Algebra II: Functions and Probability

WEBVTT
Kind: captions
Language: en

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We're on problem 70.

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And they still want us to do
some sequences and series.

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What is the n'th term in
arithmetic series below.

00:00:08.440 --> 00:00:10.800
Which means that it just
increases by a constant amount

00:00:10.800 --> 00:00:12.500
every term.

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So let's think about
it a little bit.

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The first term is 3.

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And then we increment
it by 4 each time.

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Let me write all of this.

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3, then we go to 7, then we go
to 11, and then we go to 15,

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then we go to 19.

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So we go to 3 and we're
adding 4 every time.

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So you get the sense that each
term in the sequence is going

00:00:37.490 --> 00:00:42.480
to be 4 times the n'th term.

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But that doesn't
quite work out.

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Because I've had 4 times-- if
this is the first term, 4

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times the first term is 1.

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And 3 is less than 1.

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And then 4 times 2 is 8.

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And 7 is less than 8.

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And 4 times 3 is 12.

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But this is 1 less than 12.

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So it seems like all these are
1 less than a multiple of 4.

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So the n'th term is going
to be 4n minus 1.

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And that should work out.

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The third term is 4 times
3, 12, minus 1 is 11.

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And that's the third term.

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So it's 4n minus 1,
which is choice D.

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Next problem.

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Problem 71.

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Let me copy and paste it.

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Acctually, I'll copy and
paste all of them.

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All right.

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Which expression represents f
of g of x if f of x is equal

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to x squared minus 1 and g of
x is equal to x plus 3?

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So a lot of times, these
seem daunting.

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But when you get used to it,
these become kind of fun

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problems. So f of g of x means,
in f of x , everywhere

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you see an x you have to
replace it with g of x.

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So f of g of x is equal to
g of x squared minus 1.

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Now, all I did is if I said f of
dog, wherever I see an x, I

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would replace it.

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So f of dog would be dog
squared minus 1.

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Whenever I see an x I just
replace it with whatever I

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input in right there.

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So they want to know f of g of
x , I stick a g of x there.

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So it's going to be g of
x squared minus 1.

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But what's g of x?

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Well g of x is x plus 3.

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So this is going to be equal to
x plus 3 squared minus 1.

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Which is equal to x squared
plus 6x plus 9 minus 1.

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Which is equal to x squared
plus 6x plus 8.

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Which is choice B.

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Next question.

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Given that f of x is equal to 3x
squared minus 4 and g of x

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is equal to 2x minus 6,
what is g of f of 2?

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Well here, since they gave us
an actual number, so they

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didn't leave it abstract, we
could figure out what f of 2

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is and then pop it into g.

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So what's f of 2?

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Yeah, you should be able to
see that color. f of 2.

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Wherever you see an x, you
replace it with 2.

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So that equals 3 times
2 squared minus 4.

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Which is equal to-- 3 times
2 squared, 2 squared is

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4 times 3 is 12.

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Minus 4 is equal to 8.

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So f of 2 is equal to 8.

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So g of f of 2 is
equal to g of 8.

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And what's g of 8?

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g of 8, everywhere you see
an x, just pop in an 8.

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Is equal to 2 times 8 minus 6.

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Is equal to 16 minus 6.

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So this is equal to 10.

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Choice D.

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Problem 73.

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If f of x is equal to x squared
plus 2x plus 1 and g

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of x is equal to 3 times x plus
1 squared, which is the

00:04:20.890 --> 00:04:23.990
equivalent form of f
of x plus g of x?

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So essentially they just want us
to add these two functions.

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So if we say f of x plus g of
x-- let me do that. f of x--

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and I'll do it in another
color-- plus g of x.

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Well now we just add them.

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What's f of x? f of x is x
squared-- I'll do it down

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here-- x squared
plus 2x plus 1.

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That's f of x.

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Then we're going to add
g of x to that.

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Oh, I see why it's a little--
they want us-- g of x you kind

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of have to expand this out.

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So g of x is 3 times
x plus 1 squared.

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Well that's 3 times x squared
plus 2x plus 1.

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But then from here we just have
to simplify this all out.

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And I'll do that in a
third neutral color.

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Brown color.

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x squared plus 2x plus 1 plus--
distribute the 3-- 3x

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squared plus 6x plus 3.

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And now let's see, you have an
x squared and a 3x squared.

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You add those together,
you get 4x squared.

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And I have a 2x and
I have a 6x.

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Add those together,
I get plus 8x.

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And then I have a 1
and I have a 3.

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So plus 4.

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4x squared plus 8x plus 4.

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And that's choice C.

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Next problem.

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Next page.

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All right, let me see.

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What do they want
us to do here?

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A math teacher is randomly
distributing 15 rulers with

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centimeter labels and 10 rulers

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without centimeter labels.

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All right.

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What is the probably the first
ruler she hands out will have

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centimeter labels and the second
ruler will not have

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centimeter labels?

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So, OK, let's think about
the first one.

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I'll color code it.

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So what is the probability that
the first ruler she hands

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out will have centimeter
labels?

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So there's a total of
25 total rulers.

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15 with centimeters
and 10 without.

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At least initially, there's
a total of 25 rulers.

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And they want to know the
probability that the first

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ruler she hands out will
have centimeter labels.

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So 15 have centimeters.

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So the probability
is 15 over 25.

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That's the probability
of the first one

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having centimeter labels.

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That's that one.

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But they want to know, the first
one having centimeter

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labels and the second ruler not
having-- and you can't see

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that highlight color-- and the
second ruler will not have it.

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So this is-- so we're going
to have to multiply the

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probability of the first ruler
times the probability that the

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second, without labels, and now
this is the key-- given

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that the first ruler
had labels.

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Now this might all seem really
complicated with the notation

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but when you think of it
intuitively, it should

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hopefully make some sense.

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I've given away-- so the first
ruler there's a 15 out of 25

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chance that I've given
a ruler with labels.

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Now on the second ruler, since
both of these things have to

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happen, I can assume that I've
already given away the first

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ruler having labels.

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So how many rulers
do I have now?

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Well I definitely have
24 rulers left.

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And now, how many of these
do not have labels?

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Well the first ruler I gave
away had a label.

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So I still have 10 rulers that
do not have a label.

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If you wanted to ask a similar
question, you could say what

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is the probability that the
first two ones that I hand out

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have labels?

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That would be 15 over
25 times 14 over 24.

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Because the labels would've
been decremented

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after the first one.

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Because we're assuming.

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We have to say both of
these things happen.

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Now we're saying, OK, we're
going to have one less ruler

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but I haven't reduced
the number of

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rulers without labels.

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But anyway, let's just
simplify this out.

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This becomes 15 times 10
divided by 25 times 24.

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You can do a lot of canceling
so we don't have multiply--

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let's see, divide that
by 3, get 3.

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Divide that by 3, get 5.

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Divide that by 5, you get 1.

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Divide that by 5 you get 2.

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3 times 2 is 6.

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So this is equal to 6 over 24.

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Which is equal to 1/4.

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Which is choice A.

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Next problem.

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Let's see.

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Problem 75.

00:09:22.710 --> 00:09:26.320
On a certain day the chance of
rain is 80% in San Francisco

00:09:26.320 --> 00:09:27.860
and 30% in Sydney.

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Fair enough.

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Assume that the chance of rain
in the two cities is

00:09:30.990 --> 00:09:32.000
independent.

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What is the probably that it
will not rain in either city?

00:09:36.090 --> 00:09:41.080
So it's going to be the
probability of not rain, not

00:09:41.080 --> 00:09:48.800
in San Francisco, times the
probability of not in Sydney.

00:09:48.800 --> 00:09:50.710
Because we have to have both of
these things happening and

00:09:50.710 --> 00:09:51.430
they're independent.

00:09:51.430 --> 00:09:53.710
So the probabilities will
be multiplying.

00:09:53.710 --> 00:09:55.460
So what's the probability
of not

00:09:55.460 --> 00:09:57.480
having rain in San Francisco?

00:09:57.480 --> 00:10:00.690
They gave us the chance
of having rain as 80%.

00:10:00.690 --> 00:10:03.270
So the probability of not
having rain is 20%.

00:10:03.270 --> 00:10:05.160
Or, I like to stick
to the fractions.

00:10:05.160 --> 00:10:07.330
20% is 1/5.

00:10:07.330 --> 00:10:09.830
There's a 1/5 probability
of not having rain.

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What's the probability of not
having rain in Sydney?

00:10:13.010 --> 00:10:14.200
It's 30%.

00:10:14.200 --> 00:10:16.100
Actually, I probably should've
stayed in decimal world.

00:10:16.100 --> 00:10:21.660
So let's say it's 20% in San
Francisco of not having rain.

00:10:21.660 --> 00:10:25.900
The probability of Sydney of
not having rain is 1 minus

00:10:25.900 --> 00:10:34.140
30%, or 0.7, which is a 70%
chance of not having rain.

00:10:34.140 --> 00:10:36.612
And that is equal to--
2 times 7 is 14.

00:10:36.612 --> 00:10:39.050
We have two numbers behind
the decimal point.

00:10:39.050 --> 00:10:41.760
So there's a 14% chance
that it will not

00:10:41.760 --> 00:10:44.390
rain in either city.

00:10:44.390 --> 00:10:46.560
And that is choice B.

00:10:50.190 --> 00:10:51.880
Next problem.

00:10:51.880 --> 00:10:53.220
Well actually I'm out of time.

00:10:53.220 --> 00:10:55.060
See you in the next video.


Algebra II: Binomial Expansions, Geometric Series Sum

WEBVTT
Kind: captions
Language: en

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We're on problem 67.

00:00:02.430 --> 00:00:05.690
They ask us how many terms of
the binomial expansion of x

00:00:05.690 --> 00:00:08.000
squared plus 2y to
the third to the

00:00:08.000 --> 00:00:09.500
twentieth power contain?

00:00:09.500 --> 00:00:11.890
And so right when you see, oh
my god, to the twentieth

00:00:11.890 --> 00:00:14.760
power, that will take me forever
to do, and then I'll

00:00:14.760 --> 00:00:15.930
have to count the terms. But
remember, they're just asking

00:00:15.930 --> 00:00:17.650
you how many terms.
They're not asking

00:00:17.650 --> 00:00:18.890
you to find the expansion.

00:00:18.890 --> 00:00:20.370
So you just have to say, oh boy,
let me just think about

00:00:20.370 --> 00:00:21.390
this a little bit.

00:00:21.390 --> 00:00:29.560
And think of it this way: x plus
y to the first power has

00:00:29.560 --> 00:00:31.240
how many terms?

00:00:31.240 --> 00:00:36.030
It has-- well it's
just x plus y.

00:00:36.030 --> 00:00:41.910
And so it has 2 terms.
x plus y, squared.

00:00:41.910 --> 00:00:47.510
That is equal to x squared
plus 2xy plus y squared.

00:00:47.510 --> 00:00:52.910
So that has 3 terms. If I were
to do x plus e The third

00:00:52.910 --> 00:00:55.850
power, I could do a binomial
expansion, I could do it with

00:00:55.850 --> 00:00:56.900
Pascal's Triangle.

00:00:56.900 --> 00:00:57.970
Let me do it with Pascal's
Triangle.

00:00:57.970 --> 00:00:58.980
I have the 1 and the 1.

00:00:58.980 --> 00:01:01.240
That's to the first power.

00:01:01.240 --> 00:01:04.709
The second power, those
are the coefficients.

00:01:04.709 --> 00:01:08.030
The third power, the
coefficients become 1, 3--

00:01:08.030 --> 00:01:12.220
because 1 plus 2 is 3-- 3, 1.

00:01:12.220 --> 00:01:14.070
And actually I don't even
have to expand it out.

00:01:14.070 --> 00:01:15.590
Ill do it just for a
little practice.

00:01:15.590 --> 00:01:24.880
It x to the third plus 3x
squared y plus 3xy squared

00:01:24.880 --> 00:01:25.920
plus y to the third.

00:01:25.920 --> 00:01:27.040
I just used these.

00:01:27.040 --> 00:01:31.750
But when you take any binomial
and you take it to the third

00:01:31.750 --> 00:01:36.100
power, you end up with 4 terms.
And depending on how

00:01:36.100 --> 00:01:38.520
you think of it, if you think of
it with Pascal's Triangle,

00:01:38.520 --> 00:01:40.450
every time we go down we're
just going to be adding

00:01:40.450 --> 00:01:41.150
another term.

00:01:41.150 --> 00:01:43.910
When we take the fourth power
it's going to have 5 terms. Or

00:01:43.910 --> 00:01:46.450
when you think about, when we
use the binomial expansion,

00:01:46.450 --> 00:01:48.620
and you might want to watch
some videos on that.

00:01:48.620 --> 00:01:51.090
There you also have, if you're
taking it to the nth power,

00:01:51.090 --> 00:01:53.400
you end up with n
plus 1 terms.

00:01:53.400 --> 00:01:57.380
So in this example, we have a
binomial-- a binomial just

00:01:57.380 --> 00:01:59.130
means a two-term polynomial.

00:01:59.130 --> 00:02:01.090
And we're taking it to
the twentieth power.

00:02:01.090 --> 00:02:02.900
If we were taking it to third
power, we'd have four terms.

