WEBVTT
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Unknown: Okay, this is the sample final for DS 21.
Okay,
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let's get started with question number one. And this one
is the
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natural log of 10x plus seven minus the natural log
of x is
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equal to 12. So what we'll do is first we
write it as the natural
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log of 10x plus seven, and then we're going to
divide that by
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x, and that's equal to 12. Okay, so that's equal
to 12. So what
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we'll do
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is we'll take the natural log of both sides to
get rid of the
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natural log on the left hand
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side. Okay, so when we take the natural log of
both sides,
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sorry, not the natural log, but we take the exponential,
we're
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going back. So we have the natural log of 10x
plus seven
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divided by x is equal to 12. And to get
rid of the natural log,
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we're going to take the exponential. So when we take
the
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exponential on both sides, it gets rid of the natural
log on
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the left hand side, and we have 10x plus seven
divided by x is
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equal to a race to the 12 power. Okay. We
have a race to the 12
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power. So now what we're going to do is multiply
both sides by
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x. And that gives us 10x plus seven is equal
to e raised to
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the 12 power times x, and I'm going to put
x in front of it x
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times e raised to the 12 power. All right, so
now I will
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subtract 10x from both sides. And that gives me seven
is equal
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to x e raised to the 12 power, minus 10x.
Now I'm going to
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gather like terms. I'll gather like terms, and on the
right
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hand side, so I will take the x out of
x times e raised to the
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12 power minus 10. And then I'll divide both sides
by e raised to
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the 12 power minus 10. So seven divided by and
that's e raised
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to the 12 power, minus 10. And that's going to
be equal to x.
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Now we can, what we can do here is we
can actually enter the
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problem as it exists. Let's go ahead and do that
seven. divided
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by E. F, we have yeah, here we go. e
to the 12 minus 10. Okay,
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we can also come over here to the calculator. And
we can put
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that number and so we can say seven divided by
and in
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parentheses, we can say we figure out what's in the
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denominator, the denominator will be e raised to the 12th
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power minus 10. So what we could enter here is
also seven divided
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by 160 to 162,744.7914. Okay, So we can do that.
All right. Okay,
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so let's go ahead and go to the next problem.
Are there any
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questions about this one problem?
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Okay, let's go to the next one and it says
Find a derivative of
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the function, we remember what a derivative is of a
polynomial
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function, and that's what we have here is we take
the
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exponential value, I'm going to bring it down times the
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coefficient, x two, and then we're going to raise that
to the
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three minus one. Okay? All right, and then we're gonna
look
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at that second term. That second term, we bring the
exponent
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down, which is a two, multiplied K, we're gonna put
plus go ahead
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and put a minus because they have a minus there,
bring the
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exponent down, which is a two, multiplied by a two,
and then
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that's x raised to and then what we're going to
have is two minus
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one, okay? All right, so then we'll come here, and
that'll be
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bring down the exponent of one, times 3x raised to
the one minus
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one. Okay. And if you take a derivative of a
constant, it's
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equal to zero. And the reason why it's the same
as saying 4x
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raised to the zero power, so when we bring the
zero down, and
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multiply that term, it goes away. So let's go ahead
and
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clean this up. So the derivative is equal to 3x
raised to the
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second power, minus 4x raised to the first power. Plus
3x raised
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to the zero power, anything raised to the zero power
is one.
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So that's our answer. So let's go ahead and put
that in. We'll
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have 3x. And here we're gonna raise it to the
second power.
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Minus 2x. Sorry, not when we put my x down
there first x and then
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raise that. No, we're not going to raise that till
power. Okay,
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there we go. Plus three, and that's the derivative. So
we
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have 3x squared minus 2x plus three. And I missed
messed that
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up. You should be four by here. And we should
have four here in
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our answer. Okay, does that make sense? 3x squared minus
4x plus
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three.
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Any questions? Any questions? I can hear some more. We
can ask
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everyone. Mute themselves. Okay. All right. If you have a
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question, I like to answer that question.
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Okay. All right. So let's look at the third one.
