2020 preTEST4A

1. MAT225 TEST4A Name:
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RVM (Question 1) Application of Euler’s Identity
(1) Find all three roots:​ x3
+ 8 = 0
Applying De Moivre's Theorem.
Represent a complex number ​z​ as a vector on the Complex Plane
If
z = rcis( ) (cos(θ) sin(θ)) cos(θ) isin(θ)θ = r + i = r + r
Then
cis(nθ)zn
= rn
nd z cis( )a n
1
= √
n
r n
θ+2πk
n, )εZ, n , 0⋁ ( k ≥ 2 ≤ k < n
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3. MAT225 TEST4A Name:
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(1) Double Integrals
dxdy∫
1
0
∫
2
1
2
y
e−x2
(1a) Describe the region R over which we are integrating.
(1b) Rewrite the given integral such that the area element A ydx.d = d
(1b) Evaluate the new integral over said region R.
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(2) Double Integrals
dydx∫
3
0
∫
√9−x2
0
x
(2a) Describe the region R over which we are integrating.
(2b) Rewrite the given integral such that the area element A drdθ.d = r
(2b) Evaluate the new integral over said region R.
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(3) Double Integrals
dAS = ∫ ∫√1 x y+ f 2 + f 2
(3) Find the surface area of​ in the first octant.6z = 1 − x2
− y2
(3a) Write the integral such that the area element A ydx.d = d
(3b) Rewrite this integral such that the area element A drdθ.d = r
(3c) Evaluate the new integral to find S.
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9. MAT225 TEST4A Name:
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(4) Triple Integrals
Find the volume of the solid bounded by​ and​6z = 1 − x2
− y2
.z = 0
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11. MAT225 TEST4A Name:
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(5) Line Integrals
Find the work done by the force field ​F​(x,y) = <x,2y> done on a particle
moving along the path C: x = t, y = from the point (0,0) to the point (2,8).t3
F​ = <M,N>
<M,N><dx,dy> =drW = ∫
C
F = ∫
C
dx dy∫
C
M + N
(5a) Parametrize the path C in terms of a single parameter t.
(5b) Write the Line Integral for Work in terms of t.
(5c) Evaluate your integral from t = 0 to t = 2.
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(6) Fundamental Theorem of Line Integrals
​F​ = <M,N> = <2xy, x2
+ y2
>
(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 = using f(x,y) from (5,0) to (0,4) over the path C:dx dy∫
C
M + N
x2
25 +
y2
16 = 1
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7a) Parametrize the path C​1​: ​ ​along the curve from (0,0) to (1,1) in terms of t.y = x2
(7b) Use this parametrization to find the work done:
<M,N><dx,dy> =W = ∫
C1
dx dy∫
C1
M + N
TEST4A page: 15

16. MAT225 TEST4A Name:
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17. MAT225 TEST4A Name:
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7c) Parametrize the path C​2​: ​ ​along the curve from (1,1) to (0,0) in terms of t.y = x
(7d) Use this parametrization to find the work done:
<M,N><dx,dy> =W = ∫
C2
dx dy∫
C2
M + N
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19. MAT225 TEST4A Name:
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(7) Green’s Theorem for Work in the Plane
(x, ) < , =< ,F y = M N > y2
x2
>
C: CCW once about and yy = x2
= x
(7e) Verify Green’s Theorem for Work in the Plane.
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20. MAT225 TEST4A Name:
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21. MAT225 TEST4A Name:
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(8) Surface Integrals
Given the density function ρ
(x, , ) yρ y z = x − 2 + z
find the mass of the planar region S
, 0 , 0z = 4 − x ≤ x ≤ 4 ≤ y ≤ 3
(8a) State the surface area element such that dA = dydx.S dAd = √1 x y+ f 2 + f 2
(8b) Evaluate the surface integral
(x, , )dSS = ∫ ∫
R
ρ y z
TEST4A page: 21

22. MAT225 TEST4A Name:
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23. MAT225 TEST4A Name:
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Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
x nxd
dx
n
= n−1
Trig Functions:
sin(x) os(x)d
dx = c cos(x) in(x)d
dx = − s
tan(x) (x)d
dx = sec2
cot(x) (x)d
dx = − csc2
sec(x) ec(x) tan(x)d
dx = s csc(x) sc(x) cot(x)d
dx = − c
Transcendental Functions:
ed
dx
x
= ex
a n(a) ad
dx
x
= l x
ln(x)d
dx = x
1
log (x)d
dx a = 1
ln(a) x
1
Inverse Trig Functions:
sin (x)d
dx
−1
= 1
√1−x2
cos (x)d
dx
−1
= −1
√1−x2
tan (x)d
dx
−1
= 1
1+x2 cot (x)d
dx
−1
= −1
1+x2
Product Rule:
f(x) g(x) (x) g (x) (x) f (x)d
dx = f ′ + g ′
Quotient Rule:
d
dx
f(x)
g(x) = g (x)2
g(x) f (x) − f(x) g (x)′ ′
Chain Rule:
f(g(x)) (g(x)) g (x)d
dx = f′ ′
Difference Quotient:
f’(x) =​ lim
h→0
h
f(x+h) − f(x)
TEST4A page: 23

24. MAT225 TEST4A Name:
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Reference Sheet: Anti-Derivatives You Should Know Cold!
Power Functions:
dx x∫xn
= n n−1
Trig Functions:
os(x)dx in(x)∫c = s + C in(x)dx os(x)∫s = − c + C
ec (x)dx an(x)∫s 2
= t + C sc (x)dx ot(x)∫c 2
= − c + C
ec(x)tan(x)dx ec(x)∫s = s + C sc(x)cos(x)dx sc(x)∫c = − c + C
Transcendental Functions:
dx e∫ex
= x
+ C dx∫ax
= ax
ln(a)
+ C
dx n(x)∫ x
1
= l + C dx log (x)∫ 1
ln(a) x
1
= a + C
Inverse Trig Functions:
dx sin (x)∫ 1
√1−x2
= −1
+ C dx cos(x)∫ −1
√1−x2
= + C
dx tan (x)∫ 1
1+x2 = −1
+ C dx cot (x)∫ −1
1+x2 = −1
+ C
Integration By Parts (Product Rule):
dv uv du∫u = −∫v + C
Integration By Partial Fractions Example (Quotient Rule):
∫ dx
x(x+1) = ∫ x
Adx
+∫ x+1
Bdx
TEST4A page: 24