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(Solved) de la cruz (cld2889) - HW06 - berg - (54070) This print-out should have 20 questions. Multiple-choice questions may continue on the next column or...


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de la cruz (cld2889) – HW06 – berg – (54070)
This print-out should have 20 questions.
Multiple-choice questions may continue on
the next column or page – find all choices
before answering.
001 10.0 points 1 Determine a Cartesian equation for the
curve given in parametric form by
x(t) = 4e2t , y(t) = 2e−t . 1. x2 y = 8 Determine A so that the curve
y = 8x + 20
can be written in parametric form as
x(t) = t − 3 , y(t) = At − 4 . 1. A = 9
2. A = 8 2. xy 2 = 16
3. xy 2 = 32
x2
= 8
y
x
5. 2 = 32
y
x
6. 2 = 16
y
4. 004
3. A = −9 Find parametric equations representing the
line segment joining P (−2, 4) to Q(4, −3) as 4. A = 7 (x(t), y(t)) , 5. A = −7 P = (x(0), y(0)) , 002 10.0 points Find a Cartesian equation for the curve
given in parametric form by
x(t) = 3 cos 4t , y(t) = 2 sin 4t . x2 y 2
1
−
=
9
4
36 1
x2 y 2
+
=
4
9
36 4. 4x2 − 9y 2 = 36
2 5. 4x + 9y
6. 2 = 36 003 1. (−2 + 6t2 , 4 − 7t2 )
2. (4 − 6t2 , −3 + 7t2 )
Ï€t
Ï€t 
, −3 + 7 cos2
2
2