00:02:02.900 --> 00:02:04.320
If we were taking it to the
fourth power, we'd have 5

00:02:04.320 --> 00:02:06.570
terms. So if we're taking it to
the twentieth power, we're

00:02:06.570 --> 00:02:11.390
going to have 21 terms.
And that's choice B.

00:02:11.390 --> 00:02:12.640
Next problem.

00:02:18.420 --> 00:02:21.890
All right, they want us to keep
doing these and we'll do

00:02:21.890 --> 00:02:23.200
it because they want us to.

00:02:32.400 --> 00:02:35.540
All right, what are the first
four terms of the expansion of

00:02:35.540 --> 00:02:37.540
1 plus 2x to the sixth power?

00:02:37.540 --> 00:02:39.750
And here we'll just use the
binomial expansion because I

00:02:39.750 --> 00:02:41.350
think that's what they
want us to do.

00:02:41.350 --> 00:02:43.350
And that's probably the
fastest way to do it.

00:02:43.350 --> 00:02:46.210
So let's just think of the
coefficients first. So this is

00:02:46.210 --> 00:02:47.580
going to be-- well
let's just do it.

00:02:47.580 --> 00:02:51.920
This is going to be equal
to 6 choose 0.

00:02:51.920 --> 00:02:56.070
We're just going to do the first
four terms. Times 1 to

00:02:56.070 --> 00:03:00.320
the 6 times 2x to the 0-- so I
don't have to write that--

00:03:00.320 --> 00:03:03.965
plus 6 choose 1.

00:03:03.965 --> 00:03:05.880
And actually, we'll probably
just have to figure out the

00:03:05.880 --> 00:03:07.330
third term because that's
the only time where

00:03:07.330 --> 00:03:09.380
they start to change.

00:03:09.380 --> 00:03:21.200
6 choose 1 times 1 to the fifth
power times 2x to the

00:03:21.200 --> 00:03:23.210
first power, times 2x.

00:03:23.210 --> 00:03:29.390
Plus 6 choose 2 times-- I mean
these 1s, it doesn't matter

00:03:29.390 --> 00:03:37.200
what power it is-- 1 to the
fourth times 2x squared plus--

00:03:37.200 --> 00:03:40.330
let's see, they want the first
so let's do the last one-- 6

00:03:40.330 --> 00:03:47.400
choose 3 times 1 to the third--
you have to decrement

00:03:47.400 --> 00:03:51.470
the 1 every time-- times
2x to the third.

00:03:51.470 --> 00:03:53.180
All right, and if we look
at the choices,

00:03:53.180 --> 00:03:54.740
just to save time.

00:03:54.740 --> 00:03:57.560
If we look at a choices, OK, we
agree that the first term

00:03:57.560 --> 00:03:58.470
is going to be 1.

00:03:58.470 --> 00:04:01.220
So that, we can definitely say
is going to be 1 because all

00:04:01.220 --> 00:04:01.910
the choices are 1.

00:04:01.910 --> 00:04:04.640
The second term is definitely
going to be 12x.

00:04:04.640 --> 00:04:07.050
And then we start getting
some disagreement

00:04:07.050 --> 00:04:08.075
on the third term.

00:04:08.075 --> 00:04:09.180
So let's see what
the third term.

00:04:09.180 --> 00:04:14.460
6 choose to that's equal to 6
back toward ill over to affect

00:04:14.460 --> 00:04:21.700
or you over fix minus is too
fact or you scroll down a

00:04:21.700 --> 00:04:27.380
little bit so that is equal to
6 factorial, which is 6 times

00:04:27.380 --> 00:04:30.770
5 times 4 times 3
times 2 times 1.

00:04:30.770 --> 00:04:32.730
But the 1 doesn't
change anything.

00:04:32.730 --> 00:04:36.350
Divided by 2 factorial, which
is 2 times 1, or just 2.

00:04:36.350 --> 00:04:39.510
Divided by 6 minus 2 factorial,
that's 4 factorial.

00:04:39.510 --> 00:04:44.530
Royal So times 4 times
3 times 2 times 1.

00:04:44.530 --> 00:04:46.420
That cancels out with that.

00:04:46.420 --> 00:04:48.300
The 2 and the 6 cancel out.

00:04:48.300 --> 00:04:49.030
You get 3.

00:04:49.030 --> 00:04:51.830
So you end up with 3 times 5.

00:04:51.830 --> 00:04:59.200
So the 6 choose 2 becomes 15,
3 times 5, that's 15.

00:04:59.200 --> 00:05:04.300
Times 1 to the fourth, which
is 1, times 2x squared.

00:05:04.300 --> 00:05:07.780
So times 4x squared.

00:05:07.780 --> 00:05:11.020
And so that's 15 times 4,
that's 60x squared.

00:05:11.020 --> 00:05:14.720
So the third term is going
to be 60x squared.

00:05:14.720 --> 00:05:15.700
And we're done.

00:05:15.700 --> 00:05:17.760
Choice D is the only one
that has 60x squared

00:05:17.760 --> 00:05:18.580
as the third term.

00:05:18.580 --> 00:05:18.930
So we're done.

00:05:18.930 --> 00:05:21.120
We didn't even have to go
to the fourth term.

00:05:21.120 --> 00:05:23.770
So, by carefully looking at the
choices we actually only

00:05:23.770 --> 00:05:27.120
have to evaluate one
of the terms.

00:05:27.120 --> 00:05:28.370
Next problem.

00:05:31.940 --> 00:05:33.230
Let's see.

00:05:33.230 --> 00:05:33.780
All right.

00:05:33.780 --> 00:05:34.830
Problem 69.

00:05:34.830 --> 00:05:46.750
They say-- let me copy and paste
it-- what is the sum of

00:05:46.750 --> 00:05:51.310
the infinite geometric series,
1/2 plus 1/4 plus 1/8 all the

00:05:51.310 --> 00:05:51.940
way up there.

00:05:51.940 --> 00:05:52.990
I'll be frank.

00:05:52.990 --> 00:05:54.940
I do now remember the formula.

00:05:54.940 --> 00:05:58.030
But a lot of times in life you
might forget the formula.

00:05:58.030 --> 00:06:00.050
And there's a trick
is to re-proving

00:06:00.050 --> 00:06:01.280
the formula for yourself.

00:06:01.280 --> 00:06:03.800
Just in case you forget
it when you're 32

00:06:03.800 --> 00:06:05.390
years old like me.

00:06:05.390 --> 00:06:08.700
So let's say that I'm trying to
find the infinite sum of a

00:06:08.700 --> 00:06:09.690
geometric series.

00:06:09.690 --> 00:06:12.380
So let's say that the sum-- and
I'm going to re-prove it

00:06:12.380 --> 00:06:14.800
right here in the midst of this
test. And I've done this

00:06:14.800 --> 00:06:16.730
in midst of tests and
it's saved me.

00:06:16.730 --> 00:06:17.830
So it's nice to know
the proof.

00:06:17.830 --> 00:06:20.425
And it actually impresses
people, if you know people who

00:06:20.425 --> 00:06:21.930
are impressed by proofs.

00:06:21.930 --> 00:06:28.530
So let's say the sum is equal to
a to the 0-- well actually

00:06:28.530 --> 00:06:30.930
let's say-- well, see
this is interesting.

00:06:30.930 --> 00:06:32.370
Because they're starting
at 1/2.

00:06:32.370 --> 00:06:33.910
Actually let's just do
the concrete numbers.

00:06:33.910 --> 00:06:36.150
Because we want to figure
out this exact sum.

00:06:36.150 --> 00:06:43.820
So let's say that the sum is
equal to 1/2 to the 1 1/2

00:06:43.820 --> 00:06:49.170
squared plus 1/2 to the third--
it's the whole 1/2 to

00:06:49.170 --> 00:06:53.360
the third-- plus-- and
you just keep going.

00:06:53.360 --> 00:06:54.570
Fair enough.

00:06:54.570 --> 00:06:57.460
Now let's take 1/2
times the sum.

00:06:57.460 --> 00:07:01.310
So 1/2 times this.

00:07:01.310 --> 00:07:02.810
What is this equal to?

00:07:02.810 --> 00:07:06.500
Well if I took 1/2 times this
whole thing, now 1/2, I would

00:07:06.500 --> 00:07:08.800
have to distribute it over each
of these terms. So 1/2

00:07:08.800 --> 00:07:12.350
times this term, times this
first term, 1/2 times 1/2 to

00:07:12.350 --> 00:07:14.510
the first, well that's now
going to be 1/2 squared.

00:07:17.190 --> 00:07:18.570
1/2 squared.

00:07:18.570 --> 00:07:21.291
All I did is I'm distributing
this 1/2 times each of the

00:07:21.291 --> 00:07:24.590
terms. 1/2 times 1/2 to the
first. That's 1/2 squared.

00:07:24.590 --> 00:07:27.820
Now 1/2 times 1/2 squared, well
that's going to be 1/2 to

00:07:27.820 --> 00:07:29.450
the third power.

00:07:29.450 --> 00:07:30.830
And it just keeps going.

00:07:30.830 --> 00:07:33.420
It's an infinite series.

00:07:33.420 --> 00:07:34.710
So let me ask you something.

00:07:34.710 --> 00:07:37.840
If I subtracted this from
that, what do I get?

00:07:37.840 --> 00:07:39.900
Let's see what I get.

00:07:39.900 --> 00:07:45.770
So I get S minus 1/2 S-- I'm
subtracting that from that--

00:07:45.770 --> 00:07:49.420
is equal to this minus this.

00:07:49.420 --> 00:07:51.490
Well if I'm subtracting the
green stuff from the yellow

00:07:51.490 --> 00:07:53.810
stuff, this is going to cancel
out with that, that's going to

00:07:53.810 --> 00:07:54.820
cancel out with that.

00:07:54.820 --> 00:07:55.980
And all I'm going to be
left with is this

00:07:55.980 --> 00:07:59.210
first thing right here.

00:07:59.210 --> 00:07:59.900
That's the trick.

00:07:59.900 --> 00:08:01.850
And you could prove it generally
for any base.

00:08:01.850 --> 00:08:02.980
And I thought it was interesting
because a lot of

00:08:02.980 --> 00:08:04.710
times they start with
the 0th power.

00:08:04.710 --> 00:08:07.140
And this problem was interesting
because they start

00:08:07.140 --> 00:08:08.775
with 1/2 to the first power.

00:08:08.775 --> 00:08:10.160
So anyway let's figure it out.

00:08:10.160 --> 00:08:13.970
S minus 1/2 S, well
that's just 1/2 S.

00:08:13.970 --> 00:08:17.810
1/2 S is equal to 1/2.

00:08:17.810 --> 00:08:20.380
And then you have S is-- divided
both sides by 1/2 or

00:08:20.380 --> 00:08:25.520
multiply both sides by 2 and
you get-- S is equal to 1.

00:08:25.520 --> 00:08:30.540
Now, a lot of you may know
there is a formula.

00:08:30.540 --> 00:08:33.309
Let me tell you what
the formula says.

00:08:33.309 --> 00:08:35.890
The formula that you may or may
not have memorized, is if

00:08:35.890 --> 00:08:43.840
you start at n is equal to 0 and
if you go to infinity of a

00:08:43.840 --> 00:08:47.950
to the n-- in this case a is
1/2-- this is going to be

00:08:47.950 --> 00:08:53.210
equal to 1 over 1 minus a.

00:08:53.210 --> 00:08:55.950
So your natural reaction would
have been, oh OK, well that

00:08:55.950 --> 00:09:01.640
sum that we saw, that's equal to
the sum of n is equal to 0

00:09:01.640 --> 00:09:06.860
to infinity of 1/2 to the n,
which would be equal to 1 over

00:09:06.860 --> 00:09:09.180
1 minus 1/2.

00:09:09.180 --> 00:09:09.670
What's that?

00:09:09.670 --> 00:09:13.130
1 minus 1, that's 1 over 1/2,
which would be equal to 2.

00:09:13.130 --> 00:09:16.200
And you'd be like, my God, what
did I do wrong there?

00:09:16.200 --> 00:09:18.580
And I'll show you what
you did wrong.

00:09:18.580 --> 00:09:21.110
This sum that you took the sum
of-- remember n is starting at

00:09:21.110 --> 00:09:27.600
0-- so this sum that sums up to
2 is 1/2 to 0 is 1 plus 1/2

00:09:27.600 --> 00:09:35.030
to the 1, 1/2, plus 1/2 squared,
1/4, plus 1/8, so on

00:09:35.030 --> 00:09:36.080
and so forth.

00:09:36.080 --> 00:09:37.380
That is equal to 2.

00:09:40.300 --> 00:09:42.780
This problem-- and that's why it
was a little tricky-- this

00:09:42.780 --> 00:09:45.330
problem, they didn't want this
sum, they wanted this sum.

00:09:45.330 --> 00:09:52.020
1/2 plus 1/4 plus 1/8 plus,
so forth and so on.

00:09:52.020 --> 00:09:54.000
So they wanted everything
but this 1.

00:09:54.000 --> 00:09:56.230
So if you subtracted this 1 from
both sides then you would

00:09:56.230 --> 00:09:59.040
say, oh this sum must be equal
to 1, which is the answer we

00:09:59.040 --> 00:10:00.400
got that way.

