Test 2 review
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Part I: Multiple Choice
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1. Find the domain of the function
   WW. 11.1, 2, 3, 4


2.  The equation of the tangent plane of the surface z = f(x,y) at the point 
   (a,b) is ...

    WW. 11.4, 1, 2

3. Chain rule

   WW. 11.5, 2, 3

4. The directional derivative of f(x,y,z) in the direction of v at the point P

   WW. 11.6, 2, 3, 4

 5. Find the maximum slope of the function f(x,y) at the point P=(P1, P2).

   WW. 11.6, 5, 6

6.  Find the canonical form (standard form) and classify the second order surface defined as ...

 Section 10.6 of Textbook,  Exercises. 21-28
 (and/or only completion of squares (shift) from https://math24.net/canonical-forms.html) 

7. A function z = z(x,y) is defined implicitely by

           F(x,y,z)=1

 Find its partial derivative 

   WW. 11.5, 7


8. Find the equation of the tangent plane at the point P=(P1,P2,P3) to the surface defined by

  F(x,y,z)=const 

 WW. 11.6, 11, 12 

  Review an example from our lecture notes where 
 we calculated an equation of the tangent plane to
 x^2 + y^2 +z^2 =3
at P=(1,1,1). It was done in 10:30 am class, in 1:30 pm class it was something similar.




Part II: Free Response
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1. Consider the function 
             
                       f(x,y)

 (a) Find all critical points of f(x,y)

 (b) Use the second derivative test to classify all critical points

WW. 11.7, 6

(look at "stingray" and "canyon of two lakes" in lecture notes)


2. Text problem

 WW. 11.5, 8, 9 


3. Use the Chain Rule to compute U_x at s=0, t =1, where

          U= U(x,y,z), x=x(s,t), y=y(s,t), z=z(s,t)

 WW. 11.5, 3, 4