It says just
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deposited $2,723 into an account that pays 9.8% simple interest.
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How much money will we have at the end of
nine months? We want
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to know the answer. Sir, and then we want to
know the total
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in the account. So let's go ahead and start. Let's
look at
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the interest earned. We look at the $2,723, we multiply
it by
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interest 0.098. And then we multiply that by nine divided
by
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12. Okay, and that's what we have. So I'm going
to come over
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here to my calculator, we're going to calculate that $2,723
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times point 098 times nine divided by 12. And that
gives us
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I'm sorry, let's do that again. $2,723 times point 098
times
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nine divided by 12. And that gives us 200. Point
14 notes.
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Okay. 200 point $14. And then if we want to
find out how much is
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in the account, all we have to do is out
the original amount
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plus our interests. And that'll give us our answer. And
we'll
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come over to the calculator, get that $2,723 plus 200.1
414 cents
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gives us $2,923.14 And the amount of interest earned is
214
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cents. Any questions on those two? Okay, let's go on
to the
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next one. The next question asks, a jar contains eight
red
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marbles, numbered one through eight and 10 blue marbles
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numbered one through 10. A model is drawn at random
from the jar,
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find the probability of the given event, please show your
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answer as a reduced fraction. Okay, so if I look
at this, I
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have my red marbles.
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Okay, I have different color models. And then I have
odd
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even. Okay, so I have red marbles. And how many
of them do
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I have? I have eight of them. And of those
eight, four of our
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odds, and for our event, and then I have blue
marbles. And I
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have 10 of those, and fiber optic, and fiber even.
So I have
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a total of 18, Marvel's Niner and nine aren't even.
So let's
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look at this. So they want to know the probability
that we
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have a red marble. Well, I'm going to look at
how many red
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marbles I have out of a total of 80. And
I really do see that two
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goes into eight, four times two goes into 89 times,
okay, four
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nights. So here I'm gonna enter four, nines. Okay. Now
let's
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look at this second one of probability of getting an
odd
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number. While there are a total of nine odd ones
out of 18
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total. Well, that's the same as one half and so
I'll enter one
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half Okay, now if I want a red or Ah
okay, so I'm gonna do
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probability of red or odd equal the probability of red
or minus
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the probability of red and pink Okay Okay, so if
I look at that,
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that's gonna be equal to red. So eight out of
80 odd, Nine nyati
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Eight t, and then read n OD, that's for RA
t. So I have nine
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plus eight plus nine is 77. T minus four is
13 out of 80.
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Okay. All right, so now we're talking about blow and
even. So
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we're talking about Blue, or event. So that's equal to
the
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probability that it's blue
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plus the probability that it's even minus the probability that
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it's blue. And it's even. Okay, that's equal to how
many blue?
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Do we have? We have XR 18.
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How many even do we have? We have nine out
of 80? And then
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how many blue? And even do we have? We have
a total of five
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out of 80? Okay, so that gives us 19 minus
five, that's 14.
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Okay, two goes into seven. Two goes into 14 Seven
times, and
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two goes into at nine times our answer here will
be seven
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nights. Okay. Any questions? Okay, so let's go on. Now,
let's
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look at question number five. Question number five says the
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business needs $470,000 534 F $417,534.18 years, how much
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should be deposit each month in a sinking fund that
earn 6.3%
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compounded interest. Okay, so what we can do is set
it up this
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way. We have 4000 17,400 $17,534 is equal to a
payment, and then
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we have one plus point 063 divided by 12. Okay,
and then
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you're going to raise that to the 12 times a
T and then minus
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one divided by point 063 divided by 12. Okay, so
if we look at
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this, come over to my calculator. All right, that's
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gonna be one plus point 063 divided by 12. And
then you're
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gonna raise that to the 12 times 18 minus one
divided by point
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063 divided by 12. Okay, and that's my annuity right?
