Sec2Sess26

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in direction f() travels when it increases or decreases, as seen on a graph,
indicates the effect changing x and y will have. The f() will have the effect
produced on it.

Approximation formula
<math>x_0\ </math> -> <math>\ f(x_0)</math>
<math>f(x)\ \approx \ f(x_0)\ +\ f'(x_0)(x-x_0)</math>
continues onto the taylor approximation

how can we approximate two variables?

"partial derivative" notation is a curly "d".
not a full derivative but contain only derivatives for specific variables.
Therefore "partial" rather than "whole" derivative.

A derivative is taken of one veritable but another is kept constant.
Calculated by holding one variable fixed and differentiating with respect
to the other variable.

example
<math>f(x,y)\ =\ x^3*y\ +\ y^2</math>

pd = "partial derivative"

<math>pd(x)\ =\ 3x^2*y\ +\ 0</math>
<math>pd(y)\ =\ x^3\ +\ 2yz</math>

in a pd(x,y) calculation first the pd(x) is taken and then using
the result pd(y) is taken
for all polynomial functions and all differentiable functions of several
variables that will be considered in the class pd(y,x) = pd(x,y).

recitation 1:
<math>x\ *\ y^2\ +\ x^2\ *\ y</math>

a) pd(x), pd(y), pd(x)(1,2)
b) pd(x), pd(x,y), pd(y,x)

<math>a)\ y^2\ +\ 2xy,\ 2yx\ +\ x^2,\ 2^2\ +\ 2*1*2\ =\ 8</math>
b)
pd(x)
<math>y^2\ +\ 2xy</math>
pd(y) on remainder
= 2y+2x
answers = 2y, 2y+2x, 2y+2x