MA242 Review II

Functions of two or more variables (f(x,y), f(x,y,z) ... )

Domain: All possible (x,y)

Range: All possible function values

Graph of f(x,y),   z = f(x,y)

Level curves  of f(x,y) (surfaces if f(x,y,z)). Curves where f(x,y) has the same values.    Ex. 6-10, page 752-755.

f(x,y) is continuous at (a,b) means ...           Definition, page 763

To prove f(x,y) is discontinuous at (a,b) we can set  y = k x to see whether we can get different limit.    Ex. 1-3, page 760-762.

 

Partial derivatives. Differentiate with only one variable and keep other as constant(s).

High order partial  derivatives. When f_xy = f_yx?

Applications:

Equation of the tangent plane of the graph z = f(x,y).    Ex. 1, page 779.

Equation of the tangent plane to a level surface ( f(x,y,z) = k).   Ex. 8, page 806.

Differential for approximate evaluations.  Ex. 1-2, page 790-791.

Chain rules, several cases.

f(x,y),  or f(x,y,z), x=g(t), y=h(t), z=w(t).

f(x,y),  or f(x,y,z), x=g(u,v), y=h(u,v), z=w(u,v).

f(x,y),  or f(x,y,z),  x=g(u,v), y=h(u,v), z=w(u,v),  u=q(t), v=r(t),  t = ...

Partial derivatives of implicit functions  F(x,y)=0,   F(x,y,z) = 0, ....

Direct differentiation.   Ex. 4, page 770.

Formulas.                         Ex. 8-9, page 794-795.

More application, maximum and minimum.

• Gradient of f(x,y),  f(x,y,z)
• Directional derivatives. NEED to use the UNIT direction.
• The directions of the maximum rate of change of increase or decline.  What are the maximum rate of increase or decline?
• The gradient direction is perpendicular to the tangent line and the level curves of f(x,y).  So it is the direction of the normal line.
• The gradient direction is perpendicular to the tangent plane and the level surface of f(x,y,z).  So it is the direction of the tangent plane to the level curve.    Ex. 6-7, page 804-805.

Local extremes.

Necessary condition to be a local extreme, critical points, gradient f is zero.