Problem 54 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \tan^{-1} \left(\dfrac{x - 1}{x + 1}\right) $

Answer

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Video Transcript

the domain for this function is ex cannot be won. That cannot be negative One Sorry. Um, the intercepts are 10 and zero negative pi over four. There's no symmetry. And for Assam toads, we can look at the limit as X approaches infinity over function Inverse tangent. Oh, X minus one over X post one And that's able to pile before. And we realized that if we do, the limit is X approaches. Negative infinity. It's also pirate before so the s into, um, in either direction tells her the damn. So as X approaches plus or minus infinity, we see that Why hasn't as in tow pi over four. And remember, this is hordes on horizontal asking too. And now we can look at our first derivative to look for increasing in decreasing intervals to see if we have a Miramax so live prime. And here we can just to change rule you sub. My prime is one over x squared, plus more set that equal to zero to find a political points is inconclusive. So we can't use the first river the test. But we can look for why double prime, which is negative two x over X squared twist one that's all squared, Um, said that equal to zero, and we see that we have X equals zero. All right, so now our second derivative test for Kong cavity. Let's look at zero. Anything smaller than zero is gonna be Khan gave up. Anything bigger is gonna become cave down. Remember, we're plugging into our second derivative. So whoever con cavity, another king graph with the information we have. So we know the ex cannot be negative one for the doing. We're gonna keep that in mind. All right, So let's say this is pie, and this is negative play. We haven't intercept at 10 Don't say it's about here. And zero negative pi over four. So let's say this is pie over to Nick department for somewhere right here. This is just roast kitsch, okay? And now we know that we have all we have in us until y equals pi over four. So I'm right about you, Or is that a lesson too, at why includes prior before? And now we know that it's gonna be Kong Cabo. This is an inflection point, cause it's a sign change. So is he is going to be calm, keep up. And then calling came down. But first it started on the side. So we know that our graph is going to approach negative infinity in the exes as actually purchase Negative. Infinity wise gonna go to pi over four. So it's gonna look like So I'm gonna go into pie before this line here is gonna approach, but never touch. Let's see what happens when you get close here. All right, so we know that it's going to never be negative one. So let's say this is negative one. Let's say you're right here with negative one so it can come over here, but never touch negative one. And I'm assuming it's gonna be a little higher because, um, the closer it gets, the more negative the X values get is gonna be, uh, closer to the ass in tow, and the further away is gonna be a little higher. So it's a little higher, and over here, I know that it's gonna be also the same thing. Cannot be negative one, but it's gonna be concave up if it's smaller than zero. But I'm assuming is like that. And then if it's greater than zero is gonna be calling Kate down, so it's gonna be cockeyed down. So this is a reflection 0.0 negative pie before it's gonna become came down Hit the axis, keep being concave down and approach but never touched our ass and took