PARAMETRIC EQUATIONS and VECTORS
1 — Graphing parametric equations and eliminating the parameter
dy
d2y
and
and
dx
dx 2
evaluating them for a given value of t, finding points of horizontal and
vertical tangency, finding the length of an arc of a curve
2 — Calculus of parametric equations: Finding
3 — Review of motion along a horizontal and vertical line. (The students
have studied this topic earlier in the year.)
4, 5 6 — Particle motion along a curve (vectors):
•
Finding the velocity and acceleration vectors when given the position vector;
•
Given the components of the velocity vector and the position of the particle at one
value of t, finding the position of the particle at a different value of t;
•
Finding the slope of the path of the particle for a given value of t;
•
Writing an equation for the tangent line to the curve for a given value of t;
•
Finding the values of t at which the line tangent to the path of the particle
is horizontal or vertical;
•
Finding the speed of the particle; and
•
Finding the distance traveled by the particle for a given interval of time.
1: Graphing Parametric Equations and Eliminating the Parameter
Directions: Make a table of values and sketch the curve, indicating the direction of your
graph. Then eliminate the parameter.
(a) x = 2t − 1 and y = 1 − t
Solution: First make a table using various values of t, including negative numbers,
positive numbers and zero, and determine the x and y values that correspond to
these t values.
t
x
y
–2
–1
0
1
2
3
–5
–3
–1
1
3
5
3
2
1
0
–1
–2
(
)
Plot the ordered pairs x, y , drawing an arrow on the graph to indicate its
direction as t increases.
To eliminate the parameter, solve x = 2t − 1 for t =
substitute t =
x +1
1
1
or t = x + . Then
2
2
2
1
1
1
1
x + in place of t in the equation y = 1 – t to get y = − x + .
2
2
2
2
Look at the graph of the parametric equations to see if this equation matches the
graph, and observe that it does.
Vectors
(b) x = t , y = t + 1
Solution: Since x = t , we can use only nonnegative values for t.
t
x
y
0
1
4
9
0
1
2
3
1
2
5
10
To eliminate the parameter, solve x = t for t = x 2 . Then substitute t = x 2 into
y’s equation so that y = x 2 + 1. To make this equation match the graph, we must
restrict x so that it is greater than or equal to 0. The solution is y = x 2 + 1, x ≥ 0 .
t
, –2 ≤ t ≤ 3
2
Solution: First make a table using t values that lie between –2 and 3, and
determine the x and y values that correspond to these t values.
(c) x = t2 – 2 and y =
t
x
–2
–1
0
1
2
3
2
–1
–2
–1
2
7
y
–1
−1
2
0
1
2
1
3
2
t
3
for t to find that t = 2 y , −1 ≤ y ≤ . Then
2
2
substitute 2y in place of t in the other equation so that x = 4 y 2 − 2 . To make this
To eliminate the parameter, solve y =
Vectors
3
equation match the graph, we must restrict y so that it lies between −1 and . The
2
3
solution is x = 4 y 2 − 2, − 1 ≤ y ≤ .
2
(d) x = 3 + 2 cos t , y = − 1 + 3sin t
t
0
π
2
π
3π
2
2π
x
5
3
1
3
5
y
–1
2
–1
–4
–1
Solution: To eliminate the parameter, solve for cos t in x’s equation to get
x−3
y +1
cos t =
and sin t in y’s equation to get sin t =
. Substitute into the
2
3
2
trigonometric identity cos t + sin
2
( x − 3)
t = 1 to get
2
( y + 1)
+
2
= 1 . Discuss
4
9
with the students the fact that this is an ellipse centered at the point (3, –1) with a
horizontal axis of length 4 and a vertical axis of length 6.
1 Homework
Make a table of values and sketch the curve, indicating the direction of your graph. Then
eliminate the parameter. Do not use your calculator.
1. x = 2t + 1 and y = t − 1
6. x = 2t and y = t − 1
2. x = 2t and y = t 2 , − 1 ≤ t ≤ 2
7. x = t and y =
3. x = 2 − t 2 and y = t
1
t2
8. x = 2cos t − 1 and y = 3sin t + 1
4. x = t and y = t − 3
9. x = 2sin t − 1 and y = cos t + 2
5. x = t − 2 and y = 1 − t
10. x = sec t and y = tan t
2: Parametric Equations and Calculus
Example 1 (no calculator): Given the parametric equations x = 2 t and y = 3t 2 − 2t , find
dy
d2y
.
and
dx
dx 2
Solution:
To find
Since
dy
, we must differentiate both of the parametric equations with respect to t.
dx
1
−
dy d
1 −1
dx d ⎡
= ⎡⎣3t 2 − 2t ⎤⎦ = 6t − 2 and
=
2 t ⎤ = 2 ⋅ t 2 = t 2 , then
⎦
dt dt
dt dt ⎣
2
dy
3
1
dy dt 6t − 2
2 − 2t 2 .
