POTENTIAL POPPER QUESTIONS
Work the poppers in your notes so you have more examples to study.
1. Eliminate the parameter and find a corresponding rectangular
equation: x = 3t2 and y = 2t + 1
A) 2x2 + 3y2 – 1 = 0
B) 2x – 3y +3 = 0
C) 3y2 – 4x + 1 = 0
D) 3y2 – 4x – 6y + 3 = 0
E) none of these
POTENTIAL POPPER QUESTIONS
2
2. If x = t − 1 and y = 2e , then
a)
et
t
t
b)
2e t
t
dy
=
dx
t
c)
e
t2
d)
4e t
2t − 1
e)
et
POTENTIAL POPPER QUESTIONS
3. If a particle moves in the xy-plane so that at any time t > 0, its position
vector is ln t 2 + 2t , 2t 2 , then at time t = 2, its velocity vector is
( (
⎛3 ⎞
⎜ , 8⎟
⎝4 ⎠
⎛ 5
⎞
−
,
4
⎜
⎟
⎝ 16 ⎠
a)
)
b)
)
⎛3 ⎞
⎜ , 4⎟
⎝4 ⎠
⎛1 ⎞
c) ⎜ , 8 ⎟
⎝8 ⎠
⎛1 ⎞
d) ⎜ , 4 ⎟
⎝8 ⎠
e)
POTENTIAL POPPER QUESTIONS
4. If a particle moves in the xy-plane so that at time t > 0 its position
π
⎛ ⎛
π⎞ 2 ⎞
vector is ⎜ sin ⎜ 3t − ⎟ , 3t ⎟, then at time t = , the velocity vector is
2⎠
2
⎝ ⎝
⎠
a) ( −3, 3π )
b)
( −1, 3π ) c)
( −1, 2π)
d)
( 3, 2π )
e)
( 3 , 3π )
POTENTIAL POPPER QUESTIONS
5. Find the equation of the tangent line at t = 1 to the curve given by
the parametric equations x ( t ) = 3t 2 − 4t + 2 , y ( t ) = t 3 − 4t .
a) x + 2y = 5
b) x + 2y = –5
c) 2x – y = 5
d) 2x – y = –5
e) 2x + y = 5
Suppose a particle is traveling on a path represented by a portion of the xy plane.
In this case, assume that the y-axis represents the up/down direction. Given this
information, suppose the position of the particle is given at time t in seconds by
(
)
r ( t ) = t , − 16t 2 + 40t + 10 for t > 0 until the particle strikes the ground.
Give the time when the particle strikes the ground, and determine the velocity and
acceleration of the particle at this time.
Arc Length and Speed
Section 9.8
Velocity and Speed (magnitude of velocity)
If the position of a particle at time t is given by
r(t) = (x(t), y(t))
then the velocity is given by
r’(t) = v(t) = (x’(t), y’(t))
and the speed is given by
speed =
v( t )
2
= ⎡⎣ x ' ( t ) ⎤⎦ + ⎡⎣ y ' ( t ) ⎤⎦
2
A particle is traveling on an elliptic path in the xy-plane so that its position
at time t is given by r ( t ) = 2 cos ( t ) ,3 sin ( t ) .
(
)
Give the position, velocity and
speed of the particle at time t = π/4.
3
2
1
−3
−2
−1
1
−1
−2
−3
−4
2
3
4
If a curve C is given parametrically by
(x(t), y(t)) for a ≤ t ≤ b
and x’(t) and y’(t) are continuous functions,
then the length of the curve is given by
b
L (c ) = ∫
a
[x' (t )]2 + [y' (t )]2 dt
Give an integral which represents the length of the curve given
parametrically by 2 cos ( t ) ,3 sin ( t ) for 0 ≤ t ≤ 2π.
(
)
3
2
1
−3
−2
−1
1
−1
−2
−3
−4
2
3
4
Give an integral which represents the length of the curve given
parametrically by
2 cos ( 3t ) ,3 sin ( 4 t ) for 0 ≤ t ≤ 2π
(
)
3
2
1
−3
−2
−1
1
−1
−2
−3
−4
2
3
4
Give a formula for the length of the “curve” given by the graph of
f (x) = 2x +1 for 1 ≤ x ≤ 3.
How can we find the length of the curve given by the graph of f (x) for
a ≤ x ≤ b? L( c ) =
b
∫a
2
1 + ⎡⎣ f ' ( x ) ⎤⎦ dx
Verify that the formula works for the previous example.
Give a formula for the length of the curve given by the graph of
f (x) = x2 for –1 ≤ x ≤ 1.
How can we find the length of a polar curve?
L( c ) =
β
∫
α
2
2
⎡ ρ ( θ ) ⎤ + ⎡ ρ' ( θ ) ⎤ d θ
⎣
⎦
⎣
⎦
The equations in Exercises 19–24 give the position of a particle at each time t during the time interval specified. Find the initial speed of the particle, the terminal speed, and the distance traveled.
23. x(t) = et sin t, y(t) = et cos t, from t = 0 to t = π. Find the length of the polar curve. 34. r = 1 + cos θ from θ = 0 to θ = 2π. Summary:
b
L(c) = ∫ 1 + [ f ' (x )] dx
2
a
b
L (c ) = ∫
[x' (t )] + [y' (t )] dt
2
⎡ x' ( t ) ⎤ + ⎡ y' ( t ) ⎤
⎣
⎦
⎣
⎦
β
L (c ) = ∫
α
x ∈ [a. b]
Arc length (parametric)
t ∈ [a. b]
2
a
2
Arc length (rectangular)
2
[ρ (θ )]2 + [ρ ' (θ )]2 dθ
Speed (parametric)
Arc length (polar)
θ ∈ [α, β]