:00:14.840 --> 00:00:16.683
Let me write that, that is the target.

00:00:16.683 --> 00:00:18.482
That's where they're
trying to get the box,

00:00:18.482 --> 00:00:21.515
and person A, because for
some reason, they can't

00:00:21.515 --> 00:00:24.341
push it from exactly behind the box,

00:00:24.341 --> 00:00:25.915
maybe there's not a good footing there,

00:00:25.915 --> 00:00:27.346
or I guess if they pressed
there, maybe the box

00:00:27.346 --> 00:00:30.891
squeezes a little bit,
so person A has to push

00:00:30.891 --> 00:00:33.556
in a direction that's not
exactly going in the direction

00:00:33.556 --> 00:00:35.409
of the target, so they push in a direction

00:00:35.932 --> 00:00:37.754
that looks like this,

00:00:37.754 --> 00:00:40.624
and so let me show you
this vector I'm drawing,

00:00:40.624 --> 00:00:43.109
essentially represents the
force that they're exerting.

00:00:43.109 --> 00:00:44.944
This is their force vector.

00:00:44.944 --> 00:00:46.762
This is person A's force vector.

00:00:46.762 --> 00:00:50.037
So this is person A's force vector,

00:00:50.037 --> 00:00:51.621
and we know the length of this vector,

00:00:51.621 --> 00:00:53.484
or another way to think
about it, the magnitude

00:00:53.484 --> 00:00:59.155
of vector A is 330 newtons, 330 newtons.

00:00:59.476 --> 00:01:01.151
And let's say person B, once again,

00:01:01.151 --> 00:01:05.022
because they can't push exactly

00:01:05.022 --> 00:01:06.020
in the direction of the target,

00:01:06.020 --> 00:01:08.524
maybe the box is really
soft right over here,

00:01:08.524 --> 00:01:17.154
person B is pushing at
this angle right over here,

00:01:17.154 --> 00:01:19.402
so that right over there, that vector is

00:01:19.402 --> 00:01:23.414
the force that person B is pushing onto it

00:01:23.414 --> 00:01:27.357
and the direction, and the
magnitude of that force,

00:01:27.357 --> 00:01:32.892
of that vector of person
B's pushing, is 300 newtons.

00:01:33.258 --> 00:01:34.868
And we know their angles that those make

00:01:34.868 --> 00:01:36.473
with the direction of the target.

00:01:36.733 --> 00:01:40.800
So if this is the direction
of the target right over here

00:01:40.800 --> 00:01:42.895
we know that this is a 35 degree angle,

00:01:45.418 --> 00:01:50.492
and we know that this
is a 15 degree angle.

00:01:51.040 --> 00:01:54.799
Now what I want you to
do, is pause this video

00:01:54.799 --> 00:01:57.742
and think about how much
of each of their force

00:01:57.742 --> 00:01:59.965
is going in the direction of getting

00:01:59.965 --> 00:02:02.628
the box towards the target, and then who

00:02:02.628 --> 00:02:05.616
is actually exerting more
force in that direction.

00:02:05.616 --> 00:02:08.215
We see that person A is exerting a total,

00:02:08.215 --> 00:02:10.349
a higher magnitude of
force in this direction

00:02:10.349 --> 00:02:12.491
than person B is doing in that direction.

00:02:12.491 --> 00:02:14.533
330 newtons versus 300,

00:02:14.533 --> 00:02:17.558
but who's helping the box
go more in that direction?

00:02:17.699 --> 00:02:18.895
And by how much more?

00:02:18.895 --> 00:02:20.120
And also, what's the total force

00:02:20.120 --> 00:02:22.761
now pushing the box in that direction?

00:02:23.649 --> 00:02:25.825
So I'm assuming you had a go at it,

00:02:25.825 --> 00:02:27.615
and the key here is to find the component

00:02:27.615 --> 00:02:29.343
of each of these vectors, the magnitude

00:02:29.343 --> 00:02:33.449
of each of these vectors
in this direction,

00:02:33.449 --> 00:02:35.285
in the direction towards the target.

00:02:35.633 --> 00:02:40.035
And so, let's look first at vector A.

00:02:40.035 --> 00:02:42.058
So vector A looks like this,
and I'll just draw them

00:02:42.058 --> 00:02:45.127
separately just so we can
clean it up a little bit.

00:02:45.127 --> 00:02:46.241
Vector A looks like this.

00:02:46.241 --> 00:02:49.993
We know its magnitude -
the magnitude of vector A

00:02:49.993 --> 00:02:54.557
is equal to 330 newtons, and if we say

00:02:54.557 --> 00:02:57.182
that the target is in this
direction right over here,

00:02:58.289 --> 00:03:00.838
so that's the target
is some place out here,

00:03:00.838 --> 00:03:04.170
we've already been told
that this is 35 degrees.

00:03:04.886 --> 00:03:10.927
So what we really want to
do is find the component,

00:03:10.927 --> 00:03:14.056
the magnitude of the component

00:03:14.056 --> 00:03:16.721
going in that direction right over there.

00:03:17.430 --> 00:03:18.536
The way we can do that is just

00:03:18.536 --> 00:03:20.728
with our traditional trig functions.

00:03:20.859 --> 00:03:23.419
This right over here is a right triangle.

00:03:23.487 --> 00:03:25.991
We are looking for this
side right over here.

00:03:26.277 --> 00:03:32.882
We have - let me just call that A sub X.

00:03:33.106 --> 00:03:35.562
And we already know
that the magnitude of A

00:03:35.562 --> 00:03:39.118
is 330 newtons, so the magnitude -

00:03:39.118 --> 00:03:41.223
let me just say that the magnitude -

00:03:41.223 --> 00:03:42.860
let me just write it this
way so we don't confuse.

00:03:42.860 --> 00:03:45.110
So let's just say the
magnitude of the vector

00:03:45.110 --> 00:03:48.151
in the X direction, so this
vector right over here,

00:03:48.151 --> 00:03:49.783
we can write like this.

00:03:49.861 --> 00:03:51.209
The magnitude of that, we'll just write it

00:03:51.209 --> 00:03:53.134
without the vector notation.

