Problem 35 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = x\tan x $, $ -\pi /2 < x < \pi /2 $

Answer

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Video Transcript

So we are given this function, and this is the domain that textbook gives us. So the intercepts for this are, um, every pie and, um, we're gonna see an intercept, but that's not inside of our domain. So it's disregard that, um, and we're gonna have 00 that is in our domain and still to write that down symmetry. It is an even graph. So we'll see symmetry there and ask him to, um it's every pie over to end. Well, go ahead and write that in, because on even though it's not included in our domain just so that we can visualize a little better. So pi over to end we every pie over two times, you know, an integer We're going to see an ass in two. All right, so I'm gonna start the graph. So let's say this is negative Pi over, too. And this is pi over, too. Remember that pirate ooze? Ah, essence. What? That comes along with the tinge, um, graft. So by over to a negative pi over to that's our bounce for the domain. All right? And we already said that we do have a nascent Oh, there. So everything within the ASM totes. We're going to have, um, our graph with in here. All right. And now, um, to see where increases and decreases will find the first derivative. So why prime? Why? Prime equals tangents. X plus x times seeking Squared X. All right, so that's our first derivative. And let's find this thing and derivative over at it. So why Double prime is equal to two x sequence time. Sequins squared X tangents, X times Tanja X close to seek inns squared X. All right, so those are derivatives. And now, um, so this is the first derivative so of prime. So I'm gonna test for increasing and decreasing between the bounds I have so between negative pi over, too, and is and also our intercepts that so let's put that on there. Zero. So it's he was going on there and pie over, too. And since we know the intercept, let's just put that so 00 is our intercept and it is also a point on a graph. All right, so now if we test points here into the first derivative anything betweennegative pi over two and zero, it's going to be negative in therefore decreasing anything between zero and pi over two is going to be increasing. So that's our first derivative test, All right, and now let's look at the second derivative. This is F double prime. Let's look at this same, um, between the same numbers. So negative pi over too zero in high over, too. So plugging in values into the second derivative, betweennegative Pi over two and zero, it's gonna become cave up, since it's positive. And here is, well, it's gonna be con cave up. So because there's no sign change, it's not an inflection point. And it's always gonna be con cave up between the interval Negative pi over two and pie or two. That's not included. All right, so that's a second derivative test. So what we know is that, um, it's, uh before zero. It's decreasing. And then when it hits zero, it's increasing, and it's always Khan gave up. So we know that it's gonna look something like this decrease. It's gonna be decreasing our intercept and then increase, and it's always gonna be con cave