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 GRAPHS:
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 2nd 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).