00:03:53.134 --> 00:03:55.350
So how can we think about that?

00:03:55.350 --> 00:03:58.237
Well we know cosine is
adjacent over hypotenuse,

00:03:58.237 --> 00:04:04.222
so we could write cosine
of 35 degrees is equal

00:04:04.222 --> 00:04:06.264
to the length of the adjacent side.

00:04:06.264 --> 00:04:09.260
That would just be A sub X,
without the vector over it.

00:04:09.260 --> 00:04:10.411
We're just saying that
that's the magnitude

00:04:10.411 --> 00:04:14.331
of vector A sub X, over
the magnitude of vector A,

00:04:14.489 --> 00:04:23.586
over 330 newtons, or we
could say that A sub X

00:04:23.586 --> 00:04:31.184
is equal to 330 times
the cosine of 35 degrees.

00:04:31.883 --> 00:04:35.238
And we can make the exact
same argument for B.

00:04:35.238 --> 00:04:38.389
Vector B, so let me draw it like this.

00:04:38.389 --> 00:04:41.950
Vector B, you could, maybe
I'll draw it like this

00:04:41.950 --> 00:04:43.399
just to make it a little bit -

00:04:43.399 --> 00:04:45.549
let me do it a little bit different.

00:04:45.549 --> 00:04:48.241
If this is the direction to the target,

00:04:48.241 --> 00:04:50.904
so once again, I'll just draw
a horizontal line for that.

00:04:50.904 --> 00:04:51.922
Then relative to that,

00:04:51.922 --> 00:04:54.931
vector B looks something like this.

00:04:57.337 --> 00:05:00.757
Vector B looks something like that,

00:05:00.757 --> 00:05:03.606
so that is vector B.

00:05:03.611 --> 00:05:06.049
B sub X, in the direction of the target -

00:05:06.049 --> 00:05:09.440
so we would drop a
perpendicular like that.

00:05:09.440 --> 00:05:12.455
This would be - this
right over here would be

00:05:12.455 --> 00:05:17.703
the vector B sub X, and
so what is the magnitude

00:05:17.703 --> 00:05:19.502
of B sub X going to be equal to?

00:05:19.502 --> 00:05:21.941
Now we could say that the
magnitude of B sub X -

00:05:21.941 --> 00:05:25.359
we'll just call that B sub X
without the vector notation,

00:05:25.359 --> 00:05:26.785
same exact logic.

00:05:26.798 --> 00:05:29.032
This is 15 degrees.

00:05:29.516 --> 00:05:32.352
Cosine of 15 degrees is
going to be the length

00:05:32.352 --> 00:05:33.403
of the adjacent side over

00:05:33.403 --> 00:05:35.135
the length of the hypotenuse.

00:05:35.135 --> 00:05:35.740
So we already know that the length

00:05:35.740 --> 00:05:39.070
of the hypotenuse is 300 newtons.

00:05:39.339 --> 00:05:44.298
So we could write that
cosine of 15 degrees

00:05:44.298 --> 00:05:47.637
is equal to B sub X,
length of the adjacent side

00:05:47.637 --> 00:05:49.384
over the length of the hypotenuse.

00:05:49.769 --> 00:05:53.718
Or that B sub X is equal to 300

00:05:53.718 --> 00:05:58.679
times cosine of 15 degrees.

00:05:59.323 --> 00:06:00.812
So let's get our calculator out

00:06:00.812 --> 00:06:03.280
and let's calculate what these things are.

00:06:04.484 --> 00:06:10.673
So, let's see, we have 330 times

00:06:10.673 --> 00:06:17.369
cosine of 35 degrees,
gets us to 270 newtons.

00:06:17.369 --> 00:06:19.509
So that's A sub X is 270 newtons,

00:06:19.509 --> 00:06:24.828
and B sub X is 300 times
cosine of 15 degrees.

00:06:24.828 --> 00:06:30.092
We get 289.777, so what we see is,

00:06:30.092 --> 00:06:32.334
even though B's magnitude is less

00:06:32.334 --> 00:06:36.462
than A's magnitude, the
component of vector B

00:06:36.462 --> 00:06:38.325
going the direction of the
target is actually larger

00:06:38.325 --> 00:06:40.573
than the component of
vector A going in it.

00:06:40.573 --> 00:06:43.410
So if we were rounding
to the nearest newton,

00:06:45.700 --> 00:06:48.455
this right over here, the
magnitude of this vector

00:06:48.455 --> 00:06:53.150
right over here, B sub X, that is,

00:06:53.150 --> 00:06:56.017
if we round to the nearest
newton, 290 newtons.

00:06:56.622 --> 00:07:01.329
So this is approximately
290 newtons length,

00:07:01.329 --> 00:07:02.939
or I guess you could say magnitude,

00:07:02.939 --> 00:07:06.870
while this one is a little bit shorter -

00:07:06.870 --> 00:07:08.534
it's a little bit shorter.

00:07:08.534 --> 00:07:11.029
We saw if we round to the nearest newton,

00:07:11.029 --> 00:07:12.804
it's about 270 newtons.

00:07:13.206 --> 00:07:19.373
The length of this one is
270 newtons, approximately.

00:07:19.757 --> 00:07:21.376
So if you were to say, how much more

00:07:21.376 --> 00:07:24.364
is person B pushing in the direction

00:07:24.364 --> 00:07:26.442
that we care about, it's about -

00:07:26.442 --> 00:07:28.143
well, if we want to be
a little more precise,

00:07:28.143 --> 00:07:30.654
we can subtract the two, so we can take

00:07:30.654 --> 00:07:40.541
300 cosine of 15, minus 330 cosine of 35,

00:07:40.541 --> 00:07:44.773
and we get about 19.5 newtons difference.

00:07:44.773 --> 00:07:47.022
The blue, person B, is contributing

00:07:47.022 --> 00:07:53.204
19.458 newtons more in that direction

00:07:53.204 --> 00:07:55.147
towards the target than person A is.

