Week #4: Total Derivatives and Tangent Planes, Directional Derivatives   
September 13-17

Mon Lect:   The tangent plane to the graph of a non-linear function of two variables. The famous watermelon demonstration, and the definition of directional derivatives.  Computation of directional derivatives via the formula .

Tues. discussion: Practice computing tangent planes and directional derivatives.  

Note to TAs:  I have not used the local coordinates dx, dy, dz and I don't intend to do so.  I did not discuss conditions for differentiability yet -- but the sufficient condition I intend to use is continuity of the function and of the first partial derivatives.

Wed Lect:    Computation of partial derivatives using rules for differentiation:  product rule, chain rule, etc.
Tangent plane approximation (i.e. the error between the tangent plane and the actual function.)  

Due  Thursday Sept 23.:  
Exercise A:  Find an equation for the plane containing the three points (-1, 3, 4),  (0, 2, 1),  (5, -2, 3).
Sect. 3.4 1 (a, c, f, h)   
Note: If you are not rock solid sure of being able to compute derivatives accurately, then review systematically from a standard calculus text!
Sect 3.5 Ex 1(b), 2(b), 4, 5(b), 6.

Thu Disc:

Work partial differentiations carefully step-by-step for the homework problems.  Carefully explain use of the product rule and chain rule.  Write them out carefully.  See Section 3.4.

Fri Lect:  Summary and discussions of condition for differentiability -- i.e. for the tangent plane approximation to be valid..  Reason for the directional derivative of a non-linear function to be equal to the directional derivative of its linear approximation.

Due  Thursday Sept 23.: : Sect 3.7 Ex.1

Created by Mathematica  (November 29, 2004)