preTEST3A Double Integrals Solved

1. MAT225 TEST3A Name:
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RVF (Question 1) s(t), v(t), a(t)
(1) A bullet is shot upward from the surface of the Moon such that
(t) 60t .8t
y = 1 − 0 2
[y] = meters, [t] = seconds, t≥0.
(1a) Find y‘(t)
(1b) Calculate y‘(0)
(1c) Solve for t when y’(t) = 0.
(1d) What is the maximum height?
(1e) How fast is the bullet moving when it hits the ground?
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3. MAT225 TEST3A Name:
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RVG (Question 2) a(t), v(t), s(t)
(2) Find f(x) such that f(x) is a function defined for all with these properties:
−
x > 5
(i) f ”(x) =​
1
3√x+5
(ii) tangent line to the graph of f at (4,2) makes a 45° angle with the x-axis.
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(1) Dot Products
Given the triangle ABC, A(1,1), B(4,1) and C(4,4):
(1a) Find the components of the vectors ​AB​ and ​AC​.
(1b) Calculate the magnitudes of the vectors ​AB​ and ​AC​.
(1c) Use the Dot Product of ​AB​ and ​AC​ to find the measure of angle A.
(1d) What is the area of ABC?
Δ
(1e) Let​ , does​ equal the triangle area?
s = 2
a+b+c
√s(s )(s )(s )
− a − b − c
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(2) Cross Products
Given the triangle ABC, A(1,1), B(4,1) and C(4,4):
(2a) Find the following Cross Products: ​OB​ x ​OC​, ​OC​ x ​OA​, ​OA​ x ​OB​.
(2b) Sum the following Cross Products: ​OB​ x ​OC​, ​OC​ x ​OA​, ​OA​ x ​OB​.
(2c) What is the magnitude of the sum of these Cross Products?
(2d) Is there a relationship between this magnitude and the triangle area?
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(3) Determinants
Let the vectors ​u​ = <1,2,3>, ​v​=<1,0,1> and ​w​=<2,3,4>:
(3a) Find ​u​ • (​v​ x ​w​).
(3b) What does this value measure?
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11. MAT225 TEST3A Name:
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(4) Determinants
Let the vectors ​u​ = <1,2,3>, ​v​=<1,0,1> and ​w​=<2,3,4>:
(4a) Find det(​u​,​v​,​w​).
(4b) What does this value measure?
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(5) Iterated Integrals
Consider the area in the xy-plane bounded by: ,
− √4 − y2 ≤ x ≤ √4 − y2 .
0 ≤ y ≤ 2
ydx
A = ∫
2
−2
∫
√4−y2
0
d
(5a) Draw this region labeling a vertical Riemann Rectangle with thickness dx.
(5b) Explain how to setup this integral in terms of dydx to calculate the area.
(5c) Evaluate your integral in terms of dydx.
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(5) Iterated Integrals
Consider the area in the xy-plane bounded by: ,
− √4 − y2 ≤ x ≤ √4 − y2 .
0 ≤ y ≤ 2
ydx
A = ∫
2
−2
∫
√4−y2
0
d
(5d) ReWrite this integral in terms of dxdy to calculate the area.
(5e) ReEvaluate your integral in terms of dxdy.
(5f) ReWrite this integral in terms of to calculate th area.
drdθ
r
(5g) ReEvaluate your integral in terms of​ drdθ.
r
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(6) Iterated Integrals
Find the volume of the solid bounded by the surface f(x,y)=1-xy above the triangle
bounded by y=x, y=1 and x=0.
(1 y) dxdy
V = ∫
1
0
∫
y
0
− x
(6a) Explain how to set up an integral to calculate this volume in terms of dxdy.
(6b) Evaluate your double integral.
(6c) ReWrite the integral terms of dydx.
(6d) Evaluate your double integral.
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(7) Line Integrals
Calculate a line integral to find the mass of a wire given density and the path C:
ρ
Density Function (x, ) y
ρ = F y = x
Along the path C: r(t)=<4t, 3t> such that 0 ≤ t ≤ 1
(7a) Find ds=|r’(t)|dt
(7b) Write the line integral in terms of t.
ds
∫
C
F
(7c) Evaluate your integral.
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(8) Work Done By A Conservative Field
Given the Vector Field​ (x, ) < x y , 2x xy
F y = a 2
+ y3
+ 1 3
+ b 2
+ 2 >
(8a) Find the values of and b for which F is conservative.
(8b) Using these values of a and b, find f(x,y) such that F = gradient(f).
(8c) Find the work done through F along the curve C:
(t) cos(t), y(t) sin(t), 0
x = et
= et
≤ t ≤ π
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22. MAT225 TEST3A Name:
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23. MAT225 TEST3A Name:
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Reference Sheet: Derivatives You Should Know Cold!
Power Functions:
x nx
d
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:
e
d
dx
x
= ex a n(a) a
d
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)
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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
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