WEBVTT
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So we're given this function to model concentration over time
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and were given these constraints A B and K or
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positive. Constance and B is greater than K.
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So I just picked these random numbers that do work
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for these constraints. And I plugged him into the
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function and they got this function. So if you
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want to grab that, the first thing I'm gonna
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look at is the first derivative. So see of
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tea, See, prime of tea. So the
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first derivative is gonna be negative e to the power
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of negative T plus to e to the negative to
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t. Okay. And I'm gonna set this equal
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to zero to find my critical points. And I
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see that c is equal to Ellen of two.
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All right, so Ellen of two is approximately 20.693
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So if we test values, this is Ellen,
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too. If we test values, um, into
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our first derivative for the first derivative test for less
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than Ellen to, we're going to see that it's
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increasing and anything greater than is gonna be decreasing.
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So it's a local Max. All right, So
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this could be useful for later and then see double
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prime toe. Look a ah kong cavity. See
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, double prime of tea. We're going to get
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e to the negative. T minus four e to
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the negative to t If we said that equal to
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zero, we're going to see, um, t
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equals to Ellen too, which is approximately, um
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, 1.39. And if we do the second derivative
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test of double prime and we test values around 1.39
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we see that, um, it's gonna be con
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cave down. And then Khan gave up. So
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this is an inflection point. All right, so
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we're going to use this while we graph in just
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a moment, but I'm gonna look at ah,
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one more thing, because our graph actually does have
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an accent. Oh, it has. Ah.
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Horizontal ass in tow. So let's test, um
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, the limit, as are variable t approaches infinity
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over function, which is one times e to the
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negative one t minus e to the negative to t
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. All right. And we're gonna So let's just
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say that we do, um, the limit as
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t approaches infinity, I'm gonna rewrite this in a
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way that looks easier to understand. So this is
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the same thing as one over e to the t
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because it's, ah, negative exponents bring into the
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denominator. So one over E to the T minus
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one over each of the two tea. So this
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is the same thing, all right. And if
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you take a look at this, it's every substitute
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. Wherever we see tease, we're gonna substitute infinity
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. So as t approaches infinity, we see that
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one over. A very large number is close to
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zero. So this is gonna approach zero. So
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as T approaches infinity, our why values, which
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is concentration over time, is gonna approach zero.
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So we have a horizontal ask himto at zero.
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Okay, And now we can graph. And since
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we're looking at time, I'm just gonna show the
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positive part of this cause negative time doesn't really make
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sense to me. So this is concentration overtime for
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this access. And this is just time over here
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, and we know that we do have to say
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this is approximately one. We know we have a
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maximum of Ellen, too. And if you plug
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in Ellen to into the original function, you'll see
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that it's equal to 1/4. So we know that
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about Ellen two and 1/4 is our maximum and the
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sister of sketch. So I'm gonna say this is
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Ellen too. 1/4 at this point. And this
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is our maximum. Okay? And we see that
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, um, according to the second derivative test that
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it's gonna be conquered down. And then Khan gave
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up in our inflection point is 1.39. So that's
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what we're gonna seethe. Sign change at about 1.39
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. So let's say it's right there. Okay?
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So it's gonna be con cave down. Oh,
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and we also haven't Don't forget about the ass in
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tow. The horizontal asking to is right here.
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Okay, so we have, um, con cave
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down. It's gonna first of all, increase until
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it hits are maximum local, maximum, and then
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it's gonna starts a decrease after hits the max,
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but it's still gonna be con cave down until it
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hits our inflection point where it's gonna be Khan gave
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up an approach, but never touch are asking too
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. And this is our graph. All right,
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so this is our graph for concentration over time,
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and the question asked asks us, um, what's
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happening in this graph. So we see as timing
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is increasing. So as time goes on, um
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, concentration increases until we hit the. So as
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time goes on, concentration increases until we hit the
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maximum concentration, and then after that, the concentration
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continues to decrease forever.