Okay. I
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have that on both sides. So my answer is $470,534
is equal to
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the amount of payment times 390 9.7824592 by both sides
by
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that it's $417,534 divided by 390 9.78 to four 592
And that's
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equal to your payments. Okay, so let's go ahead and
do that over
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here. Let's go to calculator, I'm going to use x
to the
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negative one. So it takes this and Dave actually divides
it as
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it raised it to the negative one half, not one
half, but negative
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one times 470,534. Okay? And this is the amount of
the
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payment. So the amount of the payment is $1,044.40. Okay,
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let's make sure I did that correct. Okay, so let's
go ahead
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and confirm that by using the apps button on the
calculator,
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so let's do that. Okay, so we have 12 payments
per year, times
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18 years. Our interest rate is 6.3. We have no
present value,
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we're looking for the payment. But we want
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$417,534. And those are 12 payments per year. And when
it
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come up here to payment and do alpha, enter, and
that gives us
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our $1,044.40.
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Okay, now, any questions on that one? All right. Let's
go to the
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next one. Question number six. Suppose the following point is
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on an exponential function that describes the exponential growth
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of decline of a customer coming to a restaurant, and
the initial
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point is 01. Find the exponential function if the
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second point is 364. Okay, so let's start off. And
we say y is
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equal to B times A raised to the x power.
All right, so that
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first point says our x values zero, our y values
one. So we
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have B A raised to the zero power, anything raised
to the
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zero power is one, so we have one is equal
to B times one.
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Okay, so one is equal to b. So b is
equal to one. And, and all in
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both cases that we're going to look at. Now let's
look at what
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happens when we have the point 364 y 64. And
that's B, A, and B
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is one. So I'm just gonna say a race to
the third power. To find
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out what a is I'm going to take the cube
root of both sides, so
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that 64 raised to the one divided by three. Okay,
and
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that's equal to a, let me just come over here
to the calculator
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and show you how that works. Okay. So I will
look at 64. And
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I'll raise it to the one divided by three. So
I'm going to take
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the cube root and that equals four. So my function
is four
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raised to the x power, because a raised to the
x power and A is
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four. So four raised to the x powers that first
one. All
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right, so if I look at the second one, the
second one, I
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have 116 1384 for my y is equal to a
raised to the negative
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seven power. Okay. All right, so that's the same as
saying 16,300
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A four is equal to and since I have that
negative, I can do one
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over a n that's raised to the seventh power. Now,
I'm going to
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take the seventh root of both sides. So 16,384 raised
to the
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one seven. And that's going to be equal to one
over A. Okay, so
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let's go to our calculator, figure out what that is.
And
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I'll come here and I'll have 16,384. And then I'm
going to
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raise that to the one divided by seven, takes the
seven through,
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and that's four. So I'll come over here, and we'll
just finish
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this off, we have four is equal to one over
A, multiply both
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sides by a, so I have or a is equal
to one, divide both sides
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by four, a is equal to one divided by four.
So my function
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this case, is one over four. And I'm going to
raise that to the x
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power. So let's come over here to our problem. And
I'm going to
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put it in parenthesis and then I'll put it 1/4.
And I'm going
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to raise that to the x power. Okay, so that's
how we would
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find that answer. Any questions?
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Okay, let's look at the next one. We have question
number
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seven. Now remember, the log rules where you can express
it
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as a sum or difference. If it's multiplication, we talk
about
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adding, if it's an exponential, we bring the exponent down.
So
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we would write this as log of 90, plus log
of x squared
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plus log y to the fifth. All right, so remember,
anytime we
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have that exponent, we can bring the exponent down. So
that's
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actually law 90
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plus two, log x plus five, log y. And we're
going to put that
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in. So we're gonna say log 90 plus two, log
x plus five log y.
