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This is the Exam of Multivariable which includes Interpret Mathematical, Integration, Region, Evaluate, Illustrate, Explanation Needed etc. Key important points are: Curve Parametrized, Equation, Plane, Starting, Ending, Perpendicular, Tangent, Point, Vector Field, Graph
Typology: Exams
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March 12, Mathematics 206a Mr. Haines 2009 Multivariable Calculus Examination #
(10) I. Let C be the curve parametrized by (^)
f ( t ) t ,^22 t^23 t^2 starting at t = 0 and ending at
t = 2. Give an equation for the plane that passes through f ( 1 )and is perpendicular to the tangent to C at that point.
A. div ( F) =
B. curl ( F) =
(10) III. Find the equation of the plane tangent to the graph of z = x^2 + y^4 + exy at the point (1, 0, 2).
(10) IV. Derivatives
A. Suppose f ( x , y , z )=( x + z + y , x^2 )and a = (1, 1, 0). Calculate the total derivative of f at a.
B. Suppose g :ℜ → ℜ^3 with rule g ( t ) =( 6 t^2 , 3 t^3 , t )and f:ℜ^3 → ℜ with rule xyz. Use the Chain Rule to calculate ( f g )′(^1 ).
(10) VII. Suppose f :ℜ^3 → ℜ with rule f ( x , y , z )= sin( x + 2 y + 3 z ).
A. Give p 1 ( x ) , the first Taylor Polynomial of f at (0, 0, 0).
B. Give p 2 ( x ) , the second Taylor Polynomial of f at (0, 0, 0).
(10) VIII. If S is the solid below the surface (^) z = 4 − x^2 − y^2 and above the x-y plane, set up but
do not evaluate an iterated integral whose value is the triple integral (^) ∫∫∫ S
f ( x , y , z ) dV.
(10) IX. Suppose C is the closed parametrized by f ( t ) = ( 3 cos t , 3 sin t )starting at t = 0 and ending at t = 2 π.
C
FdL.
C
F d x