💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!



Numerade Educator



Problem 53 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = e^{\arctan x} $


see solution


You must be signed in to discuss.

Video Transcript

All right. So for this function we have the domain is excess All riel, we haven't intercept. At 01 there is no symmetry. And for awesome toast, we can test the limit as X approaches infinity except for his infinity over function E to the inverse tan of X. Okay. And that turns out to be e to the power of pi. Over too. Which is approximately 4.8. Just so that we can graph it. Um, so that we have as X approaches infinity, we're gonna have umm and ask him to at why equals about 4.8. So just put at 4.8 and let's see what happens is X approaches. Negative infinity. So All right. So what would the answer be if this waas as X approaches Negative infinity, our answer is going to be e to the power of negative Ah pi over to which is approximately is your point to anyone So we can put why approaches about 0.21. So those are our ass in Toots. And now we can use the first derivative test to see the increasing and decreasing intervals. So why prime my prime equals e to the inverse tan of X over X squared plus one. Okay. And if we set that equal to zero, we see that it's an inconclusive Okay, so we can't use the first derivative test, but we can still look at why Double prime for con cavity. So we have e to the inverse 10 of x minus to e to the inverse 10 of x times X and that's all over X squared plus one squared. So that was just quotient rule. We set that equal to zero. We can see the, um we have X equals I have. So if we do our second derivative test double prime now, we plug in points around 1/2 to see the con cavity. What happens here is going to be con cave up here. It's gonna be con cave down, okay? And now we congrats with the information that we have. So we haven't intercepted 01 to this would be 01 We haven't us in tow at Lung. Why equals 4.8. So let's say this is why eagles 4.8 and have won it. Y equals 0.0 21. So let's say it's right here. All right. And now we can graft. So, um, we know it's gonna be Kong cave up before 1/2. So let's say that this is if this is one. So this is 1/2. All right, so this is about to have, um So here, the cone cavity is gonna switch. We have an inflection point somewhere here. We know that to the as approaches negative infinity we're gonna approach Why equals your point to anyone but never touch it, and it's gonna be con cave up. So it's gonna be like this, and it's gonna be conk it down after we get to that half. Um, and on the other side, as it approaches as exit values approach infinity, why? He's gonna approach this ass in tow. So we're gonna keep increasing until we keep calling cave up until we hit the halfway. And then we're gonna search to con cave down and approach this essence here