00:10:00.400 --> 00:10:03.140
So that's why sometimes,
memorizing formulas, this

00:10:03.140 --> 00:10:06.350
formula's a nice formula to
memorize, it's very simple.

00:10:06.350 --> 00:10:08.430
But it sometimes becomes really
confusing when you're

00:10:08.430 --> 00:10:11.630
like, oh boy, but this is when
n starts 0 but what happens

00:10:11.630 --> 00:10:12.810
when n starts at 1.

00:10:12.810 --> 00:10:14.900
And to some degree I really like
this problem because it's

00:10:14.900 --> 00:10:16.940
testing you to see if you really
understand what the

00:10:16.940 --> 00:10:18.450
formula's all about.

00:10:18.450 --> 00:10:21.480
Anyway, I'll see you
in the next video.


Algebra II: binomial Expansion and Combinatorics

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.200
In the last video, we
did problem 65,

00:00:03.200 --> 00:00:04.470
and we got the answer.

00:00:04.470 --> 00:00:06.750
And we did it in kind of an
intuitive way, but if it was a

00:00:06.750 --> 00:00:08.390
slightly harder problem, it
would've gotten a little more

00:00:08.390 --> 00:00:10.820
complicated, so I really wanted
to show you how you can

00:00:10.820 --> 00:00:13.090
do it with the binomial theorem
as well, or at least

00:00:13.090 --> 00:00:19.030
with binomial coefficients, and
then we can try to see how

00:00:19.030 --> 00:00:20.070
they relate to each other,
that they're

00:00:20.070 --> 00:00:21.270
actually the same thing.

00:00:21.270 --> 00:00:22.780
I realized that I should do
this, because I'm looking at

00:00:22.780 --> 00:00:24.320
the next couple of questions,
and they all deal with the

00:00:24.320 --> 00:00:25.450
binomial coefficient.

00:00:25.450 --> 00:00:28.620
So I figured I might as well do
it the binomial way because

00:00:28.620 --> 00:00:30.130
that's probably what
the California

00:00:30.130 --> 00:00:31.400
Standards want you to do.

00:00:31.400 --> 00:00:32.330
And it's a good way.

00:00:32.330 --> 00:00:34.230
It's good to be able to do it
intuitively and with the

00:00:34.230 --> 00:00:37.030
binomial coefficients and
understand how they're the

00:00:37.030 --> 00:00:38.020
same thing.

00:00:38.020 --> 00:00:41.050
So reading the same problem,
Teresa and Julia are among 10

00:00:41.050 --> 00:00:43.580
students who have applied
for a trip to D.C.

00:00:43.580 --> 00:00:45.390
Two students from the group
will be selected at

00:00:45.390 --> 00:00:46.890
random for the trip.

00:00:50.820 --> 00:00:53.790
What is probability that Teresa
and Julia will be the

00:00:53.790 --> 00:00:55.200
two students selected?

00:00:55.200 --> 00:00:57.160
So the way we did it last time,
we said, OK, we could

00:00:57.160 --> 00:00:59.720
pick Julia first, and there's a
1 in 10 chance, and then we

00:00:59.720 --> 00:01:02.240
would pick Teresa, and that's
a 1 in 9 chance.

00:01:02.240 --> 00:01:06.910
Or we could go the other way
around, so it would be 2 times

00:01:06.910 --> 00:01:09.960
1/10 times 1/9.

00:01:09.960 --> 00:01:11.930
2, because there's two different
ways to pick the

00:01:11.930 --> 00:01:15.240
people, and then for each of
those ways, there's a 1 in 10

00:01:15.240 --> 00:01:17.380
chance of picking the first
person and then a 1 in 9 for

00:01:17.380 --> 00:01:18.020
the second.

00:01:18.020 --> 00:01:20.310
That was the intuitive way we
did that in the last video.

00:01:20.310 --> 00:01:21.860
The way you would do it with
the binomial coefficient is

00:01:21.860 --> 00:01:25.980
you'd say, OK, I have 10 items,
10 students in this

00:01:25.980 --> 00:01:29.640
case, and I'm going to
choose two of them.

00:01:29.640 --> 00:01:33.340
And people will often read
this as 10 choose 2.

00:01:33.340 --> 00:01:37.570
And it's useful to memorize to
some degree what this means.

00:01:37.570 --> 00:01:43.500
This means 10 factorial
over 2 factorial

00:01:43.500 --> 00:01:47.750
times 10 minus 2 factorial.

00:01:47.750 --> 00:01:50.970
And so that is equal to-- and
I'll give you in a second what

00:01:50.970 --> 00:01:51.560
the intuition is.

00:01:51.560 --> 00:01:56.480
That's equal to 10
factorial over 2

00:01:56.480 --> 00:02:00.230
factorial times 8 factorial.

00:02:00.230 --> 00:02:02.480
And at first, you might say, oh,
my God, 10 factorial is a

00:02:02.480 --> 00:02:04.140
huge number.

00:02:04.140 --> 00:02:06.440
It's going to take me forever
to multiply that out, until

00:02:06.440 --> 00:02:08.870
you realize that you're dividing
it by 8 factorial.

00:02:08.870 --> 00:02:13.240
So 10 factorial, that's the same
thing as 10 times 9 times

00:02:13.240 --> 00:02:16.670
8 times 7, and you keep going.

00:02:16.670 --> 00:02:20.540
But 8 factorial is 8 times
7 times 6 and so

00:02:20.540 --> 00:02:21.310
forth and so on.

00:02:21.310 --> 00:02:24.160
So the 8 factorial, if you
divide into 10 factorial,

00:02:24.160 --> 00:02:26.050
that's going to cancel
out with that.

00:02:26.050 --> 00:02:29.520
And you're just left
with 10 times 9.

00:02:29.520 --> 00:02:32.590
So if you have to say, how many
different ways can you

00:02:32.590 --> 00:02:35.730
pick 2 people out of 10?

00:02:35.730 --> 00:02:39.800
You then get 10 times 9, which
is 90, divided by 2 factorial,

00:02:39.800 --> 00:02:43.610
which is 2 times 1, so you say
there's 45 different ways to

00:02:43.610 --> 00:02:45.070
pick 2 people.

00:02:45.070 --> 00:02:49.320
So the chances that I pick one
of those ways, pick Julia and

00:02:49.320 --> 00:02:52.490
Teresa, is 1 out of 45.

00:02:52.490 --> 00:02:56.270
There are 45 combinations of
picking 2 people out of 10.

00:02:56.270 --> 00:02:57.900
So the chances of picking
any one of those

00:02:57.900 --> 00:02:59.260
combinations is 1 out of 45.

00:02:59.260 --> 00:03:00.420
So you get to the same answer.

00:03:00.420 --> 00:03:03.240
And just to get the intuition,
this thing boils down

00:03:03.240 --> 00:03:03.990
to-- we just saw.

00:03:03.990 --> 00:03:07.660
It boils down to 10 times
9 over 2 factorial.

00:03:07.660 --> 00:03:10.090
And if you just want the
intuition, there's 10 ways to

00:03:10.090 --> 00:03:13.390
pick the first person, there's
9 ways to pick the second

00:03:13.390 --> 00:03:17.500
person, and then there's 2 ways,
or 2 factorial ways,

00:03:17.500 --> 00:03:19.300
that you could have picked
one person and

00:03:19.300 --> 00:03:20.050
then another person.

00:03:20.050 --> 00:03:21.960
So it's the same intuition
as this.

00:03:21.960 --> 00:03:24.890
But if you just wanted to kind
of plug and chug and get the

00:03:24.890 --> 00:03:26.650
answer, you say, oh,
10 choose 2.

00:03:26.650 --> 00:03:27.453
That gives me the answer.

00:03:27.453 --> 00:03:28.220
I have 10 things.

00:03:28.220 --> 00:03:28.870
I need to choose 2.

00:03:28.870 --> 00:03:31.410
That gives me 45 different
combinations and I want only

00:03:31.410 --> 00:03:32.380
one of them.

00:03:32.380 --> 00:03:34.660
So that's 1 out of 45.

00:03:34.660 --> 00:03:37.270
Next question.

00:03:37.270 --> 00:03:38.340
Let me see.

00:03:38.340 --> 00:03:40.250
Scroll up.

00:03:40.250 --> 00:03:41.060
All right.

00:03:41.060 --> 00:03:43.410
And when I saw this, I said, oh,
they definitely want us to

00:03:43.410 --> 00:03:45.010
deal with binomial
coefficients.

00:03:45.010 --> 00:03:47.020
And I'll show you two ways how
to do this problem, too.

00:03:47.020 --> 00:03:53.100
So this one says 3y minus
1 to the fourth power.

00:03:53.100 --> 00:03:56.590
And you can multiply it out just
using regular algebraic

00:03:56.590 --> 00:03:58.640
multiplication, polynomial
multiplication.

00:03:58.640 --> 00:04:00.590
It will take you some
time and you might

00:04:00.590 --> 00:04:01.790
make a careless mistake.

00:04:01.790 --> 00:04:04.270
I'm guessing they want us to
use the binomial theorem.

00:04:04.270 --> 00:04:07.880
And the binomial expansion
of this-- I'll use that

00:04:07.880 --> 00:04:11.180
combinatorics or that binomial
coefficient notation.

00:04:11.180 --> 00:04:12.510
I think you'll get
the pattern.

00:04:12.510 --> 00:04:14.870
And I've done several videos on
this where I explain deeper

00:04:14.870 --> 00:04:16.760
why it works and all of that.

00:04:16.760 --> 00:04:23.480
But it equals 4 choose 0 times
the first term to the fourth

00:04:23.480 --> 00:04:33.670
power, 3y to the fourth, times
the second term to the 0th

00:04:33.670 --> 00:04:35.030
power, so you don't
include it.

00:04:35.030 --> 00:04:35.620
That's times 1.

00:04:35.620 --> 00:04:36.790
Anything to the 0th power is 1.

00:04:36.790 --> 00:04:37.440
Fine.

00:04:37.440 --> 00:04:45.820
Plus 4 choose 1 times the first
term, so that's 3y to

00:04:45.820 --> 00:04:48.880
the third power-- I'm just
decrementing it on each term--

00:04:48.880 --> 00:04:55.880
Times the second term to the
first power, so times negative

00:04:55.880 --> 00:04:58.130
1 to the first power.

00:04:58.130 --> 00:05:00.170
And, in general, if you want
to know what should this

00:05:00.170 --> 00:05:03.390
exponent be, it's going to
be whatever is here.

00:05:03.390 --> 00:05:05.780
So this one is the
same as that.

00:05:05.780 --> 00:05:08.180
So here we had a 0, so negative
1 to the 0, I didn't

00:05:08.180 --> 00:05:10.560
even take the trouble
to write that down.

00:05:10.560 --> 00:05:11.490
Let me just keep going.

00:05:11.490 --> 00:05:12.070
I'll switch colors.

00:05:12.070 --> 00:05:14.210
So the next term, you could
probably pause it and guess

00:05:14.210 --> 00:05:16.110
it, is going to be 4 choose 2.

00:05:16.110 --> 00:05:17.825
And I know you're wondering
well, what is this 4 choose 1

00:05:17.825 --> 00:05:18.510
and 4 choose 2?

00:05:18.510 --> 00:05:21.280
Well, we saw it in the last
problem, but I'll calculate

00:05:21.280 --> 00:05:22.680
them in a second.

00:05:22.680 --> 00:05:32.290
That's going to be 3y squared
times negative 1 squared.

00:05:32.290 --> 00:05:37.330
And we have two more terms.
Plus 4 choose 3

00:05:37.330 --> 00:05:41.450
times 3y to the 1.

00:05:41.450 --> 00:05:44.020
And just so you know, the
exponents of these two should

00:05:44.020 --> 00:05:45.000
always add up to 4.

00:05:45.000 --> 00:05:47.940
So you have 4 and 0, you
have 3 and 1, 2 and 2.

00:05:47.940 --> 00:05:50.790
So in this case, you have
times negative 1 to the

00:05:50.790 --> 00:05:59.380
third-- and this 3 is that 3--
plus 4 choose 4 times negative

00:05:59.380 --> 00:06:00.630
1 to the fourth power.

00:06:03.470 --> 00:06:05.550
So what do each of these
binomial coefficients equal?

00:06:05.550 --> 00:06:11.060
4 choose 0, that is equal to--
I'll do it in yellow.

00:06:11.060 --> 00:06:17.190
That's equal to 4 factorial over
0 factorial, which, you

00:06:17.190 --> 00:06:20.570
may or may not know, is defined
to be 1, over 0

00:06:20.570 --> 00:06:25.310
factorial divided by 4
minus 0 factorial.

00:06:25.310 --> 00:06:26.790
And so that just turns
into 1, right?

00:06:26.790 --> 00:06:27.890
4 minus 0 is 4.

00:06:27.890 --> 00:06:29.530
So you get 4 factorial
over factorial.

00:06:29.530 --> 00:06:32.700
So this is equal to 1.

00:06:32.700 --> 00:06:33.440
So let me write it out.

00:06:33.440 --> 00:06:34.520
So 3y the fourth.

00:06:34.520 --> 00:06:36.340
3 to the fourth power is what?

00:06:36.340 --> 00:06:39.200
It's 81.

00:06:39.200 --> 00:06:45.080
So we get 81 y to the fourth.

