Problem 44 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = e^{-x}\sin x $, $ 0 \leqslant x \leqslant 2\pi $

Answer

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Video Transcript

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