2021 preTEST5A Final Review Packet!

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 PQxPR.
(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:
𝑟(𝑡) =< 𝑒
𝑡
𝑐𝑜𝑠(𝑡), 𝑒
𝑡
𝑠𝑖𝑛(𝑡) >
(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
𝑓(𝑥, 𝑦) = 3𝑥
2
+ 2𝑦
2
− 6𝑥 − 4𝑦 + 16
(3a) Find
δ𝑓
δ𝑥
,
δ
2
𝑓
δ𝑥δ𝑥
,
δ
2
𝑓
δ𝑥δ𝑦
(3b) Find
δ𝑓
δ𝑦
,
δ
2
𝑓
δ𝑦δ𝑦
,
δ
2
𝑓
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
𝑔(𝑥, 𝑦) = 𝑥 + 𝑦 − 1 = 0 𝑔𝑥
, 𝑔𝑦
.
(4b) Let and state
𝑑
2
= 𝑓(𝑥, 𝑦) = 𝑥
2
+ 𝑦
2
𝑓𝑥
, 𝑓𝑦
.
(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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(5) The Planimeter Theorem: An Application Of Green’s Theorem for Work
If
𝐶
∮< 0, 𝑥 >•< 𝑑𝑥, 𝑑𝑦 >= ∫
𝑅
∫(1)𝑑𝐴
and
𝐶
∮< 𝑦, 0 >•< 𝑑𝑥, 𝑑𝑦 >= ∫
𝑅
∫(− 1)𝑑𝐴
Then
∫
𝑅
∫ 𝑑𝐴 =
𝐶
∮ 𝑥𝑑𝑦 =−
𝐶
∮ 𝑦𝑑𝑥 =
1
2
𝐶
∮ 𝑥𝑑𝑦 − 𝑦𝑑𝑥 =
1
2
𝐶
∮< 𝑥, 𝑦 >•< 𝑑𝑦, − 𝑑𝑥 >
Is a Flux Integral that evaluates to the area of the region R bounded by the curve C.
(5a) Evaluate this Flux Integral parametrically over the path C:
𝑥
2
+
𝑦
2
4
= 1
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11. MAT225 TEST5A Name:
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(5) The Planimeter Theorem: An Application Of Green’s Theorem for Work
If
𝐶
∮< 0, 𝑥 >•< 𝑑𝑥, 𝑑𝑦 >= ∫
𝑅
∫(1)𝑑𝐴
and
𝐶
∮< 𝑦, 0 >•< 𝑑𝑥, 𝑑𝑦 >= ∫
𝑅
∫(− 1)𝑑𝐴
Then
∫
𝑅
∫ 𝑑𝐴 =
𝐶
∮ 𝑥𝑑𝑦 =−
𝐶
∮ 𝑦𝑑𝑥 =
1
2
𝐶
∮ 𝑥𝑑𝑦 − 𝑦𝑑𝑥 =
1
2
𝐶
∮< 𝑥, 𝑦 >•< 𝑑𝑦, − 𝑑𝑥 >
Is a Flux Integral that evaluates to the area of the region R bounded by the curve C.
(5b) Apply Green’s Theorem for Flux over the path C:
𝑥
2
+
𝑦
2
4
= 1
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13. MAT225 TEST5A Name:
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(6) Green’s Theorem for Work in the Plane
𝐹(𝑥, 𝑦) =< 𝑀, 𝑁 >=< 𝑥𝑦, 𝑥 + 𝑦 >
C: CCW once around 𝑥
2
+ 𝑦
2
= 1
<M,N><dx,dy> =
𝑊 =
𝐶
∫
𝐶
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
(6a) Parametrize the path C in terms of t.
(6b) Use this parametrization to find the work done.
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(6) Green’s Theorem for Work in the Plane
𝐹(𝑥, 𝑦) =< 𝑀, 𝑁 >=< 𝑥𝑦, 𝑥 + 𝑦 >
C: CCW once around 𝑥
2
+ 𝑦
2
= 1
<M,N><dx,dy> =
𝑊 =
𝐶
∫
𝐶
∫ 𝑀𝑑𝑥 + 𝑁𝑑𝑦
(6c) Confirm Green’s Theorem for Work.
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(7) The Divergence Theorem for Flux in Space
𝐹(𝑥, 𝑦, 𝑧) =< 𝑃, 𝑄, 𝑅 >=< 𝑥𝑧, 𝑦𝑧, 2𝑧
2
>
S: Bounded by and
𝑧 = 1 − 𝑥
2
− 𝑦
2
𝑧 = 0
𝐹𝑙𝑢𝑥 = ∫
𝑆
∫ 𝐹 𝑛
^
𝑑𝑆
(7a) Find the Flux of the vector field F through this closed surface.
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(7) The Divergence Theorem for Flux in Space
𝐹(𝑥, 𝑦, 𝑧) =< 𝑃, 𝑄, 𝑅 >=< 𝑥𝑧, 𝑦𝑧, 2𝑧
2
>
S: Bounded by and
𝑧 = 1 − 𝑥
2
− 𝑦
2
𝑧 = 0
𝐹𝑙𝑢𝑥 = ∫
𝑆
∫ 𝐹 𝑛
^
𝑑𝑆
(7b) Confirm the Divergence, aka Gauss-Green, Theorem.
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(8) Stokes' Theorem for Work in Space
𝐹(𝑥, 𝑦, 𝑧) =< 𝑃, 𝑄, 𝑅 >=<− 𝑦 + 𝑧, 𝑥 − 𝑧, 𝑥 − 𝑦 >
S: and
𝑧 = 9 − 𝑥
2
− 𝑦
2
𝑧 ≥ 0
(8a) Evaluate parametrically: W=
𝐶
∮ 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧
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(8) Stokes' Theorem for Work in Space
𝐹(𝑥, 𝑦, 𝑧) =< 𝑃, 𝑄, 𝑅 >=<− 𝑦 + 𝑧, 𝑥 − 𝑧, 𝑥 − 𝑦 >
S: and
𝑧 = 9 − 𝑥
2
− 𝑦
2
𝑧 ≥ 0
(8b) Verify Stokes’ Theorem.
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