00:06:45.080 --> 00:06:45.640
All right.

00:06:45.640 --> 00:06:46.890
Next term.

00:06:46.890 --> 00:06:49.220
Well, the first thing that you
see is you have this negative

00:06:49.220 --> 00:06:50.170
1 to the first power.

00:06:50.170 --> 00:06:52.610
This negative 1 is just going
to keep flipping the sign.

00:06:52.610 --> 00:06:54.820
So negative 1 to the first
power, that's just going to

00:06:54.820 --> 00:06:56.370
make this whole term negative
and then we can

00:06:56.370 --> 00:06:57.170
ignore it from there.

00:06:57.170 --> 00:06:58.790
So we're going to get a negative
there because of this

00:06:58.790 --> 00:07:00.055
negative 1 to the first power.

00:07:00.055 --> 00:07:02.030
And what's 4 choose 1?

00:07:02.030 --> 00:07:07.860
So that's 4 factorial over 1
factorial, which is also 1,

00:07:07.860 --> 00:07:10.470
divided by 4 minus
1 factorial.

00:07:13.210 --> 00:07:16.010
So that's 4 factorial
over 3 factorial.

00:07:16.010 --> 00:07:18.420
So it's 4 times 3 times
2 times 1 divided by 3

00:07:18.420 --> 00:07:19.880
times 2 times 1.

00:07:19.880 --> 00:07:22.190
So that just equals 4.

00:07:22.190 --> 00:07:23.300
Try it out if you don't
believe me.

00:07:23.300 --> 00:07:27.010
4 times 3 times 2 times 1
divided by 3 times 2 times 1,

00:07:27.010 --> 00:07:28.780
you're just left with a 4.

00:07:28.780 --> 00:07:31.270
So you get 4.

00:07:31.270 --> 00:07:32.820
This coefficient becomes 4.

00:07:32.820 --> 00:07:36.230
But it's 4 times 3 to
the third power.

00:07:36.230 --> 00:07:37.690
So maybe I'll just write it.

00:07:37.690 --> 00:07:45.060
The binomial coefficient is
4 times 27y to the third.

00:07:45.060 --> 00:07:46.430
And we already took care
of the negative 1.

00:07:46.430 --> 00:07:48.900
Negative 1 to the first
power is negative 1.

00:07:48.900 --> 00:07:50.500
And that's why I put
the negative there.

00:07:50.500 --> 00:07:51.100
So the next term.

00:07:51.100 --> 00:07:52.790
We have a negative 1 to the
second power, so it's going to

00:07:52.790 --> 00:07:53.710
be a positive.

00:07:53.710 --> 00:07:56.370
This is going to
turn into a 1.

00:07:56.370 --> 00:07:57.680
That turns into a 1.

00:07:57.680 --> 00:08:00.750
So it's plus-- what's
4 choose 2?

00:08:00.750 --> 00:08:05.570
4 choose 2 is-- I'm already
running out of space.

00:08:05.570 --> 00:08:06.460
I'll write it up here.

00:08:06.460 --> 00:08:10.370
It's 4 factorial over
2 factorial

00:08:10.370 --> 00:08:13.830
times 4 minus 2 factorial.

00:08:13.830 --> 00:08:17.460
4 minus 2 factorial, that's
also 2 factorial.

00:08:17.460 --> 00:08:22.520
So it's equal to 4 times 3 times
2 times 1 divided by 2

00:08:22.520 --> 00:08:26.440
factorial is 2 and
2 factorial is 2.

00:08:26.440 --> 00:08:29.080
Let's say this 2 cancels
out with that 2.

00:08:29.080 --> 00:08:32.120
This 2 cancels-- let's see, if
you cancel out that 2, that

00:08:32.120 --> 00:08:33.470
becomes a 2.

00:08:33.470 --> 00:08:36.090
So you get 2 times 3
just becomes a 6.

00:08:36.090 --> 00:08:39.350
So that third term
just becomes a 6.

00:08:39.350 --> 00:08:41.840
4 choose 2 is 6.

00:08:41.840 --> 00:08:43.440
I know this is messy.

00:08:43.440 --> 00:08:47.810
So you have a plus 6
times 3y squared.

00:08:47.810 --> 00:08:53.040
So 9 times 9y squared.

00:08:53.040 --> 00:08:54.110
And we could go keep going.

00:08:54.110 --> 00:08:56.320
And actually, the binomial
coefficient, the next one's

00:08:56.320 --> 00:09:00.400
going to be a 4, and it's
negative 1 to the third, so

00:09:00.400 --> 00:09:03.260
it's going to be a minus
4 times 3y to the 1.

00:09:03.260 --> 00:09:04.650
I mean, we could keep going.

00:09:04.650 --> 00:09:05.660
Actually, let's just
keep going.

00:09:05.660 --> 00:09:08.670
So the next term, 4 choose 3,
I'll leave that for you to

00:09:08.670 --> 00:09:10.890
calculate it, but I already know
that since we're kind of

00:09:10.890 --> 00:09:13.790
past the middle point, the
binomial coefficient is going

00:09:13.790 --> 00:09:16.020
to be the same as 4 choose 1.

00:09:16.020 --> 00:09:19.005
So it's going to be a 4 and we
have a minus 1 to the third,

00:09:19.005 --> 00:09:19.750
so this is minus 1.

00:09:19.750 --> 00:09:26.890
So it's minus 4 times 3y
to the 1, so times 3y.

00:09:26.890 --> 00:09:28.020
And then we have
the last term.

00:09:28.020 --> 00:09:30.120
4 choose 4-- you could calculate
it using this

00:09:30.120 --> 00:09:34.660
formula, but that turns out to
be 1 as well-- plus 1 times

00:09:34.660 --> 00:09:37.580
negative 1 to the fourth,
so that's just plus 1.

00:09:37.580 --> 00:09:39.010
And let's see what
this is equal to.

00:09:39.010 --> 00:09:39.980
Let's see if we can
simplify it.

00:09:39.980 --> 00:09:45.740
This is equal to 81y to the
fourth minus-- let's see, 4

00:09:45.740 --> 00:09:50.600
times 20 is 80, 4 times
7 is 28, so it'd be

00:09:50.600 --> 00:09:54.930
108, 108y to the third.

00:09:54.930 --> 00:09:59.930
6 times 9 is 54, so
plus 54y squared.

00:09:59.930 --> 00:10:03.610
Minus 12y plus 1.

00:10:03.610 --> 00:10:08.750
And if we look at the choices,
only choice A

00:10:08.750 --> 00:10:09.400
is the right answer.

00:10:09.400 --> 00:10:11.990
And actually, we could have
figured out it was choice A

00:10:11.990 --> 00:10:16.730
just by going one term deep,
because that's where all of

00:10:16.730 --> 00:10:18.130
the choices- became different,
so we could have

00:10:18.130 --> 00:10:19.420
saved a lot of work.

00:10:19.420 --> 00:10:22.240
Another option: Instead of
evaluating each of the

00:10:22.240 --> 00:10:25.970
binomial coefficients, another
option would have been to do a

00:10:25.970 --> 00:10:32.910
Pascal's triangle, where you
say, OK, if I have a plus b to

00:10:32.910 --> 00:10:35.880
the first power, the
coefficients are 1 and 1,

00:10:35.880 --> 00:10:38.500
right, 1a plus 1b.

00:10:38.500 --> 00:10:43.170
If I take it to the second
power, a plus b squared, the

00:10:43.170 --> 00:10:46.430
coefficients become
1 plus 1 is 2.

00:10:46.430 --> 00:10:49.440
You bring down a
1 both places.

00:10:49.440 --> 00:10:54.520
And that makes sense because a
plus b squared is 1a squared

00:10:54.520 --> 00:11:00.930
plus 2ab plus 1b squared.

00:11:00.930 --> 00:11:03.240
And then we could
have kept going.

00:11:03.240 --> 00:11:05.360
This would have been
to the third power.

00:11:05.360 --> 00:11:08.480
You had a 1, a 3, a 3 and a 1.

00:11:08.480 --> 00:11:10.630
And then the fourth power is
what this problem was about.

00:11:10.630 --> 00:11:13.860
So you get a 1, you get a 4.

00:11:13.860 --> 00:11:16.030
3 plus 3 is 6.

00:11:16.030 --> 00:11:19.180
3 plus 1 is 4, and
you get a 1.

00:11:19.180 --> 00:11:21.460
And so these would be the
coefficients, and in this

00:11:21.460 --> 00:11:27.070
case, we were doing 3y minus
1 to the fourth power.

00:11:30.350 --> 00:11:32.460
If we had just drawn this out
really fast, we would have

00:11:32.460 --> 00:11:39.730
said, well, this becomes 1 times
3y to the fourth plus 4

00:11:39.730 --> 00:11:45.160
times 3y to the third-- I just
keep decrementing the 3y--

00:11:45.160 --> 00:11:52.510
times negative 1 to the 1 plus
6 times 3y squared--

00:11:52.510 --> 00:11:53.340
this 6 is this one.

00:11:53.340 --> 00:11:58.790
I'm getting a little
misaligned-- 3y squared times

00:11:58.790 --> 00:12:00.700
negative 1 squared.

00:12:00.700 --> 00:12:03.230
So every time I increment the
power on negative 1 and I

00:12:03.230 --> 00:12:05.840
decrement the power on 3y.

00:12:05.840 --> 00:12:08.240
And then plus 4.

00:12:08.240 --> 00:12:09.410
Now I'm at this one.

00:12:09.410 --> 00:12:15.010
Plus 4 times 3y negative
1 to the third.

00:12:15.010 --> 00:12:23.270
And then finally plus 1 times
3y to the 0 is just 1 times

00:12:23.270 --> 00:12:25.270
negative 1 to the the
fourth power.

00:12:25.270 --> 00:12:26.720
And I would have evaluated it.

00:12:26.720 --> 00:12:28.100
And that might have been
a slightly faster

00:12:28.100 --> 00:12:28.630
way of doing it.

00:12:28.630 --> 00:12:30.130
But I wanted to do this a
couple of different ways

00:12:30.130 --> 00:12:32.360
because I think the binomial
theorem and binomial expansion

00:12:32.360 --> 00:12:35.520
is something that people forget,
and it's important to

00:12:35.520 --> 00:12:38.170
kind of have the neurons make a
lot of cross-connections so

00:12:38.170 --> 00:12:40.590
that you don't forget it when
you're 30 years old.

00:12:40.590 --> 00:12:44.040
Anyway, this was the least
number of problems I've done

00:12:44.040 --> 00:12:45.670
in a video, but I think
it was worth it.

00:12:45.670 --> 00:12:47.820
I'll see you in the
next video.


Algebra II: Functions, Combinatorics

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:02.870
We're on problem 60.

00:00:02.870 --> 00:00:10.060
Given the equation y is equal
to x to the n where x is

00:00:10.060 --> 00:00:12.920
greater than 0 and n is
less than 0, which

00:00:12.920 --> 00:00:14.340
statement is valid?

00:00:14.340 --> 00:00:18.860
So let's see, x is greater than
0 and n is less than 0.

00:00:18.860 --> 00:00:23.420
So the exponent is negative,
but the base is positive.

00:00:23.420 --> 00:00:24.860
So I'll just tell
you right now.

00:00:24.860 --> 00:00:27.510
Your gut reaction might be to
say, oh, if I have a negative

00:00:27.510 --> 00:00:29.070
exponent, maybe I somehow
end up with a

00:00:29.070 --> 00:00:30.600
negative number somehow.

00:00:30.600 --> 00:00:33.140
But you have to remember, a
negative exponent, it just

00:00:33.140 --> 00:00:34.930
tells you that it's really
just the inverse of the

00:00:34.930 --> 00:00:35.760
positive exponent.

00:00:35.760 --> 00:00:42.320
So this is equal to 1 over
x to the minus n.

00:00:42.320 --> 00:00:45.000
And if n is less than
0, then minus n is

00:00:45.000 --> 00:00:47.080
greater than 0, right?

00:00:47.080 --> 00:00:50.510
So you could say this is the
same thing as 1 over x to some

00:00:50.510 --> 00:00:53.490
other letter m, where m is minus
m, but then m would be

00:00:53.490 --> 00:00:55.540
greater than 0, right?

00:00:55.540 --> 00:00:58.720
So no matter how negative this
exponent is, especially if

00:00:58.720 --> 00:01:01.890
this base is positive, you're
going to end up with a

00:01:01.890 --> 00:01:02.580
positive number.

00:01:02.580 --> 00:01:04.769
You're going to end up with very
small fractions if this

00:01:04.769 --> 00:01:06.370
gets large enough.

00:01:06.370 --> 00:01:08.550
It could be 1 over x to the
hundredth power or something.

00:01:08.550 --> 00:01:10.350
So it could be a very
small number.

00:01:10.350 --> 00:01:11.980
It can approach zero.

00:01:11.980 --> 00:01:14.410
But it will always be
a positive number.

00:01:14.410 --> 00:01:18.250
So y is always going to
be greater than 0.

00:01:18.250 --> 00:01:21.050
Choice A.

00:01:21.050 --> 00:01:22.820
Problem 61.

00:01:22.820 --> 00:01:26.420
If x is a real number, which
best describes the value of x

00:01:26.420 --> 00:01:29.290
for which the inequality
the square root of x

00:01:29.290 --> 00:01:31.520
greater than 0 is true?