Ï€t 
Ï€t
4. 4 − 6 sin2 , 4 − 7 cos2
2
2
 −2 + 6 sin2 5. (−2 + 6t, 4 − 7t)
6. (4 − 6t, −3 + 7t)
005 x2 y 2
1
−
=
4
9
36 Find
10.0 points Q = (x(1), y(1)) and x(t), y(t) are linear functions of t. 3. 2. 9x2 + 4y 2 = 36
3. 0 ≤ t ≤ 1, where 6. A = −8 1. 10.0 points 10.0 points dy
when
dx x(t) = 3tet , y(t) = 4t + et . de la cruz (cld2889) – HW06 – berg – (54070) 1. dy
4 + et
=
dx
3et (1 + t) 2. dy
3et (1 − t)
=
dx
4 − et dy
4 + et
5.
=
dx
3et (1 − t)
dy
=
dx 2. y = 3x + 1
2 3
1
4. y = − x +
2
2 dy
3et (1 + t)
4.
=
dx
4 − et 6. 7
2 3
7
3. y = − x +
2
2 4 − et
dy
=
3.
dx
3et (1 − t) 3et (1 + 1. y = 3x + t) 5. y = 7
3
x+
2
2 6. y = 3
1
x+
2
2 4 + et
008
006 10.0 points Find an equation for the tangent line to the
curve given parametrically by
√ x(t) = 4e t , y(t) = t − ln t 2. y =
3. y = 1
x−2
2e
1
x+5
2e
1
x−3
e 1
4. y + x = 5
e
5. y + x(t) = 2 + t2 , y(t) = t2 + 2t3 . 5t
d2 y
=
1.
2
dx
6
2. d2 y
3
=
dx2
2t 3. d2 y
5
=
2
dx
6t 4. d2 y
2t
=
2
dx
3 d2 y
3t
5.
=
2
dx
2 1
x = 3
2e 6. 1
6. y + x = 3
e d2 y
3
=
dx2
t
009 007 10.0 points d2 y
Find 2 for the curve given parametrically
dx
by 3 at the point P = (x(1), y(1)).
1. y = 2 10.0 points 10.0 points Find an equation for the tangent line to the
curve given parametrically by
x(t) = e2t ,
at the point P (1, 2). y(t) = 4t2 + 3t + 2 
Find all values of t in − π2 , π2 for which the
tangent line to the graph of
x(t) = t + sin 2t ,
is vertical. y(t) = t − sin 2t , de la cruz (cld2889) – HW06 – berg – (54070)
1. t =
2. t =
3. t =
4. t =
5. t =
6. t = π
2Ï€
,
3
3
Ï€
2Ï€
− ,
3
3
Ï€
Ï€
− ,
6
6
Ï€
Ï€
− ,
3
3
5Ï€
Ï€
− , −
6
6
Ï€
5Ï€
,
6
6
010 011 y
P
x 10.0 points 1
,
t 10.0 points The curve traced out by a point P on the
circumference of a circle as the circle rolls
along a straight line shown in The region A enclosed by the line y = 3 and
the graph of the curve given parametrically by
x(t) = t − 3 y(t) = 2t + 1
t is called a Cycloid and the shaded region is
the region, A, below an arch.
If the circle has radius R, the cycloid is
given parametrically by
x(t) = R(t − sin t) , y(t) = R(1 − cos t) . Find the area of A when R = 5. is similar to the shaded region in 1. area(A) = 72π y 2. area(A) = 71π
3. area(A) = 74Ï€
x 4. area(A) = 73Ï€
5. area(A) = 75Ï€ Determine the area of A.
1. area(A) = 13
+ 3 ln 2
4 5
2. area(A) = + 3 ln 2
4
5
3. area(A) = − 3 ln 2
4
9
4. area(A) = − 3 ln 2
4
13
− 3 ln 2
5. area(A) =
4
9
6. area(A) = + 3 ln 2
4 012 10.0 points Which one of the following integrals gives
the length of the parametric curve
x(t) = 2t2 , 1. I = 2 Z 12 p Z 6 0 2. I = 2 0 3. I = Z 4. I = Z y(t) = t , 0 16t2 + 1 dt |16t2 + 1| dt 6p 16t2 + 1 dt 6
0 |16t2 + 1| dt 0 ≤ t ≤ 12 . de la cruz (cld2889) – HW06 – berg – (54070)
Z 5. I = 2 0 6. I = Z 0 2 |16t + 1| dt 12 p 013 16t2 4 rs 12 t
u + 1 dt
−4 10.0 points 4 2 b c
b
u −2 Determine the arc length of the astroid
shown in
y r 2 4 −2
−4 x 1. P : b c
Q: b t
R: u 2. t
P : u Q: b c
R: b 3. c
P : b Q: b t
R: u 4. c
P : b t
Q: u R: b 5. t
P : u c
Q: b R: b 6. P : b t
Q: u c
R: b and given parametrically by
x(t) = 3 cos3 t , y(t) = 3 sin3 t . 1. arc length = 3Ï€ 015 2. arc length = 18 Find an equation for the tangent line to the
graph of
r = cos θ − 2 sin θ 3. arc length = 3
4. arc length = 9π at θ = π/4.
5. arc length = 9
1. y =
6. arc length = 18Ï€
014 10.0 points Locate the points given in polar coordinates
by
3 
P 2, π ,
4
 among 1 
Q 2, π ,
6
 10.0 points  1 
R 1, π
2 1
1
x−
2
2 2. y = 2x − 1
2 3. y = 2x + 1
4. y = 1
x+1
2 5. y = 1
1
x+
2
2 de la cruz (cld2889) – HW06 – berg – (54070)
the polar function 1
6. y = 2x +
2
016 r = 1 + 2 cos(2θ) ?
10.0 points Find the Cartesian coordinates, (a, b),
of the point given in polar coordinates by
P (4, π/3).
1. (a, b) = (2, −4)
√
2. (a, b) = (−2, 2 3)
√
3. (a, b) = (4, 4 3)
√
4. (a, b) = (4 3, 4)
√
5. (a, b) = (2 3, 2)
√
6. (a, b) = (−4, 2 3)
√
7. (a, b) = (2 3, −2)
√
8. (a, b) = (2, 2 3)
017 1. 2. 10.0 points Find a polar equation for the curve given
by the Cartesian equation 3. y 2 = 3x .
1. r = 3 sec θ tan θ
2. 3r = sec θ cot θ
3. 3r = csc θ cot θ
4. 3r = sec θ tan θ
5. r = 3 csc θ tan θ
6. r = 3 csc θ cot θ
018 10.0 points Which one of the following is the graph of 4. 5 de la cruz (cld2889) – HW06 – berg – (54070)
2.
5. 3. 6. 019 10.0 points 4. Which one of the following could be the
graph of the polar function r = 1 − 2 sin θ ? 1. 5. 6 de la cruz (cld2889) – HW06 – berg – (54070)
6. 020 10.0 points Find the slope of the tangent line to the
graph of
r = 3eθ − 1
at θ = π/4.
1. slope = 3eπ/4
2. slope = 6e−π/4
3. slope = 6eπ/4 − 1
4. slope = 6eπ/4 + 1
5. slope =
6. slope = 1
3eπ/4
1 −1 3eπ/4 + 1 7

 


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