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March 6, Mathematics 206a Mr. Haines 2008 Multivariable Calculus Examination #
(10)I. Derivatives
A. Suppose f ( x , y , z )=( x + ez^ + y , yx^2 ) and a = (1, 1, 0). Calculate the total derivative of f at a.
B. Suppose g :โ โ โ^3 with rule g ( t )=( 6 t , 3 t^2 , t^3 )and f:โ^3 โ โ with rule f ( x , y , z )= e^ xyz. Use the Chain Rule to calculate ( f o g )โฒ(^1 ).
(10)II. If f :โ^2 โ โ has rule f ( x , y )= ln x^2 + y^2 , calculate the directional derivative of f at (2, 0) in the direction parallel to the vector 2 i + j.
(10) III. For the vector field F ( x , y , z ) =( x^2 y , z , xyz )
A. div ( F) =
B. curl ( F) =
(10) VI. Find the critical points of 1 2 2 f ( x , y ) = e + x โ y and determine whether they are local maxima, local minima, or saddle points.
(10) VII. Let C be the path in โ^4 parametrized by f ( t )= (cos t ,sin t ,cos 2 t ,sin 2 t )starting at
t = 0 and ending at t = ฯ. Compute the length of C.
(10) VIII. Suppose C is the helical path parametrized by f ( t ) = (cos t ,sin t , t ) starting at t = 0
and ending at t = 2 ฯ.
A. If F : โ^3 โ โ with F ( x , y, z ) = x + y + z^2 , Compute (^) โซ C
FdL.
B. If F : โ^3 โ โ^3 with F ( x , y, z ) = x i + y j + z^2 k. Compute (^) โซ โข C
F d x
r (^) r .