00:01:31.520 --> 00:01:36.640
So I'm assuming we're dealing
with the real numbers, and if

00:01:36.640 --> 00:01:38.470
I get an imaginary number here,
it's actually just even

00:01:38.470 --> 00:01:41.210
hard to compare it in terms
of magnitude to 0.

00:01:41.210 --> 00:01:43.610
So they're actually just asking
us what the domain is.

00:01:43.610 --> 00:01:47.540
What x's can we put in here and
get an answer where we can

00:01:47.540 --> 00:01:50.220
make some type of sense
out of the statement?

00:01:50.220 --> 00:01:53.230
So obviously, I could put
a positive x here.

00:01:53.230 --> 00:01:56.010
x could be greater than 0.

00:01:56.010 --> 00:01:58.330
Any x greater than 0 will
have a square root.

00:01:58.330 --> 00:02:01.280
And all of their square roots
will be greater than 0.

00:02:01.280 --> 00:02:03.490
I mean, you could square both
sides of that equation and you

00:02:03.490 --> 00:02:07.390
get x is greater than 0.

00:02:07.390 --> 00:02:11.280
You could put x is equal to 0
there, but then the inequality

00:02:11.280 --> 00:02:12.450
wouldn't be true, right?

00:02:12.450 --> 00:02:16.280
Because the square root
of 0 is equal to 0.

00:02:16.280 --> 00:02:20.420
So if you put x is equal to 0
there, that would work, you

00:02:20.420 --> 00:02:22.870
can take the square root of 0,
but it just won't make the

00:02:22.870 --> 00:02:23.770
inequality true.

00:02:23.770 --> 00:02:26.000
So x can't be equal to 0.

00:02:26.000 --> 00:02:27.870
So x can be greater than 0.

00:02:27.870 --> 00:02:29.860
It can't be greater than
or equal to 0.

00:02:29.860 --> 00:02:31.420
Not all values of x work.

00:02:31.420 --> 00:02:34.590
I can't put a negative
number in there.

00:02:34.590 --> 00:02:36.400
I mean, they say
no values of x.

00:02:36.400 --> 00:02:36.900
No, sure.

00:02:36.900 --> 00:02:39.720
We can put any positive number
there that works.

00:02:39.720 --> 00:02:43.060
So the answer is A.

00:02:43.060 --> 00:02:45.290
Next problem.

00:02:45.290 --> 00:02:47.100
Let me turn the page.

00:02:54.960 --> 00:02:56.210
Let me copy and paste.

00:03:02.390 --> 00:03:05.760
Which of the following
conclusions is true about the

00:03:05.760 --> 00:03:11.820
statement below: x squared is
equal to the square root of x.

00:03:11.820 --> 00:03:13.030
All right, so once again,
let's think

00:03:13.030 --> 00:03:16.080
about it a little bit.

00:03:16.080 --> 00:03:17.360
x squared is equal to the
square root of x.

00:03:17.360 --> 00:03:19.790
I mean, it definitely works for
the number 1 and it works

00:03:19.790 --> 00:03:22.300
for the number 0.

00:03:22.300 --> 00:03:26.170
If you square both sides of that
equation, you get x to

00:03:26.170 --> 00:03:31.110
the fourth is equal to x.

00:03:31.110 --> 00:03:33.760
x could be equal to 0, but if
you assume that x does not

00:03:33.760 --> 00:03:37.820
equal 0, you could divide both
sides of this equation by x

00:03:37.820 --> 00:03:42.220
and you get x to the third
is equal to 1.

00:03:42.220 --> 00:03:43.970
And the only number that
I know works for that

00:03:43.970 --> 00:03:45.240
is the number 1.

00:03:45.240 --> 00:03:48.530
But anyway, the statement
is always true.

00:03:48.530 --> 00:03:49.730
No, it's not always true.

00:03:49.730 --> 00:03:51.130
I put a 2 there.

00:03:51.130 --> 00:03:52.980
Is 2 squared equal to the
square root of 2?

00:03:52.980 --> 00:03:54.740
No, 4 does not equal the
square root of 2.

00:03:54.740 --> 00:03:56.940
So the statement is
not always true.

00:03:56.940 --> 00:03:58.415
The statement is true
when x is negative.

00:03:58.415 --> 00:03:59.700
No, x can't even be negative.

00:03:59.700 --> 00:04:01.150
You can't take the square root
of a negative number.

00:04:01.150 --> 00:04:03.400
You'd end up with imaginary
numbers and it

00:04:03.400 --> 00:04:05.600
just gets very strange.

00:04:05.600 --> 00:04:07.220
The statement is true when
x is equal to 0.

00:04:07.220 --> 00:04:07.740
Sure.

00:04:07.740 --> 00:04:09.890
It could be 0.

00:04:09.890 --> 00:04:11.520
The statement is never true.

00:04:11.520 --> 00:04:11.930
No.

00:04:11.930 --> 00:04:16.040
Clearly, it's true when x is
equal to 0 or x is equal to 1.

00:04:16.040 --> 00:04:17.070
So the answer is C.

00:04:17.070 --> 00:04:19.190
I didn't even have to do this
stuff on the side right here.

00:04:19.190 --> 00:04:20.440
But, hopefully, it
was instructive.

00:04:23.960 --> 00:04:25.210
Next problem.

00:04:28.940 --> 00:04:33.520
I copied it now I
have pasted it.

00:04:33.520 --> 00:04:34.620
All right.

00:04:34.620 --> 00:04:37.450
Let me try with the
larger pen.

00:04:37.450 --> 00:04:41.140
Abelardo wants to create several
different 7-character

00:04:41.140 --> 00:04:42.060
screen names.

00:04:42.060 --> 00:04:44.990
He wants to use arrangements of
the first three letters of

00:04:44.990 --> 00:04:48.280
his first name, ABE-- that's
why they made his name

00:04:48.280 --> 00:04:52.180
Abelardo-- followed by
arrangements of the 4 digits

00:04:52.180 --> 00:04:55.860
in 1984, the year
of his birth.

00:04:55.860 --> 00:04:58.180
How many different screen
names can he

00:04:58.180 --> 00:05:00.370
create in this way?

00:05:00.370 --> 00:05:01.950
All right, so he's got 7.

00:05:01.950 --> 00:05:02.510
So think about it.

00:05:02.510 --> 00:05:04.260
Let's just think about
each slot.

00:05:04.260 --> 00:05:07.830
So slot number 1-- that
pen's too fat.

00:05:07.830 --> 00:05:14.280
Slot number 1, slot number 1,
spot number 3, 4, 5, 6, 7.

00:05:14.280 --> 00:05:16.700
So slot number 1, how many
different choices

00:05:16.700 --> 00:05:17.600
can he put in there?

00:05:17.600 --> 00:05:21.270
He can put in an A,
a B or an E there.

00:05:21.270 --> 00:05:24.440
So he has three different
choices for slot number 1 I'll

00:05:24.440 --> 00:05:25.400
do that in a different color.

00:05:25.400 --> 00:05:28.350
He has three different choices
for slot number 1.

00:05:28.350 --> 00:05:30.600
Once he picks slot number 1, how
many choices does he have

00:05:30.600 --> 00:05:31.730
left for slot number 2?

00:05:31.730 --> 00:05:37.190
Well, he has two letters left
out of the A, B and E, right?

00:05:37.190 --> 00:05:39.410
So he has only two
options left.

00:05:39.410 --> 00:05:41.880
And once he gets to slot number
3, he has to just pick

00:05:41.880 --> 00:05:44.190
whatever leftover letter
is there in A, B and E.

00:05:44.190 --> 00:05:46.310
So he only has one option.

00:05:46.310 --> 00:05:48.700
But then when he gets to fourth
digit, now he's trying

00:05:48.700 --> 00:05:50.270
to pick from 1984.

00:05:50.270 --> 00:05:52.680
For the fourth digit, once he
has four options, he can pick

00:05:52.680 --> 00:05:55.180
the 1, the 9, the 8 or the 4.

00:05:55.180 --> 00:05:57.080
He has four options.

00:05:57.080 --> 00:05:59.190
Then on the fifth digit, he
would have picked one of the

00:05:59.190 --> 00:06:01.530
four, so he has three
of them left.

00:06:01.530 --> 00:06:03.760
By the sixth digit, he only
has two of them left.

00:06:03.760 --> 00:06:06.540
By the seventh digit, he's
forced to pick whatever's

00:06:06.540 --> 00:06:08.530
leftover, what he hasn't
picked yet.

00:06:08.530 --> 00:06:09.850
He has one digit left.

00:06:09.850 --> 00:06:13.330
So the combinations, you just
multiply all of them.

00:06:13.330 --> 00:06:17.180
3 times 2 is 6 times 1 is 6.

00:06:17.180 --> 00:06:19.870
6 times 4 is 24.

00:06:19.870 --> 00:06:24.380
24 times 3 is 72.

00:06:24.380 --> 00:06:29.130
72 times 2 is 144
times 1 is 144.

00:06:29.130 --> 00:06:32.010
So there's 144 different
screen names he can

00:06:32.010 --> 00:06:34.510
create in this way.

00:06:34.510 --> 00:06:35.760
Next problem.

00:06:38.290 --> 00:06:45.884
64.

00:06:45.884 --> 00:06:47.550
I've copied it.

00:06:47.550 --> 00:06:50.860
And now I'm pasting it.

00:06:50.860 --> 00:06:53.580
A train is made up of a
locomotive, seven different

00:06:53.580 --> 00:06:55.520
cars and a caboose.

00:06:55.520 --> 00:06:58.780
If the locomotive must be first
and the caboose must be

00:06:58.780 --> 00:07:01.520
last, how many different ways
can the train be ordered?

00:07:01.520 --> 00:07:03.350
So we don't even have
to worry about the

00:07:03.350 --> 00:07:04.870
locomotive and the caboose.

00:07:04.870 --> 00:07:06.940
They're always going to be
in the same position.

00:07:06.940 --> 00:07:09.690
The locomotive's first, caboose
is last. So we just

00:07:09.690 --> 00:07:12.170
have to worry about the seven
different cars, the order that

00:07:12.170 --> 00:07:14.990
they're in.

00:07:14.990 --> 00:07:17.360
It's really the same problem
as the last one.

00:07:17.360 --> 00:07:19.320
In some ways, it's a
little bit easier.

00:07:19.320 --> 00:07:21.010
So let's just say that's
the position.

00:07:21.010 --> 00:07:25.615
Position 1, 2, 3, 4, 5, 6, 7.

00:07:25.615 --> 00:07:28.320
And let's say we just have to
decide which car we put in

00:07:28.320 --> 00:07:32.090
each position, or how many
different options do we have.

00:07:32.090 --> 00:07:33.660
So let's say we are filling
the slots starting with

00:07:33.660 --> 00:07:34.340
position 1.

00:07:34.340 --> 00:07:38.770
Well, we have seven cars
to choose from.

00:07:38.770 --> 00:07:39.885
The seven is different
from them.

00:07:39.885 --> 00:07:42.000
That's not including the
locomotive and the caboose, so

00:07:42.000 --> 00:07:44.330
we have seven cars
to pick from.

00:07:44.330 --> 00:07:45.310
We've picked one of them.

00:07:45.310 --> 00:07:47.780
Then in slot 2, we have
six cars to pick from.

00:07:47.780 --> 00:07:50.440
Slot 3, we have five cars to
pick from because we've

00:07:50.440 --> 00:07:51.590
already picked two.

00:07:51.590 --> 00:07:53.890
And in slot 4, four
cars to pick from,

00:07:53.890 --> 00:07:55.700
three cars, two cars.

00:07:55.700 --> 00:07:57.690
And then in slot 7, we've
already picked six of the

00:07:57.690 --> 00:08:00.120
cars, so whatever's leftover,
we have to pick.

00:08:00.120 --> 00:08:01.220
So let's see what
this number is.

00:08:01.220 --> 00:08:02.990
1 times 2 is 2.

00:08:02.990 --> 00:08:05.080
2 times 3 is 6.

00:08:05.080 --> 00:08:07.600
6 times 4 is 24.

00:08:07.600 --> 00:08:11.510
24 times 5 is 120.

00:08:11.510 --> 00:08:18.550
120 times 6 is 720.

00:08:18.550 --> 00:08:21.080
720 times 7-- I'm just going
to have to do a little bit

00:08:21.080 --> 00:08:24.850
multiplication here on the
side-- 720 times 7.

00:08:24.850 --> 00:08:26.140
7 times 0 is 0.

00:08:26.140 --> 00:08:29.000
7 times 2 is 14.

00:08:29.000 --> 00:08:34.940
7 times 7 is 49 plus 1 is 50.

00:08:34.940 --> 00:08:35.740
5,040.

00:08:35.740 --> 00:08:38.850
So there's 5,040 different
ways that the

00:08:38.850 --> 00:08:40.100
train can be ordered.

00:08:42.977 --> 00:08:43.770
Let's see.

00:08:43.770 --> 00:08:45.020
Next problem.

00:08:49.570 --> 00:08:53.000
Maybe I'll just clear
all of this.

00:08:53.000 --> 00:08:54.130
All right.

00:08:54.130 --> 00:09:02.730
Let me copy the-- all right.

