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Numerade Educator

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Problem 41 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \arctan(e^x) $

Answer

see solution

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Video Transcript

So the domain for this one is exes, All riel and the intercepts are zero. Just have 10 pi over four. There is an even symmetry to this craft and the Assam totes. What's so for those? So the limit as X approaches infinity over function Inverse tangent E to the X. Let's pie or two. So with X approaches infinity, We have an s in tow at Why equals pi over too. And now, for the 2nd 1 limit as X approaches negative infinity of in verse 10 of e to the X. That's going to give us zero. So we have as X approaches Negative infinity, Why is going to be zero? Okay. And now let's look for increasing decreasing intervals. So why prime equals e to the X over Okay to the two X plus one. And if we dreaded to find the critical points, will see that this is inconclusive. This is inconclusive. Um, but we can look forward the con cavity. So the second derivative Well, give us negative e to the x times e to the two x minus one over e to the two x plus one squared crucial room equal zero so we can see X equals zero. That's a point we're gonna be looking at North. Second derivative test. This is F double prime, looking for Khon cavity. You see that? It's Khan gave up to the left of zero and Khan came down to the right of zero. All right, so using this information, we can graph, So we have zero pi over four. So I'm gonna say this is pi over four. So that is, um, our intercept. And we have Oh, we have an s in tow. A cz x approaches infinity. So we have y equals pi over too. All right, So if this is pi over four than pie over to must be here, all right? And then we have y equals zero, as are other Ask him to. And now our graph. Oh, so we see that this is an inflection point that's important, cause we have a switching con cavity. So here we have, um, as X approaches infinity, it's gonna go to, but it's gonna approach but not touch or ask them to at pi over too. So this is infinity. The exes are approaching infinity, so it's gonna go this way, and negative infinity will approach. Why equals zero