This document contains a 24-page math test with 8 multi-part questions covering topics like plane equations, velocity vectors, optimization, Lagrange multipliers, Green's theorem, divergence theorem, and Stokes' theorem. The test involves computing equations, vectors, derivatives, integrals, and verifying theorems across different coordinate systems. Students are instructed to show all work algebraically where possible.

1. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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?
TEST5A page: 1

2. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 2

3. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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?
TEST5A page: 3

4. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 4

5. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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).
TEST5A page: 5

6. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 6

7. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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?
TEST5A page: 7

8. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 8

9. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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
TEST5A page: 9

10. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 10

11. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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
TEST5A page: 11

12. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 12

13. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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.
TEST5A page: 13

14. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 14

15. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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.
TEST5A page: 15

16. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 16

17. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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.
TEST5A page: 17

18. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 18

19. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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.
TEST5A page: 19

20. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 20

21. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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
TEST5A page: 21

22. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 22

23. MAT225 TEST5A Name:
Show all work algebraically if possible.
(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.
TEST5A page: 23

24. MAT225 TEST5A Name:
Show all work algebraically if possible.
TEST5A page: 24