00:09:02.730 --> 00:09:06.240
Teresa and Julia are among 10
students who have applied for

00:09:06.240 --> 00:09:08.460
a trip to Washington, D.C.

00:09:08.460 --> 00:09:10.460
Two students from the group
will be selected at

00:09:10.460 --> 00:09:12.660
random for the trip.

00:09:12.660 --> 00:09:16.920
What is the probability that
Teresa and Julia will be the

00:09:16.920 --> 00:09:18.220
two students selected?

00:09:18.220 --> 00:09:19.160
So let's think about this.

00:09:19.160 --> 00:09:21.310
There are 10 students who have
applied for the trip.

00:09:21.310 --> 00:09:22.980
Two students from the group
will be selected at

00:09:22.980 --> 00:09:24.040
random for the trip.

00:09:24.040 --> 00:09:27.390
What is the probability Teresa
and Julia will be the two

00:09:27.390 --> 00:09:29.020
students selected?

00:09:29.020 --> 00:09:31.680
All right, so let's think about
student 1 selected and

00:09:31.680 --> 00:09:33.880
student 2 selected.

00:09:33.880 --> 00:09:38.500
What's the probability that
Teresa is the first student?

00:09:38.500 --> 00:09:42.210
So what's the probably that we
get Teresa and then we get

00:09:42.210 --> 00:09:44.630
Julia, that Teresa is the first
student selected and

00:09:44.630 --> 00:09:47.080
that Julia is the second
student selected?

00:09:47.080 --> 00:09:49.380
Well, the probability that
Teresa is the first-- well,

00:09:49.380 --> 00:09:50.260
let me take a step back.

00:09:50.260 --> 00:09:52.530
There's two ways that what
they want to happen could

00:09:52.530 --> 00:09:54.930
happen, that Teresa and
Julia will be the

00:09:54.930 --> 00:09:56.700
two students selected.

00:09:56.700 --> 00:09:58.820
Either Teresa is the first
student selected and then

00:09:58.820 --> 00:10:03.650
Julia is the second; or Julia is
the first student selected

00:10:03.650 --> 00:10:05.490
and Teresa is the second.

00:10:05.490 --> 00:10:07.790
So if we figure out the
probability of this happening

00:10:07.790 --> 00:10:10.440
and the probability of this
happening, then we can just

00:10:10.440 --> 00:10:12.770
add those probabilities together
and then we have the

00:10:12.770 --> 00:10:16.020
probability of either
of those happening.

00:10:16.020 --> 00:10:17.420
All right, so let's do that.

00:10:17.420 --> 00:10:22.616
The probability of Teresa
happening is-- let me do this

00:10:22.616 --> 00:10:24.170
in a different color.

00:10:24.170 --> 00:10:25.790
There's 10 students, so there's
a probability that we

00:10:25.790 --> 00:10:29.270
pick Teresa first. And then
once we've picked Teresa

00:10:29.270 --> 00:10:32.550
first, the probability that we
pick Julia next, well, now

00:10:32.550 --> 00:10:34.580
there's 9 students
left, right?

00:10:34.580 --> 00:10:37.400
Because you've already picked
Teresa, so there's a 1 in 9

00:10:37.400 --> 00:10:39.390
chance you pick Julia.

00:10:39.390 --> 00:10:41.900
So that is equal to 1 in 90.

00:10:41.900 --> 00:10:45.980
So there's a 1 in 90 chance
of this happening.

00:10:45.980 --> 00:10:48.120
There's a 1 in 90 chance
of that happening.

00:10:48.120 --> 00:10:51.080
And then what's the chance
of this happening?

00:10:51.080 --> 00:10:52.610
What's the chance of
that happening?

00:10:52.610 --> 00:10:53.440
Well, it's the same thing.

00:10:53.440 --> 00:11:00.170
This is 1 in 10 and this
is 1 in 90, right?

00:11:00.170 --> 00:11:04.690
And so you have a 1 in 90
chance of it happening.

00:11:04.690 --> 00:11:08.370
And so, what is the probability
of either of these

00:11:08.370 --> 00:11:09.310
things happening?

00:11:09.310 --> 00:11:11.360
Of this combination or
this combination?

00:11:11.360 --> 00:11:14.020
Well, you just add them
together, and you get 2 out of

00:11:14.020 --> 00:11:20.970
90, which is equal to 1 out
of 45, which is choice A.

00:11:20.970 --> 00:11:22.160
And I'm out of time.

00:11:22.160 --> 00:11:24.070
See you in the next video.


Algebra II: Logarithms and more

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:02.590
We're on problem 53.

00:00:02.590 --> 00:00:06.000
Which table below correctly
describes the points of the

00:00:06.000 --> 00:00:09.950
exponential function f of
x is equal to 3 to the

00:00:09.950 --> 00:00:11.620
minus x minus 2?

00:00:11.620 --> 00:00:15.330
They all input the same x-values
as far as I can tell.

00:00:15.330 --> 00:00:17.270
Yeah, they all input
the same x-values.

00:00:17.270 --> 00:00:19.000
So let's just figure out
which choice works.

00:00:19.000 --> 00:00:21.280
So if we take f of minus
2, that's the first

00:00:21.280 --> 00:00:23.680
one they give us.

00:00:23.680 --> 00:00:29.280
f of minus 2, that is equal
to 3 to the minus x.

00:00:29.280 --> 00:00:34.920
Well, x is now minus 2, so
minus minus 2 minus 2.

00:00:34.920 --> 00:00:35.960
So what's minus minus 2?

00:00:35.960 --> 00:00:38.790
Well, that's the same thing as
3 squared minus 2, so that is

00:00:38.790 --> 00:00:44.995
equal to 9 minus 2
is equal to 7.

00:00:44.995 --> 00:00:46.740
Did I get that right?

00:00:46.740 --> 00:00:49.150
Do you see any of
these choices?

00:00:49.150 --> 00:00:51.250
Well, only one of them
has that. f of minus

00:00:51.250 --> 00:00:53.020
2 is equal to 7.

00:00:53.020 --> 00:00:54.380
That's not right.

00:00:54.380 --> 00:00:55.450
That's not right.

00:00:55.450 --> 00:00:56.940
That's not right.

00:00:56.940 --> 00:00:59.080
Well, here when x is minus
2, f of x is equal

00:00:59.080 --> 00:01:01.030
to 7, choice D.

00:01:01.030 --> 00:01:03.170
We could have just evaluated the
other ones and we probably

00:01:03.170 --> 00:01:06.440
would have agreed with all
of these numbers as well.

00:01:06.440 --> 00:01:10.760
Well, that was a lot of copying
and pasting for a

00:01:10.760 --> 00:01:13.890
fairly straightforward
problem.

00:01:13.890 --> 00:01:14.980
Next problem.

00:01:14.980 --> 00:01:17.510
I already had it there
waiting for us.

00:01:17.510 --> 00:01:23.350
log base 6 of 40.

00:01:23.350 --> 00:01:27.130
So, in general, and just
inspecting it, I think they

00:01:27.130 --> 00:01:29.180
just want you to figure out
how do you figure out a

00:01:29.180 --> 00:01:33.320
logarithm of any arbitrary base
of some number if you

00:01:33.320 --> 00:01:36.040
only have a log base 10
function, which tends to be

00:01:36.040 --> 00:01:37.570
the case in your calculator.

00:01:37.570 --> 00:01:39.870
I proved this in a Khan Academy
video, but the general

00:01:39.870 --> 00:01:46.750
idea is if I have log base a
of b, that this is equal to

00:01:46.750 --> 00:01:53.460
log base 10 of b divided
by log base 10 of a.

00:01:53.460 --> 00:01:55.940
Actually, these both don't
have to be 10.

00:01:55.940 --> 00:01:57.730
They just have to be
the same base.

00:01:57.730 --> 00:01:58.920
10 tends to be convenient.

00:01:58.920 --> 00:02:01.330
The other base that tends
to be convenient is e.

00:02:01.330 --> 00:02:02.680
But anyway, let's use
that property

00:02:02.680 --> 00:02:03.660
to solve this problem.

00:02:03.660 --> 00:02:11.610
So log base 6 of 40 is equal to
log base 10 of 40 divided

00:02:11.610 --> 00:02:17.460
by log base 10 of 6,
which is choice D.

00:02:22.170 --> 00:02:23.550
All right, next problem.

00:02:32.260 --> 00:02:33.510
Copy and paste it.

00:02:37.520 --> 00:02:40.000
Just so you know, I don't like
talking while I'm copying and

00:02:40.000 --> 00:02:42.780
pasting, because I think it
slows my computer down and

00:02:42.780 --> 00:02:46.560
then I start sounding
like Optimus Prime.

00:02:46.560 --> 00:02:50.620
All right, Jonathan wrote the
equation log base 6 of x minus

00:02:50.620 --> 00:02:52.960
4 is equal to 0 on the board.

00:02:52.960 --> 00:02:56.390
He needs one clue for
problem solving.

00:02:56.390 --> 00:02:59.430
Which fact provides the correct
information that he

00:02:59.430 --> 00:03:02.340
needs to solve the equation?

00:03:02.340 --> 00:03:06.200
Well, I don't know if he
needs any information.

00:03:06.200 --> 00:03:11.180
This statement is essentially
saying that 6 to the 0th power

00:03:11.180 --> 00:03:14.230
is equal to x minus 4.

00:03:14.230 --> 00:03:16.670
Then you could add 4 to both
sides, and you would get 6 to

00:03:16.670 --> 00:03:20.080
the 0th power plus
4 is equal to x.

00:03:20.080 --> 00:03:22.560
I guess if he didn't know the
definition already of 6 to the

00:03:22.560 --> 00:03:25.990
0th power, he would have to
know that to know that it

00:03:25.990 --> 00:03:29.650
equals 1, so 1 plus 4 is
equal to x, which is 5.

00:03:29.650 --> 00:03:34.190
So I guess he has to know that
6 to the 0th power is a piece

00:03:34.190 --> 00:03:35.440
of information.

00:03:37.280 --> 00:03:40.470
Well this one wouldn't
have been useful.

00:03:40.470 --> 00:03:43.270
6 to the first power, that's
not even useful even if you

00:03:43.270 --> 00:03:44.150
didn't know it.

00:03:44.150 --> 00:03:47.430
4 minus 4 isn't useful even
if you didn't know if.

00:03:47.430 --> 00:03:50.080
But 6 minus 4, even
that's not useful.

00:03:50.080 --> 00:03:52.380
Yes, choice A, he had
to know 6 to the 0th

00:03:52.380 --> 00:03:55.860
power is equal to 1.

00:03:55.860 --> 00:03:57.840
All right.

00:03:57.840 --> 00:03:59.890
Let's see, they have
a couple more.

00:03:59.890 --> 00:04:03.640
I'm going to copy and paste all
of them at the same time.

00:04:03.640 --> 00:04:04.890
I'm getting bold.

00:04:07.080 --> 00:04:10.540
All right, like that.

00:04:10.540 --> 00:04:16.420
Let me clear all of this: clear
image, image, invert

00:04:16.420 --> 00:04:20.410
colors, and I'm back
in business.

00:04:20.410 --> 00:04:22.110
There you go.

00:04:22.110 --> 00:04:23.780
All right, I never know
whether I should use a

00:04:23.780 --> 00:04:26.420
dark-colored or light-colored
marker, because over here, the

00:04:26.420 --> 00:04:28.250
light-colored ones look good,
but on the white, the darks.

00:04:28.250 --> 00:04:29.170
So who knows?

00:04:29.170 --> 00:04:31.460
Magenta tends to be good,
maybe purple.

00:04:31.460 --> 00:04:34.860
What is the value of
log base 3 of 27?

00:04:34.860 --> 00:04:39.640
So that's just asking 3 to what
power is equal to 27?

00:04:39.640 --> 00:04:41.650
3 to the first is 3.

00:04:41.650 --> 00:04:43.060
3 squared is 9.

00:04:43.060 --> 00:04:45.910
3 to the third power
is equal to 27.

00:04:45.910 --> 00:04:49.890
So log base 3 of 27
is equal to 3.

00:04:49.890 --> 00:04:52.170
They, I guess, just want you to
know what the definition of

00:04:52.170 --> 00:04:54.290
a logarithm is in
that question.

00:04:54.290 --> 00:04:56.320
I'm noticing that there isn't
necessarily a progression of

00:04:56.320 --> 00:04:57.340
difficulty in the problem.

00:04:57.340 --> 00:05:00.620
They kind of flip between an
easy one and a hard one.

00:05:00.620 --> 00:05:01.090
Who knows?

00:05:01.090 --> 00:05:04.150
They're just a grab
bag of questions.

00:05:04.150 --> 00:05:06.980
If log base 2-- no,
not log base 2.

00:05:06.980 --> 00:05:11.060
They're saying log of 2 is
approximately equal to 0.301.

00:05:11.060 --> 00:05:13.710
When people just write a log
without a base, you usually

00:05:13.710 --> 00:05:15.840
assume it's base 10.

00:05:15.840 --> 00:05:17.350
It's kind of like what you'd
get in your calculator.

00:05:17.350 --> 00:05:19.890
If you just press log on your
calculator, it's log base 10.

00:05:19.890 --> 00:05:24.730
So 10 to the 0.3 power
is equal to 2.

