Day 19 in Mat329

• The exam
• The key -- actually I'll hold off on that, because of the "redemption policy":

  You can get half of the points you missed by correcting your mistakes. So here's the deal:

  1. Print off a fresh copy of the exam
  2. If you received full credit for a part of the test, you don't need to redo that; otherwise, redo, correctly, and I'll average your two scores.
  3. On Friday, October 2, submit both your corrected and original tests to me.
  You're free to talk to others, talk to me, etc. So there's no excuse not to do it (except those that you make up for yourself!:)

• Comments
  • General:
    • GYE: Given Your Error
    • "Justify" means you have to explain! Talk to me!

  • Problem specific:
    • The limit definition of the derivative is the most important in calculus. Learn it, and know how to use it.
    • No big news here to speak of.
    • "Discuss the behavior of this function at the origin."

      Function isn't even defined along $y=-x$, so it's not defined in a neighborhood of (0,0), and hence won't be repaired by simply setting $g(0,0)=0$. Don't forget to make your direction vector unit, and that the result of a dot product is a scalar, not a vector.

    • You need to estimate the magnitude of $\nabla f$ at (5,1). Umbrellas and bowls might help.

      Gradient points in direction of greatest ascent, not descent (think of the plane $z=x+y$, for example). The technique of "steepest descent" may confuse you....

      Notation: $P(5,1)$ denotes a point $P$; it is not the altitude (thinking of this as terrain), but rather tells us where we are in the field.

• Your homework is due today.

Problems:

• Let's start with Figure 1 on page 981.
• : Use Lagrange multipliers to find the maximum and minimum values of $f(x,y)=4x+6y$ subject to the constraint $x^2+y^2=13$.
• Examples from the text, including a surprise
• Why don't you try an example: #25