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NAME_______________________________________
I____II____III____IV____V____VI____VII____VIII____ IX____X____TOTAL ___________
February 17 Mathematics 206a Mr. Haines
2005 Multivariable Calculus
Examination #
(12)I. Suppose f ( x , y )= x sin( xy )
A. =
( x , y ) x
f
B. =
( x , y ) y
f
C. =
2 x y x y
f
D. =
2 x y y x
f
(8)II. Suppose
3 3 f :^ โ โโ with rule f ( x , y , z )= ( xyz , xy , x ).
A. Calculate Jf (2, 2, 2), the Jacobian matrix of f at (2, 2, 2).
B. Find a point at which Df (2, 2, 2,), the total derivative of f , has the value (0, 8, 2).
(12) IV. Suppose
3 3 F : โ โโ with rule (^ , , ) ( , , )
2 2 2 2 F x y z = x y x โ y and
3 3 G : โ โโ with
rule (^ , , ) ( , , )
2 2 2 G x y z = x + y + z x + y + zz.
A. Calculate the Jacobian matrix of F at the point (1, 2, 1).
B. Calculate the Jacobian matrix of the function G at the point F (1, 2, 1).
C. Calculate the Jacobian matrix of the function G ฮฟ F at the point (1, 2, 1).
(10) V. Find the equation of the tangent plane at the point (0, 1, 1) to the surface with equation:
3 2 3 x โ y + yz =.
(10) VI. Suppose f ( x , y , z ) xy x y z 5 x
2 2 = + โ โ and a = (1, 1, 1).
Compute the directional derivative of f at a in the direction parallel to the line
x (t)= (t + 1, t + 2, t + 3).
(8) VIII. Suppose 2 2
2 2 ( , ) x y
x y f x y
A. What is the domain of f?
B. ( , )
lim
(,) ( 0 , 0 )
f x y x y โ
does not exist, and is so nasty that given any number you pick from
โ1 to 1 you can find a line along which to approach (0, 0) and get your number.
Find a value for k so that if you approach (0, 0) along the line y = kx the limit
of f(x,y) will be 4/.
(8) IX Give an example [a sketch is sufficient] of:
A. An open set in
2 โ that is not bounded.
B. A closed set in
2 โ that is bounded.
