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A function f(x,y) is continuous at a point (a,b) if (3 conditions)
 a) f(x,y) is defined at (a,b)
 b) lim(x,y)>(a,b) f(x,y) exists
 c) lim(x,y)>(a,b) f(x,y) = f(a,b)

Interior point P of shape R
Has a disk centered at point P which contains only points inside R

Boundary point Q of R
Every disk centered at Q has a point inside of R and a point outside of R



f is differentiable at (a,b) if
if and are defined and continuous at (a,b)

If a function is differentiable at (a,b)
it is continuous at (a,b)


f(x,y)≃
 or
 ΔZ +f(a,b)


f has its
1) maximum rate of increase
2) rate of zero increase
3) maximum rate of decrease
when:
1) in the direction of the gradient. The rate of increase is the magnitude of the gradient
2) in any direction orthogonal to the gradient
3) in the direction of the negative gradient. The rate of decrease is the negative of the magnitude of the gradient

What is the chain rule for z(x,y) with respect to t

The implicit differentiation
can be rewritten as

if F(x,y,z)=K and ⊽F(a,b,c)≠0 then ⊽F(a,b,c) is
orthogonal to the tangent plane to the level surface at (a,b,c)

Critical points (on the interior of the domain of f) exist when
or both do not exist


if D(a,b)>0 and <0, there exists
a local max

if D(a,b)>0 and >0, there exists
a local min

if D(a,b,)<0, there exists
a saddle point

if D(x,y)=0, there exists
inconclusive u hoe

What are the steps for finding the absolute max and min on a closed and bounded region of a function with two variables (not lagrange)?
 1) Determine value of f at all critical points
 2) Find all extrema of f on the boundary
 3) The greatest value is the absolute max, least value is the absolute min