00:07:55.147 --> 00:07:56.390
But if we wanted to talk about,

00:07:56.390 --> 00:08:00.123
what is the total force
going in that direction,

00:08:00.123 --> 00:08:02.446
then we would take the
sum of these two things.

00:08:02.446 --> 00:08:07.798
So we would - the total
force in that direction

00:08:07.798 --> 00:08:10.633
is going to be 560 newtons

00:08:10.633 --> 00:08:12.181
if we round to the nearest newton.

00:08:12.181 --> 00:08:14.106
So if you add this blue component

00:08:14.106 --> 00:08:17.138
to this magenta component,
you get this one

00:08:17.138 --> 00:08:22.582
right over here, which is 560 newtons.

00:08:22.582 --> 00:08:27.173
So this whole vector right
over here, it's magnitude -

00:08:28.726 --> 00:08:35.755
so I could write that as
the magnitude of A sub X,

00:08:35.755 --> 00:08:41.486
plus B sub X, which is
the same thing as A sub X,

00:08:41.486 --> 00:08:43.978
plus B sub X - I already
said these are the equivalent

00:08:43.978 --> 00:08:46.722
of the magnitude of each of
these vectors is equal to

00:08:47.521 --> 00:08:55.149
- I could write approximately
equal 560 newtons.


Total displacement while hiking

WEBVTT
Kind: captions
Language: en

00:00:00.516 --> 00:00:02.523
Keita left camp three days ago

00:00:02.523 --> 00:00:04.188
on a journey into the jungle.

00:00:04.188 --> 00:00:06.436
The three days of his
journey can be described

00:00:06.436 --> 00:00:10.308
by displacement, distance
and direction vectors,

00:00:10.308 --> 00:00:12.807
or displacement vectors,

00:00:12.807 --> 00:00:15.343
and displacement is
distance with the direction,

00:00:15.343 --> 00:00:17.632
and the vectors are d1, d2, and d3.

00:00:17.632 --> 00:00:19.015
They list them right over here.

00:00:19.015 --> 00:00:20.648
The distances are given in kilometers.

00:00:20.648 --> 00:00:24.048
How far is Keita from camp
at the end of day three?

00:00:24.048 --> 00:00:26.126
So let's just think
about what is happening.

00:00:26.126 --> 00:00:28.715
On day one, let's say this
is his starting point.

00:00:28.715 --> 00:00:32.331
His displacement, he starts
here and he goes there,

00:00:32.331 --> 00:00:34.252
but if you break it down based on,

00:00:34.252 --> 00:00:38.208
I guess if you call this direction

00:00:38.208 --> 00:00:39.628
Let me draw a little compass here.

00:00:39.628 --> 00:00:41.405
If you say that this is north,

00:00:41.405 --> 00:00:43.845
this is east,

00:00:43.845 --> 00:00:46.389
this is west, and that this is south,

00:00:46.389 --> 00:00:48.014
you can break it down by how much

00:00:48.014 --> 00:00:49.741
he went in the east direction

00:00:49.741 --> 00:00:51.766
and how much he went
in the north direction.

00:00:51.766 --> 00:00:54.734
So this is saying he went
seven in the east direction.

00:00:54.734 --> 00:00:58.153
One, two, three, four, five, six, seven.

00:00:58.153 --> 00:00:59.570
Seven in the east direction,

00:00:59.570 --> 00:01:01.666
and he went eight in the north direction.

00:01:01.666 --> 00:01:05.578
One, two, three, four,
five, six, seven, eight.

00:01:05.578 --> 00:01:07.146
Just like that.

00:01:07.146 --> 00:01:09.314
This is seven and this is eight.

00:01:09.314 --> 00:01:15.954
Then on day two, he went six units,

00:01:15.954 --> 00:01:17.100
I guess these are kilometers,

00:01:17.100 --> 00:01:18.922
six kilometers to the east,

00:01:18.922 --> 00:01:21.707
one, two, three, four, five, six,

00:01:21.707 --> 00:01:25.322
and he went two kilometers
to the north, one, two.

00:01:25.322 --> 00:01:28.403
So he ends up right over here.

00:01:28.403 --> 00:01:30.556
Six and two, and then finally day three,

00:01:30.556 --> 00:01:33.721
the component of his displacement

00:01:33.721 --> 00:01:36.890
that is to the east is two,

00:01:36.890 --> 00:01:38.843
and the component of his displacement

00:01:38.843 --> 00:01:42.837
that is to the north is nine.

00:01:42.837 --> 00:01:45.588
And so to figure out
how far he is from camp

00:01:45.588 --> 00:01:47.531
at the end of day three,
we just have to figure out

00:01:47.531 --> 00:01:49.275
what is his total displacement?

00:01:49.275 --> 00:01:51.204
What is the length of the vector

00:01:51.204 --> 00:01:53.732
that is the sum of all of these?

00:01:53.732 --> 00:01:57.963
So what is the length of
this vector right over here?

00:01:57.963 --> 00:02:01.949
Let's call this d sub t for total,

00:02:01.949 --> 00:02:01.977
displacement d sub t for total.

00:02:01.977 --> 00:02:04.097
Displacement total, total displacement.

00:02:04.097 --> 00:02:06.344
And you can see how this is arranged,

00:02:06.344 --> 00:02:08.824
that our total displacement
vector is just going

00:02:08.824 --> 00:02:11.058
to be the sum of d1,

00:02:11.058 --> 00:02:13.537
let me do those in the appropriate colors.

00:02:13.537 --> 00:02:25.233
It's just going to be the
sum of d1, d2, and d3.

00:02:25.617 --> 00:02:26.683
And if you're summing these vectors,

00:02:26.683 --> 00:02:28.434
you can just add the
corresponding component.

00:02:28.434 --> 00:02:33.034
So for example, the total
displacement is going to be equal

00:02:33.034 --> 00:02:35.258
to the sum of the horizontal,

00:02:35.258 --> 00:02:38.401
I should say the displacement
in the east direction.

00:02:38.401 --> 00:02:49.718
So it's going to be
seven plus six plus two,

00:02:49.888 --> 00:02:54.079
and then in the north direction you have

00:02:54.079 --> 00:03:02.345
eight plus six, or sorry,
eight plus two plus nine.

