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April 15, Mathematics 206a Mr. Haines 2009 Multivariable Calculus Final Examination
(8) I. Give a parametrization for:
A. The line segment from the point (2, 4, 5) to the point (3, 6, 7).
B. The plane rectangular surface in R^3 with corners at the points (0, 2, 0), (4, 2, 0), (4, 2, 3), and (0, 2, 3).
The volume of a right circular cylinder of height h and radius r is V = π r^2 h.
The volume of a sphere of radius r is 3 3
V = π r.
The cylindrical coordinate conversion formulas are x = r cos θ ; y = r sin θ; z = w.
(12) II. If F ( x , y, z ) = -y i + x j + 5 k and C is the curve parametrized by c ( t )= ( 2 cos t , 2 sin t , 5 )with 0 ≤ t ≤ 2 π.
A. Explain why C is or is not a closed curve.
B. Calculate the value of ∫ •
C
F d x
C. Explain why your answer to Part B proves that curl F is not the zero vector.
(8) IV. Give examples of
A. two vectors in R^3 whose cross product is i + j + k.
B. a vector field on ℜ^3 with divergence xy + 6 x + 2.
(4) V. Suppose f ( x , y )= ( x^3 y , 2 x^2 + y ). Write a formula for D f (1,1 ) , the derivative of f at
(1,1).
(4) VI. If f :ℜ^2 → ℜ has rule f ( x , y )= x^2 + 2 xy + 3 y^2 , calculate the directional derivative of f
at (1, 2) in the direction parallel to the vector 3 i + 4 j.
(10) IX. Given the vector field F (x, y, z) = (2 , z^2 , 2 yz ) and the path C in ℜ 3 parametrized by c
(t) = (^)
t (^) t , t 2
,sin 2
(^4) π with 0 ≤ t ≤ 2 , calculate the path integral (^) ∫ • C
F d x.
(10) X. Compute (^) ∫ • C
F d x where C is the counterclockwise-oriented boundary of the triangular
region in the first quadrant bounded by the curves x = 0, y = 0, and x + y = 1
A. if F ( x, y ) = ( y^2 , y + x^2 ).
B. if F ( x, y ) = (2 x^4 + 3 y, 3 x + 2 y^5 ).
(5) XI. If S is the solid bounded by the surfaces 9 = x^2 + y^2 , z = 2 , and z = 7 , set up, but do not
evaluate, the iterated integral that results from changing the triple integral ∫∫∫ + + S
( x^2 y^2 z^2 ) dxdydz to cylindrical coordinates.
(5) XII. If f :ℜ^2 → ℜ with rule f ( x , y )= x^2 + 3 xy − y^2 calculate the Hessian of f at (1, 2).