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All right. So similarly, for natural law, we do the
same
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thing. So on this one, we will have the natural
log of four
210
00:28:23.680 --> 00:28:31.310
plus the natural log of u raised to the 10th
power, plus the
211
00:28:31.440 --> 00:28:37.108
natural log of W raised to the fifth power. And
that's equal to
212
00:28:37.200 --> 00:28:39.240
the natural log of four
213
00:28:41.400 --> 00:28:52.737
plus 10, natural log of u plus five natural log
of w. And let's
214
00:28:52.920 --> 00:29:03.433
go ahead and put that in natural log four plus
10 natural log of
215
00:29:03.600 --> 00:29:18.240
u plus five natural log of W. All right, so
let's look at this
216
00:29:18.480 --> 00:29:23.000
other one. Now when we talk about division, that implies
217
00:29:23.040 --> 00:29:30.440
subtraction. So we would write this particular log expression
218
00:29:30.440 --> 00:29:44.654
as log of 30 sets plus log of x raised
to the 10 power minus log
219
00:29:45.360 --> 00:29:52.553
of y raise to the 19th power. Okay, so then
we can rewrite
220
00:29:52.680 --> 00:30:06.965
that as log 36 plus 10 log of x x
plus minus sorry, 90, log of
221
00:30:07.200 --> 00:30:14.754
y. So let's go ahead and put that in. All
right, so I'm going
222
00:30:14.880 --> 00:30:17.080
to have log of 36
223
00:30:20.640 --> 00:30:39.643
plus 10 log of x minus nine t log of
y. And this last one, we
224
00:30:39.960 --> 00:30:48.812
would have natural log of two plus natural log of
u raised to
225
00:30:48.960 --> 00:30:50.040
the fifth power
226
00:30:51.600 --> 00:30:58.129
minus natural log of W raised to the fifth power.
And we can
227
00:30:58.240 --> 00:30:59.280
rewrite that
228
00:31:01.320 --> 00:31:12.372
as natural log of two plus five natural log of
u minus five,
229
00:31:13.080 --> 00:31:20.004
natural log of w. So let's go ahead and do
that here. Natural
230
00:31:20.120 --> 00:31:35.393
log of two plus five natural log of u minus
five natural log of
231
00:31:35.640 --> 00:31:43.465
W. Okay, so that's, that's how you would do question
number
232
00:31:43.600 --> 00:31:44.160
seven.
233
00:31:44.400 --> 00:31:51.598
Any questions that? Okay, let's go to the next one.
Question
234
00:31:51.720 --> 00:31:56.240
Eight, determine the equation of the quadratic formula with the
235
00:31:56.240 --> 00:32:01.118
vertex of one and five, and it passes through three.
So let's
236
00:32:01.200 --> 00:32:05.444
first write the equation with the vertex. So we have
y is
237
00:32:05.520 --> 00:32:13.985
equal to a x minus one. And then we're going
to square that. And
238
00:32:14.120 --> 00:32:20.579
then we're going to add five. So now let's put
in the point 03, y
239
00:32:20.680 --> 00:32:26.698
is equal to three. And that's equal to A x
is equal to zero,
240
00:32:27.040 --> 00:32:34.950
so zero minus one. And that's where plus five. So
that's three
241
00:32:35.080 --> 00:32:43.344
is equal to a. And this is one plus five,
I subtract five from
242
00:32:43.480 --> 00:32:51.899
both sides, and that's negative two is equal to a.
All right,
243
00:32:52.080 --> 00:32:56.798
and answer is y is equal to negative two times
the quantity
244
00:32:56.880 --> 00:33:04.075
X minus one squared plus five. All right, so we're
going to
245
00:33:04.200 --> 00:33:10.772
clear that. And this one, number nine, it says write
a simplex
246
00:33:10.880 --> 00:33:14.600
matrix with the following standard maximization problem,
247
00:33:15.200 --> 00:33:19.597
maximize F is equal to 3x minus two Y's subject
to the
248
00:33:19.680 --> 00:33:26.960
constraints 9x plus five y is less than or equal
to 36 1x plus
249
00:33:27.080 --> 00:33:31.712
seven wise less than or equal to 27. So if
I put in my
250
00:33:31.800 --> 00:33:37.680
constraints, and I put in my slack variables, my first
251
00:33:37.680 --> 00:33:43.975
equation, the coefficient would be nine. For X, five for
y, I
252
00:33:44.080 --> 00:33:48.171
have a slack variable one, I don't have a slack
variable two,
253
00:33:48.640 --> 00:33:52.810
and I don't have the F. So I'll have zero
here. But my answer
254
00:33:52.880 --> 00:33:57.520
will be 36. Let's look at the second constraint, the
255
00:33:57.520 --> 00:34:03.766
coefficient is one 4x. seven for Y. I don't have
a slack
256
00:34:03.880 --> 00:34:10.210
variable, one, but I have a slack variable s two.
