Review for 2nd Test

First test will be 10 problems of 10 points each for 100 points total. Show your work and/or explain your answers.

1. Sketch the level curves of f( x,y) = y-x² corresponding to levels k = 1, 2, 3.
2. Show that the following limit does not exist by evaluating it along the x and y axes:

lim ( x,y) ® ( 0,0)  x2+4y2 4x2+y2

3. Find the second partial derivatives of f( x,y) = xln( y)
4. What types of functions satisfy the partial differential equation

¶2u ¶x¶y =  ¶2u ¶y¶x

5. Show that u( x,y) = sin( 2x+3y) satisfies

9 ¶2u  ¶x2 +4 ¶2u ¶y2 = 12 ¶2u ¶y¶x

6. Find the equation of the tangent plane to f( x,y) = x²y at ( 2,3) .
7. Use the chain rule to evaluate dw/dt if w = x²+y² and

x =  cos( t) +sin( t)  y =  cos( t) -sin( t)

8. Find the derivative of f( x,y) = x³y² in the direction of the vector á 5,12 ñ .
9. Find the equation of the tangent plane to

x2+y2-z2 = 1

at the point ( 1,1,1) .
10. Find the extrema of f( x,y) = x³-3xy+y³.
11. Find the point on the curve x²-xy+y² = 1 which is farthest from the origin.
12. Problems 57 - 60 on page 978 are important. Example: Find the least squares line for the data set

( 0,0) ,( 2,3) ,( 5,7)