The document contains a 24-page math test with 9 multi-part questions covering topics like planes, vectors, optimization, line integrals, and surface integrals. The test involves computing equations, derivatives, integrals, and verifying theorems regarding vector fields, flux, and work. Students are instructed to show all work algebraically where possible to solve problems involving geometry, calculus, and vector calculus concepts.

1. MAT225 TEST5A Name:
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(1) Equation of a Plane
Let P(0,1,0), Q(2,1,3), R(1,-1,2).
(1a) Compute ​PQ​x​PR​.
(1b) Find the equation of the plane through P, Q and R in the form ax+by+cz=d.
(1c) What is the angle formed by this plane and the xy-plane?
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(2) Velocity Vectors
Consider the curve given by the position vector:
(t) < cos(t), sin(t)r = et
et
>
(2a) Find the velocity vector for this trajectory.
(2b) Find the speed for a particle moving along this trajectory.
(2c) What is the angle between the position vector and the velocity vector?
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(3) Optimization
(x, ) x y x y 6f y = 3 2
+ 2 2
− 6 − 4 + 1
(3a) Find​ , ,
δf
δx
δ f2
δxδx
δ f2
δxδy
(3b) Find​ , ,
δf
δy
δ f2
δyδy
δ f2
δyδx
(3d) Classify and determine the relative extrema of the f(x,y).
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(4) LaGrange Multipliers
Minimize the square of the distance from the line x + y = 1 to the point (0,0).
(4a) Let and state gx, gy.(x, )g y = x + y − 1 = 0
(4b) Let and state fx, fy.(x, )d2
= f y = x2
+ y2
(4c) State and solve a system of 3 equations for x,y and .λ
(4d) What is the minimum value of d?
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8. MAT225 TEST5A Name:
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(5) Double Integrals
e dydxM = ∫
1
0
∫
2
2x
4 y2
(5a) Find the region R over which we are integrating in the xy-plane.
(5b) Rewrite the given integral in terms of dxdy.
(5c) Evaluate this new integral to find the mass M of the planar region R.
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(6) Fundamental Theorem of Line Integrals
​F​ = <M,N> = os(x) sin(y), in(x) cos(y)< c s >
(6a) Show that ​F​ is a Conservative Vector Field.
(6b) Find the Potential Function f(x,y) for the Vector Field ​F​.
(6c) Evaluate W =​ over the path C:dx dy∫
C
M + N
ine Segment from (0,− ) to ( , )L π 2
3π
2
π
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< y,F y = M N > x x + y >
C: CCW once around x2
+ y2
= 1
<M,N><dx,dy> =W = ∫
C
dx dy∫
C
M + N
(7a) Parametrize the path C in terms of t.
(7b) Use this parametrization to find the work done.
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< y,F y = M N > x x + y >
C: CCW once around x2
+ y2
= 1
<M,N><dx,dy> =W = ∫
C
dx dy∫
C
M + N
(7c) Confirm Green’s Theorem for Work.
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(8) The Divergence Theorem for Flux in Space
(x, , ) < , , =< z, z, zF y z = P Q R > x y 2 2
>
S: Bounded by andz = 1 − x2
− y2
z = 0
lux n dSF = ∫∫
S
F
︿
(8a) Find the Flux of the vector field F through this closed surface.
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(8) The Divergence Theorem for Flux in Space
(x, , ) < , , =< z, z, zF y z = P Q R > x y 2 2
>
S: Bounded by andz = 1 − x2
− y2
z = 0
lux n dSF = ∫∫
S
F
︿
(8b) Confirm the Divergence, aka Gauss-Green, Theorem.
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(9) Stokes' Theorem for Work in Space
(x, , ) < , , =<− , ,F y z = P Q R > y + z x − z x − y >
S: andz = 9 − x2
− y2
z ≥ 0
(9a) Evaluate​ W= dx dy dz∮
C
P + Q + R
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23. MAT225 TEST5A Name:
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(9) Stokes' Theorem for Work in Space
(x, , ) < , , =<− , ,F y z = P Q R > y + z x − z x − y >
S: andz = 9 − x2
− y2
z ≥ 0
(9b) Verify Stokes’ Theorem.
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