Okay, and
257
00:34:10.320 --> 00:34:16.582
I don't have an F, but I do have an
answer of 27. Finally, I take
258
00:34:16.920 --> 00:34:21.288
what I'm going to maximize, and I move everything to
the left.
259
00:34:21.360 --> 00:34:25.099
So the three which is positive on the right becomes
negative on
260
00:34:25.160 --> 00:34:31.524
the left, so a negative three for the x, then
plus two y
261
00:34:31.640 --> 00:34:37.702
becomes a negative two y sorry, becomes a positive two
y on the
262
00:34:37.800 --> 00:34:42.522
left, so I have a two for the Y. I
don't have a slack variable
263
00:34:42.600 --> 00:34:46.254
one, nor do I have a slack variable two, but
I have an F.
264
00:34:47.240 --> 00:34:50.820
And that's equal to zero because if I move everything
over, I
265
00:34:50.880 --> 00:34:55.602
just have zero on the right. So that's my initial
simplex math
266
00:34:56.120 --> 00:35:06.416
matrix. Any questions on that? Okay, let's go to the
next
267
00:35:06.600 --> 00:35:11.075
storm. Now this is a Venn diagram. And we're trying
to
268
00:35:11.160 --> 00:35:14.818
find a probability. So we're looking at the numbers. But
the
269
00:35:14.880 --> 00:35:20.230
first thing we're what what, what makes going from a
set to a
270
00:35:20.320 --> 00:35:24.840
probability is we take the number of occurrences of a
of a
271
00:35:24.920 --> 00:35:30.746
set, whatever the set we have, and then we divided
by the grand
272
00:35:30.840 --> 00:35:34.620
total of all the possibility. So the first thing we
want to do is
273
00:35:34.680 --> 00:35:38.806
figure out what are all the possibilities. So add all
the
274
00:35:38.880 --> 00:35:50.101
numbers up. So eight plus nine, plus one plus five,
plus 50 plus
275
00:35:50.280 --> 00:35:56.648
seven, plus 12. Plus four, and you see the little
four out
276
00:35:56.760 --> 00:36:00.774
here. Yeah, we have to include that too. So we
have a total of
277
00:36:00.840 --> 00:36:05.829
61. Now, if I look at A intersect B, intersect
C, that's
278
00:36:05.920 --> 00:36:10.483
this little area right here, where I have a one.
So I'm going
279
00:36:10.560 --> 00:36:26.494
to have one divided by 61. And that's equal to
0.016. And I'm
280
00:36:26.760 --> 00:36:44.478
gonna have 0.016. All right, and that's my answer. All
right. So
281
00:36:44.800 --> 00:36:48.892
let's look at question number 11. If there are no
questions on
282
00:36:48.960 --> 00:36:56.155
10, are there any questions on 10? Okay, let's look
at 1111
283
00:36:56.280 --> 00:37:00.964
says we want to solve a system of equations, x
minus y is equal
284
00:37:01.040 --> 00:37:07.217
to two 6x plus four, y is equal to 92.
