Day 26, Mat128: Calculus A

• If you need to Zoom to class, here's the link. Old Zooms are here.

• Your exams are returned.
  • Max: 79%
  • Median: 67.5%

  These didn't go as well as I'd like, so this is what we'll do:

  1. I'm returning your exams; don't mark on them.
  2. You may revise any problem on which you didn't receive a perfect for half of the points back.
  3. You have until next Monday at class time to revise and return them.
  4. You should resubmit both your original, unmarked exams, and the corrections (on separate sheets).
  5. Here is the exam, if you want to print off the necessary pages again. Or you can do them on your own paper (one problem per page).
  6. You do not need to resubmit problems if they're correct (or you do not wish to redo them, for whatever reason).
  7. At the end, I'll add 5% points to your score (after revisions). That will get us some As, if folks revise.
  A couple of things I need to point out:
  • Problem 1:

    I'm a little curious about the lack of factoring. It's not critical, but does help clarify some expressions. In this case it was the term $2x-2$.

    You should probably write that $2(x-1)$, to show off that it's 0 at $x=1$.

    On a related note, I was pleased to see a lot of your tangent lines written in point-slope form (and, as usual, a little distressed to see the time wasted rewriting those in slope-intercept form).

    \[ y=f(x_0)+f'(x_0)(x-x_0) \] is the tangent line of $f$ at $(x_0,f(x_0))$ of slope $f'(x_0)$.

    Everything is staring you in the face: where, the function value there, and the instantaneous rate of change there.

  • Problem 2:

    Pretty good, overall.

    • every point where the function "touches the mirror" is on the inverse function (around $x=-2$).
    • Every identifiable point should be carefully reflected (e.g. intercepts)
    • Some of you marked points on the function of your own (e.g. closest approach) and mirrored them -- good work!

  • Problem 3:

    I'd promised you one problem straight from your text: this was the one.

    The tangent line is staring you in the face. What's the sign of its slope? Can you estimate its slope?

    I had one student who had the wrong value, but knew it -- and told me so. That's what you should do: you do the best you can, but then, if you know you're wrong, you fess up!

  • Problem 4:
    • L'Hopital's rule is only used for indeterminate forms. So the first thing to do is find out if the limit is indeterminate or not: \[ \frac{0}{0} \] or \[ \frac{\infty}{\infty} \] or \[ 0\cdot \infty \] or the like. If not, you should just evaluate the limit! It's determinate, meaning you can determine it!
    • Don't drop the limits. Every one of these problems is a limit. Why would you drop them? That's what this is all about!
    • You need to learn your functions. This was a test of two things: your understanding of L'Hopital's rule, and your understanding of inverse functions (arctan and ln, in particular).

  • Problem 5:
    • $f'(x)$ has two zeros
    • $f''(x)$ has three zeros (one of which is an integer, which you can see -- that turns a cubic into a quadratic).
    • For the revisions, make sure you find their true (and approximate) values.

  • Problem 6:

    We've seen this one several times. Checking to see if you understand the functions \[ h(x)=\frac{10 (x-2)^{2/3}}{x} \] \[ h'(x)=\frac{10 (6-x)}{3 \sqrt[3]{x^2 (x-2)}} \]

    Fractional powers can be tricky, but this one isn't. The domains are about the zero denominators, not about the root functions.

• We have just one more new section of material: section 3.5

• Please do the preview of 3.5 (submit for next time); your 3.4 worksheet for today will be due next Monday.

• Exam 3

We continue with optimization ("applied optimization").

• Let's start with the preview. It's another example involving finding the optimal volume of a box, with a different set of constraints.

• Then we'll look at another example optimization problem (Activity 3.4.5):

  I like to use a method I call UPCE ("oopsie") for story problems:

  • Understand
  • Plan
  • Carry out
  • Evaluate

• Your Worksheet for 3.4 is just a set of three problems (two of them from your textbook). You should
  1. UPCE: Understand, Plan, Carry out, Evaluate
  2. Draw pictures
  3. Give meaningful names (letters) for variables.
  4. Establish reasonable domains for variables
  5. Establish a reasonable coordinate system
  6. Doodle relationships!
  7. Look for critical points of the target function
  8. Check the boundaries

• A Chain Rule tutorial
• Our free, on-line textbook, called Active Calculus.