Thanks for watching along with these. Looking at it now it definitely looks like e^(-x^3) -1 (I think, starts at 1 instead of 0...won't impact the derivative though, other than the starting index I suppose). I've got two more videos set to come out and then there's likely to be a decent dry spell because, unfortunately, I've run into some family health issues that have eaten all my video planning/making time! Definitely hope to be back at it sooner than later though.

@@turksvids sorry to hear that. Best wishes to you and your family.

thanks! side question: are you doing AP Precalc at your school this year/next year? Thoughts on it either way?

@@turksvids we are doing it this year and I’m teaching an honors version of it (meaning I teach some advanced concepts besides the ones there). I’m not in love with some of the topics particularly the semi log plots and regressions. I honestly think all those topics are stats based and don’t really fit in a Precalculus course. It also feels like we have to dance around the derivative and concavity without talking about them from a calculus perspective but I can make a better argument as to why we have to teach those concepts using things like finite differences and average rate of change.

It also doesn’t make sense that they’re not including parametric equations and vectors in this year’s exam. I guess it’s the pilot year so they’re to figure out where to make adjustments going forward

In general the coefficient of (x-a)^n is nth derivative at a over n!, since all the derivatives of e^x are just e^x, I think we're looking at e^2/4! hope this helps! please check our my other videos and share with others!

@@turksvids the function is e^(4x) tho, not e^x

@@turksvids i was thinking about writing the taylor series centered at 2 given that i know the taylor series centered at 0 but i don’t know how to transform the series to a new center

ah, i did the wrong function, but told you how to do it for any function, find the fourth derivative, plug in 2, and divide by 4!.

Yup! This is year 21 I think. Hard to believe, life goes fast!

Uh...Imagine that a surface is made up of a LOT of parallel rollercoaster tracks all parallel to the y-axis. If you're on one track and stay on that track, f_y is just the normal slope of the track/curve you're used to from Calc 1 (but the variable is y instead of x) and f_yy is just the concavity of that track/curve. Now, freeze some moment on your track, like you took a picture of it and can see it just like a normal tangent line figure. If you then move to a parallel track while holding the y-coordinate constant, the slope of the tangent line in your picture will likely change. f_yx tells you if that change will be positive (the slope increases) or negative (the slope decreases). In the context of "the photograph" it's like you're moving closer to or farther from the camera but not moving left or right. This is a pretty tough thing to visualize and describe, but if you can "put yourself in the rollercoaster car" and imagine holding the tangent line as like a broom pole, I find this helps to think about it this way. Hope this helps! (I did not really proof read this...)

@@turksvids thank you so much 😊🙏