So we're gonna look at
285
00:37:07.320 --> 00:37:13.057
the matrix, the augmented matrix, I go over to edit,
I do
286
00:37:13.200 --> 00:37:17.761
second x to the negative one, move over to edit,
hit enter,
287
00:37:18.240 --> 00:37:23.740
and I have a two by three matrix, I have
the coefficient
288
00:37:23.840 --> 00:37:29.418
of one for the first one and negative one for
y. And our
289
00:37:29.520 --> 00:37:34.480
answer to our second equation, we have a coefficient of
six for
290
00:37:34.560 --> 00:37:43.215
EPs, four for y, and 92, for our answer, so
4x minus y equals
291
00:37:43.360 --> 00:37:48.431
two, I have a coefficient of one, negative one, and
then a
292
00:37:48.520 --> 00:37:53.322
tick. Okay, so what I'm going to do is come
out of here, second
293
00:37:53.720 --> 00:37:58.597
mode. And then I'll go back again, second, x to
the negative
294
00:37:58.680 --> 00:38:04.463
one, come over to now. And then I'm going to
move down to our
295
00:38:04.560 --> 00:38:12.273
RAF row reduced echelon form of the matrix. Select the
matrix,
296
00:38:12.560 --> 00:38:17.948
hit Enter. Now you see this identity matrix here. This
gives
297
00:38:18.040 --> 00:38:22.326
us the information. So in the one column in the
X column, I
298
00:38:22.400 --> 00:38:27.474
have a one I move over to the answer, and
it tells me that x
299
00:38:27.560 --> 00:38:32.754
is equal to 10. All right, so I go to
the y column, I have the
300
00:38:32.840 --> 00:38:38.313
one move it over, and y is equal to eight.
And that's how I find
301
00:38:38.400 --> 00:38:46.782
my answer. Okay. Now, let's look at 1212. We're looking
at the
302
00:38:46.920 --> 00:38:53.767
limit. If you recall, the limit exists if I approach
it on the
303
00:38:54.360 --> 00:38:58.965
left, and I approach it on the right, and I
get the same value.
304
00:39:00.040 --> 00:39:04.958
So as I'm approaching it on the left, I'm coming
close to and
305
00:39:05.040 --> 00:39:10.591
I'm coming to the value 10. I'm approaching the value
10. As I'm
306
00:39:10.680 --> 00:39:15.561
approaching it from the right, I'm approaching the value 10.
So
307
00:39:15.640 --> 00:39:23.897
the limit is 10. Now, notice that there's an open
circle at
308
00:39:24.040 --> 00:39:28.723
that function. So when I try to figure out what
the answer is,
309
00:39:28.960 --> 00:39:35.056
the answer is there is not. Okay, there's no f
of z f of 20.
310
00:39:35.760 --> 00:39:40.244
F of 20 is just open circle, meaning there doesn't
exist. So
311
00:39:40.320 --> 00:39:44.728
I'm gonna put D and E. If it had been
a value, I would have put
312
00:39:44.800 --> 00:39:58.101
the value. Any questions? Okay. Let's look at 13 Okay,
13 says
313
00:39:58.360 --> 00:40:03.280
our revenue function is 9x squared, and the cost function
314
00:40:03.280 --> 00:40:09.290
is negative 2x square plus $6,875. What is the number
of
315
00:40:09.400 --> 00:40:14.040
items required to break even breakeven means that the revenue
316
00:40:14.040 --> 00:40:25.240
is equal to the cost. And you have 9x squared
is equal to
317
00:40:25.440 --> 00:40:39.922
negative 2x squared plus 6875. All right, so what I'm
going to
318
00:40:40.160 --> 00:40:46.325
do is, I'm going to add 2x squared to both
sides. So 9x
319
00:40:46.440 --> 00:40:55.769
squared plus 2x square gives me 11x square. And that's
equal to
320
00:40:55.920 --> 00:41:02.215
6875. Now we're gonna divide both sides by 11. And
that gives
321
00:41:02.320 --> 00:41:13.944
me x squared is equal to 6875 divided by 11.
And that's
322
00:41:20.760 --> 00:41:35.754
6875 divided by 11 is 625 625. So I'm going
to take the square
323
00:41:36.000 --> 00:41:43.198
root of both sides. So x is equal to 25.
Square root of 625.
