**MATH 252: NOTES FOR QUIZ 3**

TABLE OF CONTENTS (under construction)

**BIG FILE: **Sections 16.3-16.9 notes (handwritten); see links and
comments below

**SMALLER FILES (if you cant get the big file):**

**SECTIONS 16.3-16.9: PARTIAL DIFFERENTIATION AND
APPLICATIONS**

Section 16.3 of Swokowski
: Partial Derivatives

*MATHEMATICA ***GRAPHICS GALLERY:**

Math252Ch16Gallery.pdf
(there are some ideas from later in Ch.16)

**SOME MATHEMATICA**

**PRACTICE WITH PARTIAL DERIVATIVES:**

**A function whose mixed partials are unequal** (credits: Ali Eftekhari, Tom Vogel)

http://www.math.tamu.edu/~tom.vogel/gallery/node18.html#SECTION00045000000000000000

**A non-differentiable function with partial
derivatives everywhere**

http://www.math.tamu.edu/~tom.vogel/gallery/node14.html#SECTION00041000000000000000

Section 16.4 of Swokowski
: Increments and Differentials

Section 16.5 of Swokowski
: Chain Rules

**Implicit Differentiation from Math 150:** Section 3.7 (pdf)

Section 16.6 of Swokowski
: Directional Derivatives (DDs)

**Great animation on DDs! (Gradient in green.)**

http://math.bu.edu/people/paul/225/dir_derivative.mov

**A non-differentiable function with all
directional derivatives (DDs)**

http://www.math.tamu.edu/~tom.vogel/gallery/node17.html#SECTION00044000000000000000

Section 16.7 of Swokowski
: Tangent Planes and Normal Lines

Section 16.8 of Swokowski
: Extrema of Functions of Several Variables (Revised
11/1/07)

**See Section 15.7 on this site for some graphs.**

http://www.grossmont.edu/jennyvandeneynden/archive/M281_links.htm

** IN THE DEFINITION
OF "CRITICAL POINT (CP)" (16.8.3):**

A critical point (CP) must be in the domain of *f*.

** DEFINITION OF
"SADDLE POINT" FROM THE HARPER COLLINS DICTIONARY OF MATHEMATICS:**

A point on a surface that is a maximum in one
planar cross-section and a minimum in another. (Visualizing a hyperbolic
paraboloid like *z* = *y*^2 - *x*^2
near the origin helps.)

(The definition may vary. Are degenerate
"ties" allowed along a cross-section, like for horizontal lines?
Also, for example, are the points along the *y*-axis saddle points if we have the graph of the "snake
cylinder" *f*(*x*,*y*) = *x*^3 ? That's debatable. Using the Harper Collins
definition, I don't believe they would be; the thing just doesn't look like a
"saddle" along the *y*-axis.
But it is true that there are higher and lower points "immediately
around" those points. Incidentally, *D* = 0 everywhere for this function, so the 2^{nd} Derivative
Test says nothing.) See: http://en.wikipedia.org/wiki/Saddle_point

** OPTIONAL COMMENTS ON
EXTENDING THE SECOND DERIVATIVE TEST TO FUNCTIONS OF MANY VARIABLES:**

If you have a nice function of *n* variables, you will construct an *n* x *n*
real symmetric matrix consisting of 2nd-order partial derivatives; such a
matrix only has real eigenvalues. If, when analyzing a critical point (CP)

the determinants of all the upper left square
submatrices (1 x 1, 2 x 2, etc.) are positive, the matrix is called positive
definite, and all of its eigenvalues are positive. The CP corresponds to a
local min.

the determinants of the upper left square
submatrices (1 x 1, 2 x 2, etc.) alternate in sign from negative to positive,
etc., the matrix is called negative definite, and all of its eigenvalues are
negative. The CP corresponds to a local max.

if the determinants of the upper left square
submatrices (1 x 1, 2 x 2, etc.) are all nonzero, and neither of the two above
configurations occur, then the CP corresponds to a saddle point (SP).

Section 16.9 of Swokowski
: Lagrange Multipliers (Revised 11/1/07)

**Optimize f(x,y) = 2x^2 + 4y^2 subject to x^2 +
y^2 = 1.**

http://math.bu.edu/people/paul/225/lagrange_animation.html

http://math.bu.edu/people/paul/225/lagrange_animation_qt.mov

(Quicktime format)

(by Paul Blanchard)

Quiz 3 (Sections 16.3-16.9) Notes: 51 pages (16.3-16.9) (2.3 MB), but if we
include bonus *Mathematica* graphics /
practice pdfs (61 pages; 3.3 MB
total).