00:05:24.730 --> 00:05:28.670
If log base 10 of 2 is
approximately equal to 0.301,

00:05:28.670 --> 00:05:33.500
and log base 10 of 3 is
approximately equal to 0.477,

00:05:33.500 --> 00:05:40.390
what is the approximate
value of log of 72?

00:05:40.390 --> 00:05:42.610
So let's see, I think we just
have to do some prime

00:05:42.610 --> 00:05:44.390
factorization here.

00:05:44.390 --> 00:05:50.710
So 72, that's equal to the log--
let's think about it.

00:05:50.710 --> 00:05:55.810
That's the log of-- let
me think about it.

00:05:55.810 --> 00:05:58.800
This is 36 times 2.

00:05:58.800 --> 00:06:03.330
Well, I don't know if
that even helps us.

00:06:03.330 --> 00:06:09.380
3 to the third, that's 27.

00:06:09.380 --> 00:06:14.500
This is 9 times 8.

00:06:14.500 --> 00:06:21.440
So this is log of 3 to
the third times 2

00:06:21.440 --> 00:06:23.800
squared; is that right?

00:06:23.800 --> 00:06:26.380
So this is log of 9.

00:06:26.380 --> 00:06:28.070
I made a mistake.

00:06:28.070 --> 00:06:29.500
Ignore that.

00:06:29.500 --> 00:06:34.320
So this is the same thing as
the logarithm of 9 times 8.

00:06:34.320 --> 00:06:35.570
Fair enough.

00:06:35.570 --> 00:06:41.640
So that's the same thing
as the log of 9

00:06:41.640 --> 00:06:43.300
plus the log of 8.

00:06:46.450 --> 00:06:47.500
Fair enough.

00:06:47.500 --> 00:06:53.770
Well, that's the same thing as
the log of 3 squared plus the

00:06:53.770 --> 00:06:56.360
log of 2 to the third.

00:06:56.360 --> 00:06:59.050
2 to the third is equal to 8.

00:06:59.050 --> 00:06:59.700
Let's keep going.

00:06:59.700 --> 00:07:00.910
I think you see where
it's going.

00:07:00.910 --> 00:07:01.540
3 squared is 9.

00:07:01.540 --> 00:07:04.210
OK, so log of 3 squared, that's
the same thing as 2

00:07:04.210 --> 00:07:08.090
times the log of 3.

00:07:08.090 --> 00:07:10.080
The log of 2 to the third power,
that's the same thing

00:07:10.080 --> 00:07:13.530
as 3 times the log of 2.

00:07:13.530 --> 00:07:16.830
We know what the log of 3 is and
we know the log of 2 is.

00:07:16.830 --> 00:07:24.770
The log of 3 is 0.477, so we
have 2 times 0.477 plus 3

00:07:24.770 --> 00:07:31.450
times the log of 2, plus
3 times 0.301.

00:07:31.450 --> 00:07:36.670
So 2 times 0.477-- let
me do it here.

00:07:36.670 --> 00:07:41.590
So 0.477 times 2.

00:07:41.590 --> 00:07:47.200
2 times 7 is 14 plus 1 is 15.

00:07:47.200 --> 00:07:51.530
2 times 4 is 8 plus 1 is 9,
three numbers behind the

00:07:51.530 --> 00:07:54.040
decimal point, 0.954.

00:07:54.040 --> 00:07:58.210
Then 3 times point 0.301,
that's easier.

00:07:58.210 --> 00:08:00.950
That's 0.903.

00:08:00.950 --> 00:08:03.060
If you add them together, I
can already tell you, it's

00:08:03.060 --> 00:08:04.450
going to be that right there.

00:08:04.450 --> 00:08:07.305
It's going to be 1.857.

00:08:07.305 --> 00:08:09.720
It's choice D.

00:08:09.720 --> 00:08:15.670
Problem 58.

00:08:15.670 --> 00:08:18.950
If x is a real number, for
what values of x is the

00:08:18.950 --> 00:08:24.200
equation 3x minus 9 over 3 is
equal to x minus 3 true?

00:08:24.200 --> 00:08:24.700
All right.

00:08:24.700 --> 00:08:27.120
Well, if x is a real
number, let's just

00:08:27.120 --> 00:08:28.780
play around with it.

00:08:28.780 --> 00:08:29.830
Maybe it's just a truism.

00:08:29.830 --> 00:08:30.890
Well, I know it is a truism.

00:08:30.890 --> 00:08:32.390
Look at it.

00:08:32.390 --> 00:08:33.309
Let me write it.

00:08:33.309 --> 00:08:39.250
3x minus 9 over 3 is
equal to x minus 3.

00:08:39.250 --> 00:08:42.429
Let's multiply both sides
of this equation by 3.

00:08:42.429 --> 00:08:49.350
You'll get 3x minus 9 is
equal to 3x minus 9.

00:08:49.350 --> 00:08:50.990
So we can stop there.

00:08:50.990 --> 00:08:54.190
So if I take 3 times a number
and subtract 9 from it and if

00:08:54.190 --> 00:08:56.680
I said that's the same thing as
taking 3 times a number and

00:08:56.680 --> 00:09:00.030
subtracting 9 from it, well,
then x could be any number.

00:09:00.030 --> 00:09:03.780
So it works for all
values of x.

00:09:03.780 --> 00:09:04.550
You can keep solving.

00:09:04.550 --> 00:09:05.740
Let's just add 9
to both sides.

00:09:05.740 --> 00:09:08.760
You get 3x equal to 3x.

00:09:08.760 --> 00:09:11.315
Divide both sides by 3, you
get x is equal to x.

00:09:11.315 --> 00:09:12.880
Well, that's true for
any real number x.

00:09:12.880 --> 00:09:14.420
It's equal to itself.

00:09:14.420 --> 00:09:18.090
So that works for
all numbers x.

00:09:18.090 --> 00:09:19.850
Let's see what else they
have to throw at us.

00:09:24.960 --> 00:09:27.560
All right, let's do one more.

00:09:32.692 --> 00:09:37.580
I've copied, and now
I have pasted.

00:09:37.580 --> 00:09:41.320
On a recent test, Jeremy wrote
the equation x squared minus

00:09:41.320 --> 00:09:44.450
16 divided by x minus 4
is equal to x plus 4.

00:09:44.450 --> 00:09:46.165
Which of the following
statements is correct about

00:09:46.165 --> 00:09:47.860
the equation he wrote?

00:09:47.860 --> 00:09:51.180
So let's play with it a little
bit. x squared minus 16 over x

00:09:51.180 --> 00:09:54.050
minus 4 is equal to x plus 4.

00:09:56.630 --> 00:09:58.960
So immediately when you see
something like this, in order

00:09:58.960 --> 00:10:04.925
to do this, what he had to say
is that x squared minus 16 is

00:10:04.925 --> 00:10:08.800
x plus 4 times x minus 4.

00:10:08.800 --> 00:10:11.200
That's over x minus 4.

00:10:11.200 --> 00:10:13.880
Then to simplify from this to
that, he essentially canceled

00:10:13.880 --> 00:10:16.660
out both of these things.

00:10:16.660 --> 00:10:17.960
That's fine.

00:10:17.960 --> 00:10:19.960
That's fine, assuming
that this number

00:10:19.960 --> 00:10:21.210
down here is non-zero.

00:10:23.650 --> 00:10:25.280
0 over 0 is undefined.

00:10:25.280 --> 00:10:28.500
Anything over a 0
is undefined.

00:10:28.500 --> 00:10:32.610
His simplification is true as
long as x does not equal 4.

00:10:32.610 --> 00:10:36.330
Because if x equaled 4, then you
would have an 8 times a 0

00:10:36.330 --> 00:10:41.590
divided by a 0, and 8 times
a 0 divided by a 0 is not

00:10:41.590 --> 00:10:45.530
defined, so this doesn't work
when x is equal to 4.

00:10:45.530 --> 00:10:49.970
If you think about it, if I were
to graph this equation or

00:10:49.970 --> 00:10:53.630
his original equation, it
would look just like the

00:10:53.630 --> 00:10:57.190
equation of x plus 4, except
it would have a hole at 4

00:10:57.190 --> 00:10:59.380
because it's not defined
at x is equal to 4.

00:10:59.380 --> 00:11:01.880
So they're not exactly
the same.

00:11:01.880 --> 00:11:05.145
The graphs would be exactly the
same except at the point x

00:11:05.145 --> 00:11:06.390
is equal 4.

00:11:06.390 --> 00:11:10.340
So the point is b.

00:11:10.340 --> 00:11:11.660
All right.

00:11:11.660 --> 00:11:12.620
I'm out of time.

00:11:12.620 --> 00:11:15.080
See you in the next video.


Algebra II: Logarithms Exponential Growth

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.300
We're on problem 49.

00:00:02.300 --> 00:00:07.980
Let me copy and paste
it into the window.

00:00:07.980 --> 00:00:12.080
I've copied it and now
let me paste it.

00:00:12.080 --> 00:00:15.110
There you go, problem 49.

00:00:15.110 --> 00:00:18.915
Let me pick a good highlighter
color, magenta is good.

00:00:18.915 --> 00:00:22.190
A student showed the following
steps in his solution to the

00:00:22.190 --> 00:00:25.810
equation below, but his answer
was not correct.

00:00:25.810 --> 00:00:28.070
Then at the bottom they say, in
which step did he make his

00:00:28.070 --> 00:00:28.910
first error?

00:00:28.910 --> 00:00:34.410
So let's see, log base 5 of 2x
squared, minus 3x, plus 1,

00:00:34.410 --> 00:00:39.180
minus log base 5 of x, minus 1,
plus log base 5 of 125 is

00:00:39.180 --> 00:00:39.830
equal to 6.

00:00:39.830 --> 00:00:41.570
Fair enough.

00:00:41.570 --> 00:00:45.180
All right, now it looks like
in this first step, he kept

00:00:45.180 --> 00:00:46.910
this the same.

00:00:46.910 --> 00:00:48.640
You can't mess up there.

00:00:48.640 --> 00:00:51.490
Then he said log base
5 of 125 is 3.

00:00:51.490 --> 00:00:52.110
Yeah, that's right.

00:00:52.110 --> 00:00:55.900
5 to the third power
is equal to 125.

00:00:55.900 --> 00:00:58.300
So that is right as well.

00:00:58.300 --> 00:01:01.630
Let's see, here it looks like he
took the 2x squared, minus

00:01:01.630 --> 00:01:05.640
3x plus 1, and he factored
them into 2x minus

00:01:05.640 --> 00:01:06.430
1 and x minus 1.

00:01:06.430 --> 00:01:07.520
Let's see if that's right.

00:01:07.520 --> 00:01:09.980
2x times 2x is 2x squared.

00:01:09.980 --> 00:01:15.540
2x times minus 1 is minus 2x.

00:01:15.540 --> 00:01:18.406
Minus 1 times x is minus 3x.

00:01:18.406 --> 00:01:21.870
And you have a minus 1 times a
minus 1, you get a plus 1.

00:01:21.870 --> 00:01:26.080
So you get 2x squared,
minus x, minus 2x, so

00:01:26.080 --> 00:01:27.310
minus 3x, plus 1.

00:01:27.310 --> 00:01:28.470
So this is right.

00:01:28.470 --> 00:01:30.680
So so far so good.

00:01:30.680 --> 00:01:32.390
What does he do next?

00:01:32.390 --> 00:01:33.880
All right, in the next
step he keeps this

00:01:33.880 --> 00:01:38.070
first term the same.

00:01:38.070 --> 00:01:39.680
2x minus 1, x minus 1.

00:01:39.680 --> 00:01:41.140
Fair enough.

00:01:41.140 --> 00:01:43.490
He keeps this term the same.

00:01:43.490 --> 00:01:46.720
He just subtracts 3
from both sides.

00:01:46.720 --> 00:01:49.550
He subtracts 3 from both sides
and this becomes 0 and then

00:01:49.550 --> 00:01:51.770
this becomes 3.

00:01:51.770 --> 00:01:53.030
Fair enough.

00:01:53.030 --> 00:01:54.690
What does he do in
the next step?

00:01:54.690 --> 00:01:55.940
This is interesting.

00:02:01.370 --> 00:02:05.720
He essentially said, if I am
subtracting 2 logarithms with

00:02:05.720 --> 00:02:08.280
the same base, that's
the equivalent of

00:02:08.280 --> 00:02:09.479
dividing the two things.

00:02:09.479 --> 00:02:12.010
So he kind of did a
quick step here.

00:02:12.010 --> 00:02:17.170
So his logic here is that log--
I don't even have to

00:02:17.170 --> 00:02:24.190
write the base 5-- but log of
2x, minus 1, times x minus 1,

00:02:24.190 --> 00:02:30.490
minus log of base 5, times x
minus 1, that that would be

00:02:30.490 --> 00:02:40.450
equal log of 2x minus 1, times
x minus 1, all of that

00:02:40.450 --> 00:02:41.780
over x minus 1.

00:02:45.490 --> 00:02:46.840
That seems fair enough.

00:02:46.840 --> 00:02:49.660
So that should cancel out.

00:02:49.660 --> 00:02:53.460
Oh, but that's where he
made his mistake.

00:02:53.460 --> 00:02:56.190
What you end up with left over
when you cancel out the 2x

00:02:56.190 --> 00:02:58.330
minus 1's, isn't one
of the minus 1's.