00:03:02.698 --> 00:03:04.420
And so our total displacement vector

00:03:04.420 --> 00:03:06.269
if we were to write it in this form,

00:03:06.269 --> 00:03:08.996
is going to be, this is 13 plus two,

00:03:08.996 --> 00:03:11.509
so it's going to be 15 to the east,

00:03:11.509 --> 00:03:13.733
or the component of displacement

00:03:13.733 --> 00:03:15.540
in the eastern direction is 15,

00:03:15.540 --> 00:03:18.582
and in the northern direction
is 10 plus nine, 19.

00:03:18.582 --> 00:03:21.805
So that's this vector right over here.

00:03:21.805 --> 00:03:26.308
It's 15, it's component in the east is 15,

00:03:26.308 --> 00:03:30.778
and to the north is 19.

00:03:30.916 --> 00:03:32.383
Let me make that clear.

00:03:32.383 --> 00:03:34.704
So this distance right over here,

00:03:34.704 --> 00:03:38.624
or if I were to make this
a triangle the length

00:03:38.624 --> 00:03:40.944
of this side of the triangle here is 19

00:03:40.944 --> 00:03:46.521
and that's his total displacement
in the northern direction,

00:03:46.521 --> 00:03:47.695
and his total displacement

00:03:47.695 --> 00:03:57.111
in the eastern direction is that 15.

00:03:57.326 --> 00:03:59.488
So what's going to be the length
of our displacement vector

00:03:59.488 --> 00:04:01.809
or what's the magnitude of
our displacement vector?

00:04:01.809 --> 00:04:05.258
The magnitude of our
total displacement is,

00:04:05.258 --> 00:04:06.969
well it's the Pythagorean theorem.

00:04:06.969 --> 00:04:07.954
This is a right triangle.

00:04:07.954 --> 00:04:09.552
15 squared plus 19 squared is going

00:04:09.552 --> 00:04:11.089
to be the magnitude squared.

00:04:11.089 --> 00:04:12.945
Or we could say that
the magnitude is equal

00:04:12.945 --> 00:04:18.185
to the square root of 15
squared plus 19 squared,

00:04:18.185 --> 00:04:21.218
plus 19 squared.

00:04:21.218 --> 00:04:23.402
So let me get my calculator out.

00:04:23.402 --> 00:04:25.241
All right, so I have the square root

00:04:25.241 --> 00:04:32.561
of 15 squared plus 19 squared gives, 24,

00:04:32.561 --> 00:04:35.018
let's see, they want us to
round to the nearest tenth,

00:04:35.018 --> 00:04:37.240
24 point two.

00:04:37.240 --> 00:04:39.897
So 24 point two kilometers.

00:04:39.897 --> 00:04:41.281
Then they say what direction is Keita

00:04:41.281 --> 00:04:43.329
from camp at the end of day three?

00:04:43.329 --> 00:04:44.490
Round to the nearest degree.

00:04:44.490 --> 00:04:46.994
Your answer should be
between zero and 180 degrees.

00:04:46.994 --> 00:04:50.386
So distance I'm assuming, this
would just be the convention,

00:04:50.386 --> 00:04:51.938
they should be a little bit more precise

00:04:51.938 --> 00:04:53.123
here in their wording

00:04:53.123 --> 00:04:54.313
or a little bit less ambiguous.

00:04:54.313 --> 00:04:57.905
The convention is, the angle relative to,

00:04:57.905 --> 00:05:00.305
if we were thinking of
the coordinate axes,

00:05:00.305 --> 00:05:01.778
the positive X axis,

00:05:01.778 --> 00:05:04.057
on this you could say
the eastern direction,

00:05:04.057 --> 00:05:07.073
so it's really this angle right over here,

00:05:07.073 --> 00:05:08.281
let's call that theta.

00:05:08.281 --> 00:05:09.800
How could we figure out what theta is?

00:05:09.800 --> 00:05:11.535
It's part of this right triangle.

00:05:11.535 --> 00:05:13.128
We know this side has length 19

00:05:13.128 --> 00:05:14.903
and we know this side has 15.

00:05:14.903 --> 00:05:16.936
So we know the opposite side theta

00:05:16.936 --> 00:05:18.519
and we know the adjacent side theta,

00:05:18.519 --> 00:05:21.559
so what trig function involves
opposite and adjacent?

00:05:21.559 --> 00:05:25.336
Well tangent involves
opposite and adjacent,

00:05:25.336 --> 00:05:31.992
so we could write that
the tangent of theta is

00:05:31.992 --> 00:05:34.183
equal to 19 over 15,

00:05:34.183 --> 00:05:41.529
is equal to opposite over
adjacent, 19 over 15,

00:05:41.529 --> 00:05:43.768
and to solve for theta we can just say,

00:05:43.768 --> 00:05:46.436
now we've got to make sure that our angle,

00:05:46.436 --> 00:05:48.333
if we just take the inverse tangent here,

00:05:48.333 --> 00:05:50.486
is actually the angle we're looking for,

00:05:50.486 --> 00:05:52.565
but inverse tangent, the convention is,

00:05:52.565 --> 00:05:54.014
it will give you an angle that is

00:05:54.014 --> 00:05:57.150
between negative pi over two and pi,

00:05:57.150 --> 00:05:58.092
if we're thinking in radiants,

00:05:58.092 --> 00:05:59.370
but since we're thinking in degrees,

00:05:59.370 --> 00:05:59.766
it'll give you an angle

00:05:59.766 --> 00:06:02.120
between negative 90
degrees and 90 degrees,

00:06:02.120 --> 00:06:04.032
and this angle is clearly in that.

00:06:04.032 --> 00:06:07.256
It's actually between zero and 90 degrees.

00:06:07.256 --> 00:06:08.800
So when we take inverse tangent,

00:06:08.800 --> 00:06:10.728
we know that we're going
to get the right angle.

00:06:10.728 --> 00:06:12.336
Otherwise we would have
to make some adjustments.

