WEBVTT
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to pipe. The intercepts are why, um let's
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sit either extra wide is your NBC that we haven't
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intercepted. 00 We have no symmetry. And Assam
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toast. Let's test. Um, we do have
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one as X approaches infinity. So it's X approaches
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infinity. What is our ask himto it. Why
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gonna be now? This is a horizontal lesson toe
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. So testing as limit of X approaches infinity of
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her function Eat e to the negative X sign eggs
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. All right now equals zero. So we have
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ah, horizontal at Santo at y equals zero.
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Okay. And now we conduce the first derivative,
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um, test to look for increasing and decreasing intervals
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. So why prime equals negative e to the negative
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x sign IX plus e to the negative x co
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sign next. That's just power rule and set that
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equal to zero. We have critical point at X
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equals pi over four. So if we do the
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first derivative test, um, testing values in our
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first derivative around pi over four, we can see
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that increasing smaller than fire before and decreasing If it's
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greater than so. If it's going from increasing to
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decreasing, you know that this is a local,
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um, maximum value. So And if we do
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plug into our original function, if you plug in
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pi over four into our original function to see the
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Y value, we're going to get approximately 0.32.
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So this is our local max. And now we
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can solve our second derivative to look at Kong cavity
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. So why Double prime is equal to negative to
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e to the negative x co sign X said that
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equal to zero take our, um of double prime
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. This is our 2nd 0 good. Insulted us
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. Yet X equals pi over too. So if
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we look at by over two and our second derivative
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test, we see that it's calm, keep down
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for values below pirate too. And if is greater
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than fire for to conk a pup concrete down and
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then you can keep up. So that means that
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this is an inflection point at pi over tube.
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Okay. And now we can growth. The first
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thing I would do is grab the ass in tow
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. And since we're only looking at, um,
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our domain from 0 to 2 pi, I'm just
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gonna drop that positive first court quicker. Okay,
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So our ask himto is a Y equals zero.
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So it's the X axis. This is a horizontal
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. I think so. And we haven't intercepted 00
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So that's right here. And, uh, let's
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say this is pie and that's to play. So
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we have a local maximum value at and let's say
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this is one. So we have a local maximum
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at Pi over four in approximately zero plane 32.
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So that's gonna be about this is pi over to
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This is about prior before 0.30 to say it's about
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there isn't just a rough sketch, and we have
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an inflection point high over, too. That's about
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here. So somewhere here, um and now we
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can see that it's gonna be increasing and decreasing after
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it hits our local maximum value. And it's gonna
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be Kong cave down, and it's gonna now decrease
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and continue being conquered down until it's our inflection point
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. Now it's gonna be Khan gave up an approach
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or ask himto but never touch. Does the values
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get closer and closer to, um, negatives?
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I mean, um, zero