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You're welcome.

Glad it was helpful!

Complex rational expressions are tricky for some. Not all of my students know when components can move or cancel when there are fractions inside of fractions. I want to make sure everyone knows why we're justified in taking shortcuts sometimes.

In this video, I was explaining how to answer this in terms of rational function end behavior. Most of students in my calc class had me for precalculus as well - we spend a lot of time discussion polynomial and rational function behaviors. Since this function BEHAVES like a rational function whose degree in the numerator is greater than the degree in the denominator, it will not have a horizontal asymptote. Moreover, the right side branch will increase towards infinity as x approaches infinity, and we can say this function will behave the same way.

Because Calculus is awesome.

@@mitchehrman I’m scared 💔

Thanks. By the time you got to this, did you already factor out the 2 from the second derivative of the denominator? When recording the video, I simplified the first derivative in terms of sine and cosine as 4(sin(2x)/cos^(2x)) thinking something would cancel. From there, I didn't want to go through the quotient rule in the video knowing that this problem was one I'd need to discuss together with students in class.

​@mitchehrman After taking the first derivative I just factored out 2 and was left with 2tan(2x)sec^2(2x) / x Then I just took the second derivative, without simplifying the first one. And then I just substituted 0 in and got 4 as the answer. P.S. sorry for my English, I'm currently living in the uk and still learning it 😅

Interesting. I'm not familiar with that result. I recall that function is special, in that the the indefinite integral doesn't exist. Perhaps, an intuitive description might help explain... similar to the description why the infinite sum of natural numbers is -1/12?

@@mitchehrman I did a revision in the question and I was wrong about the meaning of integral, because instead of considering negative the area below the XX axis, I considered negative areas to the left of the YY axis. Sorry for the misunderstanding.

are you actually serious????

in (insert country) we did that in kindergarten

Thanks. That's a good approach after you got the derivative of the numerator. I hadn't worked through the problem ahead of time and anticipated some sine or cosine terms would have canceled. They didn't, and I knew I'd be going over it in class anyway to further clarify to my students.

Agreed. I've seen it spelled both ways, but I suspect L'Hopital is *more* correct.

Agreed. Delta-Epsilon definition (and later proofs) is a more rigorous standard. This video was recorded as a review for high school students completing an entry level calc course on an applied engineering and business track. The formal definition was beyond the scope of this general review.

But x*cos(x) is unbounded so this example doesn't follow my justification in the video. The squeeze probably explains the behavior of this limit a little better.

@@mitchehrman just rewatched and you did say bounded my mistake!

It's good to remember that it's there, but when you execute the integral, the constant subtracts itself because of the Fundamental Theorem. Be careful though, sometimes you'll integrate more than once within a problem, and the constant for each level integration might be different and shouldn't be ignored.

The concavity doesn't change at A. If you look just to the left of that point, the function is concave up, if you look at the point the function is concave up, and if you look to the right of the point it is still concave up. Only the SLOPE changes at point A, so it is not an inflection point.

me tomorrow

Me in about 3 weeks.

​@@cocainejeezusHow do you fail 3 weeks in advance 💀

☠️

@@cocainejeezusthen study💀

Question 9 d

Agreed. A more appropriate technique here would be to use the Squeeze Theorem since the function cos(x) oscillates between -1 and 1. Generally speaking though, the limit of 1/sqrt(x) is enough to force the overall limit to 0.

Nope, sorry. I used Kuta Software to create the worksheets I referenced in this video; www.kutasoftware.com/buy.html

this is a school course lol.

Come to india

😂😂😂

he knows he said he just wanted to show how you would do it before learning chain rule

Thanks Will. Things good?

@@mitchehrman yea things are doing good, finally got out of my POS car and got something more fun, still have the job at faxon

I don't think he did - I've not seen it. This clip was from two years ago.

It could be on btube

@@DGISALWAYS butler techs youtube?

@@willdawson8460 could be. Haven't checked there in years

Thanks Will. How's that guitar coming along? Did you ever revisit that project?

oops i spelled intensively immensely wrong