00:06:12.336 --> 00:06:14.231
So theta is going to be equal

00:06:14.231 --> 00:06:21.520
to the inverse tangent of 19 over 15,

00:06:21.582 --> 00:06:24.713
which is equal to, let's
get our calculator out,

00:06:24.713 --> 00:06:26.322
let's make sure that we are in,

00:06:26.322 --> 00:06:28.026
yep we are in degree mode,

00:06:28.026 --> 00:06:36.330
and so let's take the inverse
tangent of 19 divided by 15,

00:06:36.330 --> 00:06:38.281
which is equal, let's see they want us

00:06:38.281 --> 00:06:39.610
to round to the nearest degree,

00:06:39.610 --> 00:06:43.393
so 52 degrees if we round
to the nearest degree.

00:06:43.393 --> 00:06:47.601
52 degrees, which also looks
right, right over here.

00:06:47.601 --> 00:06:50.377
This looks like a little bit
more than a 45 degree angle,

00:06:50.377 --> 00:06:53.017
which 52 degrees seems to fit the bill,

00:06:53.017 --> 00:06:54.892
and we are done.


Power series of ln(1+x_)

WEBVTT
Kind: captions
Language: en

00:00:00.346 --> 00:00:02.233
Voiceover:We have an infinite series here,

00:00:02.233 --> 00:00:03.566
and the first thing I'd like you to try

00:00:03.566 --> 00:00:06.700
is to pause this video and
see if you can express this

00:00:06.700 --> 00:00:09.233
as an infinite geometric series,

00:00:09.233 --> 00:00:11.966
and if you can express it as
an infinite geometric series,

00:00:11.966 --> 00:00:15.833
see what its sum would be given
an interval of convergence.

00:00:15.833 --> 00:00:18.700
Figure out over what
interval of xs would your

00:00:18.700 --> 00:00:20.167
infinite geometric series converge

00:00:20.167 --> 00:00:22.333
and what would that sum actually be.

00:00:22.968 --> 00:00:24.433
I'm assuming you've given a go at it,

00:00:24.433 --> 00:00:26.533
so let's try to work
through this together.

00:00:27.023 --> 00:00:28.434
The first thing I want to do is

00:00:28.434 --> 00:00:30.100
let me just factor out a common factor.

00:00:30.100 --> 00:00:32.566
This might simplify it in
terms of trying to express it.

00:00:32.566 --> 00:00:33.035
Let's see.

00:00:33.035 --> 00:00:34.366
If I factor out, it
looks like all of these

00:00:34.366 --> 00:00:36.767
are divisible by three x squared.

00:00:36.767 --> 00:00:41.812
I can rewrite this as
three x squared times

00:00:42.627 --> 00:00:50.467
one minus x to the third
power plus x to sixth power

00:00:50.900 --> 00:00:56.100
minus x to the ninth power,

00:00:56.100 --> 00:00:59.466
and a pattern is starting to emerge.

00:00:59.832 --> 00:01:02.566
Let me actually close the
brackets with the same color,

00:01:02.566 --> 00:01:04.133
with that pink right over there.

00:01:04.367 --> 00:01:04.703
Let's see.

00:01:04.703 --> 00:01:09.100
This looks like we are taking
powers of x to the third,

00:01:09.100 --> 00:01:10.166
so let me write it that way.

00:01:10.166 --> 00:01:14.035
This is the same thing
as three x squared times,

00:01:14.035 --> 00:01:16.647
we could write this first term,

00:01:16.647 --> 00:01:18.300
or I guess I could say the zeroth term.

00:01:18.300 --> 00:01:21.833
This is x to the third to the zeroth power

00:01:21.833 --> 00:01:27.500
then minus, this is x to the
third to the first power,

00:01:27.500 --> 00:01:31.766
and then plus this is x to
the third to the second power,

00:01:31.766 --> 00:01:33.192
and then I think you see what's going on.

00:01:33.192 --> 00:01:35.679
This is x to the third to the third power,

00:01:35.679 --> 00:01:37.300
and of course, we can keep going.

00:01:37.300 --> 00:01:38.500
But now we have to worry about

00:01:38.500 --> 00:01:41.167
the switching of signs
that we keep having.

00:01:41.167 --> 00:01:42.900
This would be negative one.

00:01:43.100 --> 00:01:44.500
This is positive, which is the same thing

00:01:44.500 --> 00:01:46.500
as negative one in the zero power.

00:01:46.500 --> 00:01:49.017
This is negative, which is
negative one to the first power,

00:01:49.017 --> 00:01:50.500
so let's actually write it this way.

00:01:50.500 --> 00:01:55.468
We can write it as three x squared times

00:01:56.033 --> 00:01:58.833
this first term we could
write as negative one,

00:01:58.833 --> 00:02:01.966
or we could just write as
negative x to the third

00:02:01.966 --> 00:02:04.179
to the zeroth power,

00:02:04.179 --> 00:02:05.833
and then you're going to say plus.

00:02:05.833 --> 00:02:08.957
With plus, we could say
negative x to the third

00:02:08.957 --> 00:02:10.633
to the first power.

00:02:10.633 --> 00:02:13.033
Negative one to the first
power is negative one.

00:02:13.371 --> 00:02:15.866
x to the third to the first
power is x to the third

00:02:15.901 --> 00:02:20.433
plus negative x to the
third to the second power

00:02:20.433 --> 00:02:24.966
plus negative x to the
third to the third power.

00:02:24.966 --> 00:02:26.766
That's this term right over here.

00:02:26.766 --> 00:02:28.500
Negative one to the third
power is negative one,

00:02:28.500 --> 00:02:31.233
and of course, x to the third
to the third is x to the ninth

00:02:31.233 --> 00:02:32.667
and it keeps going.

00:02:32.702 --> 00:02:36.567
This makes it a lot clearer
what our common ratio is.

00:02:36.567 --> 00:02:42.134
Our common ratio here is
negative x to the third.

00:02:42.500 --> 00:02:45.433
Over what interval would this converge?

00:02:45.433 --> 00:02:47.567
It's going to converge if our common,

00:02:47.567 --> 00:02:50.300
if the absolute value of our common ratio

00:02:50.300 --> 00:02:52.833
is less than one.

