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)
A non-differentiable function with partial derivatives everywhere
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.)
A non-differentiable function with all directional derivatives (DDs)
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.
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.
(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).