00:02:58.330 --> 00:03:03.500
You're left with log
of 2x minus 1.

00:03:03.500 --> 00:03:07.750
This x minus 1 and this
x minus 1 cancel out.

00:03:07.750 --> 00:03:09.710
He should be left
with just the 2x

00:03:09.710 --> 00:03:11.900
minus 1 in the logarithm.

00:03:11.900 --> 00:03:14.190
But instead, somehow he got
rid of the 2x minus 1 and

00:03:14.190 --> 00:03:16.340
ended up with just an
x minus 1 here.

00:03:16.340 --> 00:03:19.115
So step 3 is where he made
his first mistake.

00:03:25.630 --> 00:03:26.010
I don't know.

00:03:26.010 --> 00:03:29.910
He somehow intuitively knew that
you could divide x minus

00:03:29.910 --> 00:03:37.390
1 into this, because of log of
A minus log of B is equal to

00:03:37.390 --> 00:03:39.960
log of A over B.

00:03:39.960 --> 00:03:41.900
But I don't know, he simplified
wrong or something.

00:03:41.900 --> 00:03:42.930
Who knows?

00:03:42.930 --> 00:03:44.180
Next problem.

00:03:48.950 --> 00:03:50.850
All right, let me copy and
paste this one too.

00:03:50.850 --> 00:03:58.430
I'm starting to like this notion
of copying and pasting.

00:03:58.430 --> 00:03:59.790
I don't have to rewrite
the problem.

00:03:59.790 --> 00:04:02.660
And I can mark it up and do all
sorts of things with it.

00:04:02.660 --> 00:04:04.690
All right, a certain radioactive
element decays

00:04:04.690 --> 00:04:09.000
over time, according to the
equation y is equal to A times

00:04:09.000 --> 00:04:11.870
1/2 to the t divided
by 300th power.

00:04:11.870 --> 00:04:17.600
Where A equals the number of
grams present initially, t

00:04:17.600 --> 00:04:19.829
equals time in years.

00:04:19.829 --> 00:04:22.980
If 1,000 grams were present
initially, how many grams will

00:04:22.980 --> 00:04:25.740
remain after 900 years?

00:04:25.740 --> 00:04:30.220
So we get y is equal to--
1,000 grams were present

00:04:30.220 --> 00:04:31.920
initially, so that's A.

00:04:31.920 --> 00:04:39.640
So 1,000 times 1/2 to the-- and
they say, what is time?

00:04:39.640 --> 00:04:42.340
How many grams will remain
after 900 years?

00:04:42.340 --> 00:04:44.120
t is time in years.

00:04:44.120 --> 00:04:50.420
So 1/2 to the 900
divided by 300.

00:04:50.420 --> 00:04:52.120
I just substituted
A and t into the

00:04:52.120 --> 00:04:54.300
equations they gave us.

00:04:54.300 --> 00:04:57.340
1,000 is A, and 900 is t.

00:04:57.340 --> 00:04:59.780
So let's just simplify
this thing.

00:04:59.780 --> 00:05:05.920
So we get this is equal to
1,000 times 1/2 to the--

00:05:05.920 --> 00:05:07.490
what's 900 divided by 300?

00:05:07.490 --> 00:05:09.740
It's just 3.

00:05:09.740 --> 00:05:11.740
So that equals 1,000.

00:05:11.740 --> 00:05:13.090
What's 1/2 to the third power?

00:05:13.090 --> 00:05:15.261
It's 1/8.

00:05:15.261 --> 00:05:18.000
It's 1/2 times 1/2 times 1/2.

00:05:18.000 --> 00:05:23.590
So we get 1,000 divided
by 8, which is 125.

00:05:23.590 --> 00:05:27.990
125 grams are left
in 900 years.

00:05:27.990 --> 00:05:31.360
That is choice C.

00:05:31.360 --> 00:05:34.360
Next problem.

00:05:34.360 --> 00:05:37.030
All right, let me see
if I have to copy

00:05:37.030 --> 00:05:39.590
and paste this one.

00:05:39.590 --> 00:05:41.520
Yeah, sure, they have a
chart, even better.

00:05:45.360 --> 00:05:48.700
I've copied it, and
there we go.

00:05:48.700 --> 00:05:50.410
Look at that.

00:05:50.410 --> 00:05:52.620
All right.

00:05:52.620 --> 00:05:55.920
Bacteria in a culture are
growing exponentially with

00:05:55.920 --> 00:05:58.420
time as shown in the
table below.

00:05:58.420 --> 00:06:01.090
Which of the following equations
expresses the number

00:06:01.090 --> 00:06:06.090
of bacteria y present
at any time t.

00:06:06.090 --> 00:06:12.510
OK, so growing exponentially
tells us that if I say, the

00:06:12.510 --> 00:06:15.910
amount of bacteria after any
days, so let's say y would be

00:06:15.910 --> 00:06:19.050
the amount of bacteria,
exponential form tends to be

00:06:19.050 --> 00:06:21.530
you'd have your initial amount--
like A in that

00:06:21.530 --> 00:06:25.770
previous example-- times some
number that is kind of the

00:06:25.770 --> 00:06:27.960
exponent factor, I guess
you could call it.

00:06:27.960 --> 00:06:29.240
The last one it was 1/2.

00:06:29.240 --> 00:06:32.580
When you do continuous compound
interest, it'll be e.

00:06:32.580 --> 00:06:36.570
It'll be how fast something is
either growing or decaying.

00:06:36.570 --> 00:06:38.720
I don't know, that'll be x.

00:06:38.720 --> 00:06:40.310
I'm just making up a letter.

00:06:40.310 --> 00:06:43.760
It's usually x to some power t
that's usually a function of

00:06:43.760 --> 00:06:46.740
time, some constant
times time.

00:06:46.740 --> 00:06:48.230
This is the general form
of any kind of

00:06:48.230 --> 00:06:50.190
exponential growth or decay.

00:06:50.190 --> 00:06:51.800
You might see different letters
than that, but that's

00:06:51.800 --> 00:06:53.730
the general idea.

00:06:53.730 --> 00:06:58.310
So anyway, if we look at all of
these choices, well, if you

00:06:58.310 --> 00:06:59.800
just look at the thing,
you're seeing it's

00:06:59.800 --> 00:07:02.840
doubling every day.

00:07:02.840 --> 00:07:04.520
The amount of bacteria is
doubling every day.

00:07:04.520 --> 00:07:08.680
So that tells us that's going
to be some power of 2.

00:07:08.680 --> 00:07:11.460
So y is going to be-- the amount
of bacteria after any

00:07:11.460 --> 00:07:14.980
day is going to be equal
to some initial amount.

00:07:14.980 --> 00:07:16.760
Actually, we could just
think about it, right?

00:07:16.760 --> 00:07:22.680
We have some initial amount
times 2 to some constant times

00:07:22.680 --> 00:07:23.450
the day's power.

00:07:23.450 --> 00:07:25.640
But I think if we just
take 2 to the day's

00:07:25.640 --> 00:07:28.920
power-- let's see.

00:07:28.920 --> 00:07:32.520
At 0 days you have
a 100 bacteria.

00:07:32.520 --> 00:07:36.950
So y is equal to 100-- I think
this gets you there; and

00:07:36.950 --> 00:07:38.970
instead of writing days,
let's write time--

00:07:38.970 --> 00:07:41.110
times 2 to the time.

00:07:41.110 --> 00:07:42.370
Sometimes there's a constant
here, but this is

00:07:42.370 --> 00:07:43.560
straightforward enough.

00:07:43.560 --> 00:07:49.650
When t is equal to 0, this 2
to the t becomes 1, and you

00:07:49.650 --> 00:07:51.010
have 100 here.

00:07:51.010 --> 00:07:55.750
When t is 1, then
this becomes 2.

00:07:55.750 --> 00:07:57.920
2 times 100 is 200.

00:07:57.920 --> 00:08:01.390
You could say this is
t and this is y.

00:08:01.390 --> 00:08:03.080
When t is 0, y is 100.

00:08:03.080 --> 00:08:05.120
When t is 1, y is 200.

00:08:05.120 --> 00:08:08.210
When t is 2, y is 400.

00:08:08.210 --> 00:08:13.210
So it is choice B.

00:08:13.210 --> 00:08:15.550
Probably the easiest way is
just to look at these.

00:08:15.550 --> 00:08:18.560
And say, wow, there's only
one-- well, two of these

00:08:18.560 --> 00:08:20.460
choices, well, three of
them have kind of

00:08:20.460 --> 00:08:21.335
an exponential form.

00:08:21.335 --> 00:08:23.310
We're just like, oh, well, this
doesn't take into account

00:08:23.310 --> 00:08:24.710
the initial condition.

00:08:24.710 --> 00:08:27.530
At day 0 I have 100 bacteria.

00:08:27.530 --> 00:08:29.790
This one, I don't know, it
says at day 0, I have 200

00:08:29.790 --> 00:08:30.720
bacteria, which is just wrong.

00:08:30.720 --> 00:08:31.660
I have 100.

00:08:31.660 --> 00:08:33.630
You take t equals 0, and
you're left with what

00:08:33.630 --> 00:08:35.179
you have at day 0.

00:08:35.179 --> 00:08:36.520
So depending on how you
look at it, you

00:08:36.520 --> 00:08:38.820
get to B either way.

00:08:38.820 --> 00:08:40.919
Next question.

00:08:40.919 --> 00:08:42.539
I'm running out of space.

00:08:42.539 --> 00:08:44.130
Oh, there you go.

00:08:44.130 --> 00:08:48.830
All right, next question.

00:08:48.830 --> 00:08:50.700
They write, I'll write
the whole thing here.

00:08:57.470 --> 00:09:01.160
If the equation y to the 2x
is graphed, which of the

00:09:01.160 --> 00:09:03.920
following values of x would
produce a point

00:09:03.920 --> 00:09:08.730
closest to the x-axis?

00:09:08.730 --> 00:09:13.880
So if you think about it, let's
just think about how

00:09:13.880 --> 00:09:16.590
that graph would even look.

00:09:16.590 --> 00:09:17.610
Let me draw it here actually.

00:09:17.610 --> 00:09:19.390
I'll draw it in a
darker color.

00:09:19.390 --> 00:09:21.230
I can do it right on
top of the white.

00:09:23.730 --> 00:09:25.810
Let's just get an intuition.

00:09:25.810 --> 00:09:28.380
What's 2 to the 0?

00:09:28.380 --> 00:09:32.140
2 to the 0 is 1, right?

00:09:32.140 --> 00:09:35.060
Its y-intercept would
be 1 at x equals 0.

00:09:35.060 --> 00:09:37.995
2 squared, then 2
to the 2 power.

00:09:37.995 --> 00:09:41.565
And you were looking at
4, we'll go there.

00:09:45.510 --> 00:09:53.500
Sorry, let's say if this is 1,
this would be 2, 2 to the

00:09:53.500 --> 00:09:54.990
first power.

00:09:54.990 --> 00:09:57.920
Then 2 to the second
power would be 4.

00:09:57.920 --> 00:10:01.970
Then 2 to the third
power would be 8.

00:10:01.970 --> 00:10:03.440
Actually, 4 would be like.

00:10:03.440 --> 00:10:04.700
So it's going to look
something like that.

00:10:04.700 --> 00:10:05.250
We don't have to worry.

00:10:05.250 --> 00:10:07.310
It's going to go
up really fast.

00:10:07.310 --> 00:10:10.100
Now what happens as you go
into the negative powers?

00:10:10.100 --> 00:10:15.180
So 2 to the negative
1 power is 1/2.

00:10:15.180 --> 00:10:17.240
You're going to get
right there.

00:10:17.240 --> 00:10:20.740
2 to the negative 2 is 1/4.

00:10:20.740 --> 00:10:22.090
So you're just going to get
smaller and it's just going to

00:10:22.090 --> 00:10:22.930
asymptote to 0.

00:10:22.930 --> 00:10:27.320
So that's what the graph
is going to look like.

00:10:27.320 --> 00:10:29.230
I didn't even have to worry
about negatives, because there

00:10:29.230 --> 00:10:31.480
are no negative numbers here.

00:10:31.480 --> 00:10:34.390
So really, the smallest of these
are going to produce a

00:10:34.390 --> 00:10:37.750
point closest to the x-axis.

00:10:37.750 --> 00:10:39.090
The smallest of these
values, you want to

00:10:39.090 --> 00:10:41.100
get as small as possible.

00:10:41.100 --> 00:10:44.490
Let's see, 3/4 is
bigger than 1/4.

00:10:44.490 --> 00:10:46.540
So that's not right.

00:10:46.540 --> 00:10:48.720
5/3 is bigger than 1/4.

00:10:48.720 --> 00:10:50.380
8/3 is bigger than 1/4.

00:10:50.380 --> 00:10:53.740
So the answer is 1/4, A.

00:10:53.740 --> 00:10:57.780
2 to the 1/4 power is going to
be a little bit more than 1.

00:10:57.780 --> 00:10:59.850
It's going to be something
like that.

00:10:59.850 --> 00:11:03.310
Anyway, next question.

00:11:03.310 --> 00:11:04.370
Oh, I'm all out of time.

00:11:04.370 --> 00:11:06.210
See you in the next video.