00:02:52.833 --> 00:02:55.133
We are going to converge

00:02:57.433 --> 00:03:01.500
if the absolute value of our common ratio,

00:03:01.500 --> 00:03:03.766
the absolute value of our common ratio,

00:03:03.766 --> 00:03:08.567
which is negative x to the
third is less than one.

00:03:08.567 --> 00:03:11.033
Or another way of saying
this is the same thing,

00:03:11.033 --> 00:03:13.700
the absolute value of a
negative is going to be

00:03:13.700 --> 00:03:15.766
the same thing as the
absolute value of a positive,

00:03:15.766 --> 00:03:17.633
so that's the same way of
saying the absolute value

00:03:17.633 --> 00:03:20.568
of x to the third is less than one,

00:03:20.568 --> 00:03:23.499
or saying that x to the
third is less than one

00:03:23.499 --> 00:03:27.166
and is greater than negative one.

00:03:27.166 --> 00:03:28.500
The way that's going to happen,

00:03:28.500 --> 00:03:32.100
if you take the cube roots
of both sides of this,

00:03:32.653 --> 00:03:34.369
or all the sides of this inequality,

00:03:34.369 --> 00:03:36.833
you're going to get that
x is going to be between

00:03:36.833 --> 00:03:38.167
negative one and one.

00:03:38.167 --> 00:03:41.500
This right over here is our
interval of convergence,

00:03:41.715 --> 00:03:45.800
interval of convergence.

00:03:46.434 --> 00:03:49.433
If we restrict our xs to that,

00:03:49.433 --> 00:03:51.100
what is this going to sum to?

00:03:51.100 --> 00:03:53.568
This infinite geometric
series, our common ratio,

00:03:53.568 --> 00:03:56.033
its absolute value is less than one,

00:03:56.033 --> 00:03:58.500
and so this is going to sum to,

00:03:58.500 --> 00:04:02.100
this is going to be
equal to our first term,

00:04:02.100 --> 00:04:02.967
I guess we could say,

00:04:02.967 --> 00:04:04.679
or the thing that's multiplying
them by this whole thing,

00:04:04.679 --> 00:04:05.561
but if you multiply it out,

00:04:05.561 --> 00:04:07.166
this would be our first term.

00:04:07.166 --> 00:04:09.700
It's going to be three x squared,

00:04:09.700 --> 00:04:13.833
all of that over one
minus our common ratio,

00:04:13.833 --> 00:04:16.366
so one minus negative x to the third,

00:04:16.366 --> 00:04:19.733
that's just going to be
one plus x to the third.

00:04:20.100 --> 00:04:22.100
Everything we've done so far

00:04:22.100 --> 00:04:25.433
is we've shown that this,

00:04:25.433 --> 00:04:26.367
let me actually write it this way,

00:04:26.367 --> 00:04:29.301
that this thing is equal to this thing

00:04:29.301 --> 00:04:32.200
over the interval of convergence.

00:04:32.300 --> 00:04:36.966
Let me write, copy and paste
it, so write like that,

00:04:36.966 --> 00:04:39.433
over the interval of convergence.

00:04:39.433 --> 00:04:41.366
If x is between negative one and one,

00:04:41.366 --> 00:04:43.400
these two things are the same.

00:04:43.633 --> 00:04:46.567
Now, we can start to put
our calculus hat on here

00:04:46.567 --> 00:04:48.433
because this looks interesting.

00:04:48.433 --> 00:04:50.366
This, you might remember,

00:04:50.366 --> 00:04:52.633
this looks like the derivative
of something that's familiar.

00:04:52.633 --> 00:04:54.674
One plus x to the third,
what's the derivative of that?

00:04:54.674 --> 00:04:56.634
That's three x squared.

00:04:56.972 --> 00:05:01.766
It looks like this right over
here is the derivative of

00:05:01.766 --> 00:05:05.300
the natural log of one plus x to the third

00:05:05.300 --> 00:05:08.100
or the absolute value of
one plus x to the third.

00:05:08.100 --> 00:05:09.567
If you don't believe me,

00:05:09.567 --> 00:05:12.502
let's take the
anti-derivative of this thing

00:05:12.502 --> 00:05:13.684
right over here.

00:05:13.684 --> 00:05:15.100
In fact, for fun,

00:05:15.100 --> 00:05:17.967
let's take the anti-derivative
of both sides of this,

00:05:17.967 --> 00:05:21.240
and if we do that, then we
will have shown essentially

00:05:21.240 --> 00:05:25.567
a geometric series representation

00:05:25.567 --> 00:05:28.166
of whatever the anti-derivative
of this thing is.

00:05:28.166 --> 00:05:29.966
I encourage you to pause the video again

00:05:29.966 --> 00:05:32.034
and try to take the anti-derivative

00:05:32.034 --> 00:05:33.933
of both sides of this equation.

00:05:36.521 --> 00:05:37.966
We're going to take the anti-derivative

00:05:37.966 --> 00:05:39.433
of the left-hand side,

00:05:39.433 --> 00:05:41.821
and we're going to take
the anti-derivative,

00:05:41.821 --> 00:05:44.800
the anti-derivative of
the right-hand side.

00:05:44.900 --> 00:05:46.033
Now on the left-hand side,

00:05:46.033 --> 00:05:48.112
I mentioned that it looks like we have

00:05:48.527 --> 00:05:50.100
an expression and its derivative.

00:05:50.100 --> 00:05:52.866
That just calls out for u-substitution.

00:05:53.234 --> 00:05:57.333
If we say that u is equal
to one plus x to the third,

00:05:57.566 --> 00:05:58.933
let me write this down,

00:05:59.366 --> 00:06:02.967
u is equal to one plus x to the third,

00:06:02.967 --> 00:06:04.866
then what's du going to be?

00:06:05.366 --> 00:06:11.668
Then du is going to be
equal to three x squared dx.

00:06:11.834 --> 00:06:15.200
Notice, we have u and then du.

00:06:15.568 --> 00:06:19.134
du is this right over here.

