Curve Parametrized - Multivariable - Exam, Exams of Mathematics

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

2012/2013

Uploaded on 03/07/2013

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NAME_______________________________________
I____II____III____IV____V____VI____VII____VIII____ IX____ X____ TOTAL __________
(10) (10) (10) (10) (10) (10) (10) (10) (10) (10) (100)
March 12, Mathematics 206a Mr. Haines
2009 Multivariable Calculus
Examination #2
(10) I. Let C be the curve parametrized by
=2
,
3
22
,)(
2
2
3
t
tttf starting at t = 0 and ending at
t = 2 . Give an equation for the plane that passes through )1(
f
and is
perpendicular to the tangent to C at that point.
(10) II. For the vector field
(
)
1,sin,),,( zxyzyx =F, compute
A. div (F) =
B. curl (F) =
pf3
pf4
pf5

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NAME_______________________________________

I____II____III____IV____V____VI____VII____VIII____ IX____ X____ TOTAL __________

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.

(10) II. For the vector field F ( x , y , z )=( xy ,−sin z , 1 ), compute

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.

A. If F : ℜ^2 → ℜ with F ( x , y ) = x + y , Compute ∫

C

FdL.

B. If F : ℜ^2 → ℜ^2 with F ( x , y ) = x i + y j. Compute ∫ •

C

F d x