Day 16 in Mat329

• We have a test this Thursday. We'll start with some new material, but reserve a half-hour at the end for a review.
• Homeworks returned:
  • 14.4: graded 14, 22, 37

    Don't forget your units (#22)

  • 14.5: graded 4, 13, 39

    Clear that denominator in #4!

• Your quiz is returned, as well.
  • chain rule
  • Explain the intuitive meaning.... and compute....
  • Seen that problem before? How do we guarantee differentiability of $f$ at $(a,b)$?

• Now how can we determine whether we have a minimum or a maximum or a saddle at a given critical point? If the second partials are continuous on a disk with center $(a,b)$, a critical point, then define \[ D = D(a,b) = f_{xx}(a,b) f_{yy}(a,b) - [f_{xy}(a,b)]^2. \] There are three cases:
  1. If $D>0$ and $f_{xx}(a,b)>0$, then $f(a,b)$ is a local minimum.
  2. If $D>0$ and $f_{xx}(a,b)<0$, then $f(a,b)$ is a local maximum.
  3. If $D<0$, then $f(a,b)$ is neither a local maximum nor minimum (in this case, $(a,b)$ is called a saddle point.
  So now we compute $D$:
  • $f_{xx}(x,y)=2+2y$
  • $f_{xy}(x,y)=2x$
  • $f_{yx}(x,y)=2x$ (ah ha! we should have known better: this is a polynomial, after all....)
  • $f_{yy}(x,y)=2$
  Hence $D=4(1+y)-4x^2$. In two cases, $D$ is negative, and we have a saddle; in the other case, at the origin, $D=4$ and $f_{xx}=2>0$, so we have a local minimum.

• If we're interested in absolute extrema, then we need to consider the boundary of a region, as well. There is an extreme value theorem for multivariate functions -- a natural extension of the EVT for univariate functions (p. 975).

• Now let's take a look at some examples. I want to start with one of the most important, however: #55, p. 979, which illustrates the importance of a standard problem in the linear algebra.
  
  • #3, p. 977
  • #14, p. 978
  • #32 (using the EVT)
  • #50, p. 979
  • #52

Types of problems to expect:

• defining and drawing functions
• contour plots
• computing limits
• checking continuity
• computing partial derivatives, including by
  • limit definition
  • formulas
  • approximately, using tables of data or contours.

• higher partial derivatives
• Differentiability at a point and tangent planes
• differentials and increments
• linear function (i.e. tangent plane) approximation
• quadratic function (i.e. "tangent bowl") approximation
• behaviors of quadratic functions
• chain rule

Here's last fall's exam 1 (Note: that exam went through 14.6).

Problems:

• Let's start with Figure 1 on page 981.

• An example: 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