00:06:19.680 --> 00:06:23.833
This expression right over
here could be rewritten as,

00:06:23.833 --> 00:06:25.366
let me go over here,

00:06:25.366 --> 00:06:34.833
this could be rewritten as
the integral of du over u,

00:06:36.079 --> 00:06:36.767
or I could say,

00:06:36.767 --> 00:06:37.700
actually, let me write this,

00:06:37.700 --> 00:06:42.887
one over u du,

00:06:43.071 --> 00:06:45.900
which of course is equal to,

00:06:45.900 --> 00:06:48.633
which is going to be
equal to the natural log

00:06:48.633 --> 00:06:52.033
of the absolute value of u,

00:06:52.033 --> 00:06:55.046
the natural log of the absolute value of u

00:06:55.046 --> 00:06:57.669
plus some constant.

00:06:59.566 --> 00:07:02.909
We of course know that u
is one plus x to the third,

00:07:02.909 --> 00:07:06.100
so this is going to be
equal to the natural log

00:07:06.100 --> 00:07:09.167
of the absolute value of
one plus x to the third,

00:07:09.167 --> 00:07:12.700
one plus x to the third plus c,

00:07:13.038 --> 00:07:14.366
plus c.

00:07:14.366 --> 00:07:18.034
Now, we're restricting
our domain for x being

00:07:18.034 --> 00:07:19.434
between negative one and one.

00:07:19.434 --> 00:07:24.359
So for that domain, this
thing is always going to be,

00:07:25.298 --> 00:07:28.720
this whole thing actually is
always going to be positive,

00:07:29.643 --> 00:07:31.601
so what we can do,

00:07:33.566 --> 00:07:35.568
we don't have to write
the absolute value sign,

00:07:35.568 --> 00:07:38.267
so this is going to be
equal to the natural log,

00:07:38.302 --> 00:07:39.666
let me write it,

00:07:39.766 --> 00:07:43.501
the natural log of one
plus x to the third,

00:07:43.501 --> 00:07:46.533
one plus x to the third plus c,

00:07:47.042 --> 00:07:48.500
plus c.

00:07:48.500 --> 00:07:50.301
That's this left-hand side,

00:07:50.301 --> 00:07:51.966
and the right-hand side is actually

00:07:51.966 --> 00:07:52.900
a lot more straightforward.

00:07:52.900 --> 00:07:55.233
This is just a straightforward polynomial.

00:07:55.233 --> 00:07:56.673
Now, as you can imagine,
we're going to get

00:07:56.673 --> 00:07:58.033
some type of constant there,

00:07:58.033 --> 00:07:59.361
so let me differentiate them a little bit.

00:07:59.361 --> 00:08:01.502
Let me call this one c one,

00:08:01.502 --> 00:08:04.567
and then on the right-hand
side, what do we get?

00:08:04.567 --> 00:08:08.300
The anti-derivative of
this is going to be,

00:08:08.300 --> 00:08:08.634
let's see,

00:08:08.634 --> 00:08:11.767
the anti-derivative of x
squared is x to the third

00:08:11.767 --> 00:08:14.000
divided by three.

00:08:14.100 --> 00:08:16.566
This first term, the anti-derivative,

00:08:16.566 --> 00:08:19.734
is just going to be x to the third power.

00:08:19.967 --> 00:08:22.566
The derivative of x to the
third is three x squared.

00:08:22.566 --> 00:08:25.368
Now, this term right over here,

00:08:25.368 --> 00:08:27.766
negative three x to the fifth,

00:08:27.766 --> 00:08:29.835
the anti-derivative of x to the fifth

00:08:29.835 --> 00:08:34.968
is x to the sixth over six,

00:08:35.192 --> 00:08:36.833
x to the sixth over six,

00:08:36.833 --> 00:08:38.233
but then we have that three over here.

00:08:38.233 --> 00:08:39.833
Three over six is two,

00:08:39.833 --> 00:08:44.767
so it's negative x to the sixth over two.

00:08:45.013 --> 00:08:46.367
Actually, let me do that
in a different color,

00:08:46.367 --> 00:08:47.933
just so we can keep track of it.

00:08:48.566 --> 00:08:51.100
This one right over here is negative,

00:08:51.100 --> 00:08:54.500
the anti-derivative is negative
x to the sixth over two,

00:08:54.946 --> 00:08:56.033
and then, let's see,

00:08:56.033 --> 00:08:57.302
I'm running out of colors.

00:08:57.302 --> 00:08:59.433
The anti-derivative of x to the eighth

00:08:59.433 --> 00:09:00.900
is x to the ninth over nine,

00:09:00.900 --> 00:09:03.566
so it's going to be plus x to the ninth,

00:09:03.566 --> 00:09:04.966
and then we have this three.

00:09:04.966 --> 00:09:06.866
Three over nine is three.

00:09:07.500 --> 00:09:09.800
And I think you see a pattern happening.

00:09:09.967 --> 00:09:13.166
And then let's just do
one more of these for fun.

00:09:13.166 --> 00:09:15.566
x to the 12th over 12,
but we have this three,

00:09:15.566 --> 00:09:20.433
so negative x to the 12th over four,

00:09:20.433 --> 00:09:22.433
and then we keep going,

00:09:22.433 --> 00:09:24.366
and then we're of course going to have,

00:09:24.366 --> 00:09:25.700
we're going to have some constant.

00:09:25.700 --> 00:09:28.073
Actually, let me put the
constant in the front.

00:09:28.953 --> 00:09:31.033
Let me copy and paste that,

00:09:31.033 --> 00:09:34.433
or cut and paste that
so I have some space.

00:09:34.433 --> 00:09:39.500
Let me write over here.

00:09:39.500 --> 00:09:41.313
I'll put some other constant, c two,

00:09:41.313 --> 00:09:43.166
it doesn't have to be the same one,

00:09:43.166 --> 00:09:44.500
plus all of this.

00:09:44.500 --> 00:09:45.567
Now to simplify this,

00:09:45.567 --> 00:09:47.703
I could subtract c one from both sides

00:09:47.703 --> 00:09:49.300
or essentially from c two,

00:09:49.300 --> 00:09:54.700
and then I'm going to
have the natural log of

00:09:54.700 --> 00:09:57.035
one plus x to the third power.

