Day 5 in MAT229

For all of you here today, you're welcome to zoom in as well, so that we can share screens if some problem arises, or if we want to show off something you've done.

It's always weird to have a holiday so close to the beginning of the semester, but it's certainly good to honor MLK! I had the luxury of listening to a few of King's speeches. Wow.

Listening to him read his Letter from a Birmingham Jail was very moving, in particular.

It's going to be a recap of some problems from Calc I, with a new collection of functions (these exponentials, and inverse functions).

1. when you submit your pdfs, please do them as a single pdf. It's just a lot easier for the grader that way (and I always tell my students that you want to keep your grader in a good mood!:).

2. please submit a pdf, rather than a Notebook. For that first assignment I made it an option to submit a .nb file, but I won't do so going forward.

   It turns out to be easy to grade a pdf within Canvas, but not a Notebook. So save your Notebook as a pdf, and submit that.

   However: I want your notebooks, if there's something you want to share with me!

   If you want to submit a notebook for me to look at, because of some Mathematica issue(s), just email it to me. That will be easier, and I can get right back to you.

3. One more thing: it's helpful, if you email me, to put "MAT229" in the message subject line.

We want you to use Mathematica, but it's not mandatory.

• Estimation: for these problems, we might hope that you'd zoom in to get that "1 decimal place" correct. Some of you went farther, and actually solved for the locations of the extrema; but sometimes it's just a "back of the envelope" answer we need.

• Linear approximation. A tangent line can serve as a linear approximation to a function: if the point of tangency is $(x_0,y_0)$, then the tangent line is perfect for $f(x)$ at that point. But, as we move away from $x=x_0$, we find that frequently $|f(x)-(mx+b)|$ gets larger (i.e., the approximation gets worse).

  Engineers frequently need to assure that their answers (which are nearly always approximations) are within a "tolerance" -- e.g. within 0.25 -- of the correct answer.

  So the question is "How far can I wander from $x_0$ and still be close enough to the right value?"

• That wild function $\frac{x^{x/5}}{(1+x^2)}$

  has a domain which is is open: $(0,\infty)$. Some of you gave as an upper limit whatever you plotted in your graph -- but nothing stops $x$ from increasing forever. It can't be negative, however, as you discovered.

You can add text lines to add comments about how you did things.