324
00:41:43.320 --> 00:41:50.289
So let me just show you second x squared, and
then that gives me
325
00:41:50.400 --> 00:42:00.157
the square root of 625. Is 25. So that's my
answer. Okay. And
326
00:42:00.320 --> 00:42:05.950
then let's look at any questions on 13. All right,
let's look at
327
00:42:06.040 --> 00:42:15.561
14. Now, this one right here gives us our objective
function,
328
00:42:15.920 --> 00:42:21.400
and then our constraints. Now remember, the maximum or the
329
00:42:21.400 --> 00:42:26.045
minimum will occur at the corner points. So the first
thing I'm
330
00:42:26.120 --> 00:42:30.722
going to do is hit the STAT button. And then
I'll hit enter.
331
00:42:31.560 --> 00:42:35.767
And mine I have no values, if you have values
just go up to
332
00:42:35.840 --> 00:42:40.918
the top glare, and then enter. Okay, so I'm going
to put in the
333
00:42:41.000 --> 00:42:44.892
values of the corner points of the objective function. So
I
334
00:42:44.960 --> 00:42:52.470
have 08. So the x is zero, and six, five,
the x is six, and
335
00:42:52.600 --> 00:42:59.166
then I have nine, zero. And I also have 00.
I'll go over to
336
00:42:59.360 --> 00:43:04.516
list two, if you have something and wants to go
to the top, get
337
00:43:04.600 --> 00:43:08.760
the clear button. And then enter. I'll put the
338
00:43:08.760 --> 00:43:18.677
corresponding y values 850, and zero. Now I'm going to
come to
339
00:43:18.840 --> 00:43:21.790
list three, if you like I said, if you have
something there,
340
00:43:21.840 --> 00:43:25.699
just hit the clear button, go to the top, hit
the clear button in
341
00:43:25.760 --> 00:43:30.126
the Enter. Now I'm gonna go back up to the
top here. This is
342
00:43:30.200 --> 00:43:36.019
important. I'll hit five, and then second one for list
one,
343
00:43:36.920 --> 00:43:43.536
plus three, and then second, two for list two, and
it'll evaluate
344
00:43:43.800 --> 00:43:48.281
all my that all of the objective function at each
of those
345
00:43:48.360 --> 00:43:53.591
points. Now I want the maximum value. The maximum value
if I
346
00:43:53.680 --> 00:44:00.606
look is 45. It's at two points. Those two points
are six comma
347
00:44:00.720 --> 00:44:10.988
phi, or nine, comma zero. Okay? Those are the two
points that
348
00:44:11.160 --> 00:44:22.845
give me a maximum value. All right. All right. So
then, let's
349
00:44:23.040 --> 00:44:27.407
look at and I can actually I know it says
it's not right, but
350
00:44:27.720 --> 00:44:37.148
yeah, we can do it this way. Okay, so now
if we want the
351
00:44:37.320 --> 00:44:42.357
minimum, the minimum value is at zero, and it's the
point that
352
00:44:42.440 --> 00:44:48.620
we're looking for is zero comma zero. Okay. And that's
going to
353
00:44:48.720 --> 00:44:56.390
be our answer for number 14. Any questions on that?
So we're
354
00:44:56.520 --> 00:45:00.720
evaluating it at the corner points, the object function at
355
00:45:00.720 --> 00:45:05.791
the corner points because this is bound, okay I can
find a
356
00:45:05.880 --> 00:45:15.907
maximum and a minimum. The last one talks about I
have nine
357
00:45:16.080 --> 00:45:31.498
skirts I have seven blouses and I have a number
of shoots so
358
00:45:31.760 --> 00:45:42.452
there are nine skirts seven blouses and nine pair of
shoes
359
00:45:46.560 --> 00:45:50.213
so when I look at the number of ways that
have different
360
00:45:50.320 --> 00:46:02.674
outfits, all I have to do is multiply those together
so, I'll
361
00:46:02.880 --> 00:46:08.520
come over here and to the calculator nine times seven
362
00:46:09.280 --> 00:46:21.619
times nine. So, I have 567 outfits, okay. And that
is how
363
00:46:21.840 --> 00:46:25.892
you do the sample exam and the final is very
similar to that.
364
00:46:26.840 --> 00:46:34.640
Any questions? Okay, so I will stop the video