00:09:57.035 --> 00:09:58.766
One plus x to the third power,

00:09:58.766 --> 00:10:00.309
this is kind of neat what we've just done,

00:10:00.309 --> 00:10:01.967
with a little bit of integration,

00:10:01.967 --> 00:10:05.308
is equal to c two minus c one.

00:10:05.308 --> 00:10:07.578
This is some constant
minus some other constant,

00:10:07.578 --> 00:10:11.866
so that's just going to be
some arbitrary constant,

00:10:12.566 --> 00:10:14.266
plus all of this business.

00:10:16.901 --> 00:10:20.166
We can even figure out what
the constant is going to be

00:10:20.166 --> 00:10:22.301
by trying out some values of x

00:10:22.301 --> 00:10:25.733
that's in our restricted domain.

00:10:26.013 --> 00:10:28.700
x equals zero is between
negative one and one,

00:10:28.700 --> 00:10:31.566
so let's see what happens
when x is equal to zero

00:10:31.566 --> 00:10:32.800
to solve for c.

00:10:33.166 --> 00:10:35.268
If x is equal to zero,

00:10:37.038 --> 00:10:48.050
we get natural log of
one is equal to c plus,

00:10:48.050 --> 00:10:49.900
all of these terms are going to be zero,

00:10:49.900 --> 00:10:52.301
zero to the third power
minus zero to the sixth

00:10:52.301 --> 00:10:53.101
on and on and on,

00:10:53.101 --> 00:10:54.701
plus zero plus zero,

00:10:55.024 --> 00:10:55.766
or another way,

00:10:55.766 --> 00:10:57.233
and natural log of one, of course,

00:10:57.233 --> 00:10:58.166
[either what] power is one,

00:10:58.166 --> 00:10:59.700
well, that's zero,

00:10:59.700 --> 00:11:01.100
so c must be zero.

00:11:01.100 --> 00:11:02.700
C is equal to zero.

00:11:02.700 --> 00:11:05.102
This thing right over
here is equal to zero.

00:11:05.102 --> 00:11:08.569
What we have just done, using
a little bit of integration,

00:11:08.569 --> 00:11:10.367
is starting with a ...

00:11:10.367 --> 00:11:12.500
let's just appreciate what went on.

00:11:12.500 --> 00:11:15.167
Starting with an
arbitrary infinite series,

00:11:15.167 --> 00:11:18.167
we showed it could be represented
as a geometric series.

00:11:18.167 --> 00:11:19.634
We defined it into rule of convergence

00:11:19.634 --> 00:11:21.501
over which this would converge,

00:11:21.501 --> 00:11:23.167
over which the common
ratio's absolute value

00:11:23.167 --> 00:11:25.100
is less than one,

00:11:25.100 --> 00:11:26.300
and then using that,

00:11:26.300 --> 00:11:28.100
we expressed its sum,

00:11:28.100 --> 00:11:30.434
and then we took the
anti-derivative of both sides

00:11:30.434 --> 00:11:36.166
to figure out an expansion
for the natural log

00:11:36.166 --> 00:11:41.100
of one plus x to the third power,

00:11:41.100 --> 00:11:45.000
which, at least in my
mind, that was pretty neat.

00:11:46.033 --> 00:11:48.700
Natural log of one plus
x to the third power

00:11:48.700 --> 00:11:52.634
is x to the third power
minus x to the sixth over two

00:11:52.634 --> 00:11:55.433
plus x to the ninth over three,

00:11:55.433 --> 00:11:56.500
so on and on and on.

00:11:56.500 --> 00:11:59.500
Actually, just to give
ourselves some closure here,

00:11:59.500 --> 00:12:01.700
let's write it in sigma notation.

00:12:01.700 --> 00:12:06.039
We could write the natural
log of one plus x to the third

00:12:07.023 --> 00:12:08.633
over our restricted domain,

00:12:08.633 --> 00:12:11.633
where the absolute value
of x is less than one,

00:12:11.633 --> 00:12:19.333
is equal to the sum from, let's say,

00:12:19.986 --> 00:12:24.434
n equals one to infinity

00:12:24.434 --> 00:12:30.700
of x to the third to the nth power,

00:12:30.700 --> 00:12:33.567
so to the first power, the
second power, third power,

00:12:33.567 --> 00:12:34.766
over n.

00:12:34.766 --> 00:12:36.366
This is x to the third over one,

00:12:36.366 --> 00:12:39.100
x to the third squared over two,

00:12:39.100 --> 00:12:42.100
oh, and I of course have
to throw in, let's see,

00:12:42.100 --> 00:12:44.446
this first one is,

00:12:44.446 --> 00:12:46.233
we're going to have to
care about the sign,

00:12:46.233 --> 00:12:48.500
so let me throw in a negative one.

00:12:48.500 --> 00:12:48.835
Let's see.

00:12:48.835 --> 00:12:51.900
Negative one to the first
power should be negative,

00:12:51.900 --> 00:12:53.101
but here, it's positive,

00:12:53.101 --> 00:12:57.033
so I'll say negative
one to the n plus one,

00:12:57.033 --> 00:12:58.900
negative one to the n plus one power.

00:12:58.900 --> 00:12:59.833
Does that work?

00:12:59.833 --> 00:13:00.500
I think it does.

00:13:00.500 --> 00:13:01.966
When n is equal to one,

00:13:01.966 --> 00:13:04.766
this thing just becomes one.

00:13:04.766 --> 00:13:06.234
This is x to the third over one.

00:13:06.234 --> 00:13:07.802
When n is equal to two,

00:13:07.986 --> 00:13:10.966
this becomes negative,
which it needs to be,

00:13:10.966 --> 00:13:13.500
and then this becomes x to
the sixth, and we're over two,

00:13:13.500 --> 00:13:14.700
and so there you go.

00:13:14.700 --> 00:13:16.041
We are done.

00:13:16.041 --> 00:13:18.333
I found that pretty satisfying.