6
=
=
=
t
−1
dx dx
t 2
dt
To find
d2y
dy
dx
, we must differentiate
with respect to t and divide by
:
2
dx
dt
dx
d ⎡ dy ⎤ d ⎡ 3
1
2 − 2t 2 ⎤
1
1
−1
−1
6
t
⎢
⎥
2
⎥ 9t 2 − t 2 9t 2 − t 2
d y dt ⎣ dx ⎦ dt ⎣⎢
⎦
=
=
=
= 9t − 1 .
=
−1
dx
dx
dx
dx 2
t 2
dt
dt
dt
Example 2 (no calculator): Given the parametric equations x = 4cos t and y = 3 sin t, write
3π
an equation of the tangent line to the curve at the point where t =
.
4
Solution:
dy
d
⎡⎣3sin t ⎤⎦
3cos t
dy dt
3
dt
=
=
=
= − cot t .
− 4 siin t
dx dx d
4
⎡⎣ 4 cos t ⎤⎦
dt dt
⎛
⎛ 2⎞ 3 2
3π dy
3
3
2⎞
.
When t =
,
= − −1 = , x = 4 ⎜ −
⎟ = − 2 2 and y = 3⎜
⎟=
2
4
4 dx
4
⎝ 2 ⎠
⎝ 2 ⎠
( )
(
)
3 2 3
= x+2 2 .
2
4
(Remind students that they may leave their tangent line equations in point-slope form.)
Therefore the tangent line equation is: y −
Vectors
Example 3 (no calculator): Find all points of horizontal and vertical tangency given the
parametric equations x = t 2 + t , y = t 2 − 3t + 5.
Solution:
dy
d 2
⎡t − 3t + 5⎤
⎣
⎦ 2t − 3
dy dt
=
= dt
=
d 2
2t + 1
dx dx
⎡t + t ⎤
⎣
⎦
dt
dt
.
A horizontal tangent will occur when
dy
= 0 , which happens when 2t – 3 = 0
dx
horizontaltangent
tangentoccurs
occurs at
at tt ==
(and 2t + 1 ≠ 0 ), sosoa ahorizontal
3
3
. Substituting t = into the given
2
2
⎛ 15 11⎞
equations, we find that a horizontal tangent will occur at ⎜ , ⎟ . A vertical tangent
⎝ 4 4⎠
dy
is undefined, which happens when 2t + 1 = 0 (and 2t – 3 ≠ 0), so a
will occur when
dx
1
1
vertical tangent occurs at t = − . Substituting t = − into the given equations, we find
2
2
⎛ 1 27 ⎞
that a vertical tangent will occur at ⎜ − , ⎟ .
⎝ 4 4⎠
Example 4 (no calculator): Set up an integral expression for the arc length of the curve
given by the parametric equations x = t2 + 1, y = 4t3 – 1, 0 ≤ t ≤ 1. Do not evaluate.
Solution:
For parametric equations x = f(t) and y = g(t), a ≤ t ≤ b, the formula for arc length is:
s=
b
∫a
2
2
⎛ dx ⎞ ⎛ dy ⎞
⎜⎝ dt ⎟⎠ + ⎜⎝ dt ⎟⎠ dt .
For our problem,
dx d 2
dy d
= ⎡⎣t + 1⎤⎦ = 2t and
= ⎡⎣ 4t 3 − 1⎤⎦ = 12t 2 , so the arc
dt dt
dt dt
length is given by the integral expression s =
∫ 0 ( 2t )
1
2
( )
+ 12t 2
2
dt =
1
∫0
4t 2 + 144t 4 dt .
2 Homework
Do not use your calculator on the following problems.
dy
d2y
On problems 1–5, find
.
and
dx
dx 2
1. x = t 2 , y = t 2 + 6t + 5
2. x = t 2 + 1, y = 2t 3 − t 2
3. x = t , y = 3t 2 + 2t
4. x = ln t , y = t 2 + t
5. x = 3sin t + 2, y = 4 cos t − 1
6. A curve C is defined by the parametric equations x = t 2 + t − 1, y = t 3 − t 2 . Find:
dy
(a)
in terms of t.
dx
(b) an equation of the tangent line to C at the point where t = 2.
7. A curve C is defined by the parametric equations x = 2 cos t , y = 3sin t . Find:
(a)
dy
in terms of t.
dx
(b) an equation of the tangent line to C at the point where t =
π
.
4
On problems 8–10, find:
dy
in terms of t.
dx
(b) all points of horizontal and vertical tangency.
(a)
8. x = t + 5, y = t 2 − 4t
9. x = t 2 − t + 1, y = t 3 − 3t
10. x = 3 + 2 cos t , y = − 1 + 4 sin t , 0 ≤ t < 2π
On problems 11–12, a curve C is defined by the parametric equations given. For
each problem, write an integral expression that represents the length of the arc of the
curve over the given interval.
11. x = t 2 , y = t 3 , 0 ≤ t ≤ 2
12. x = e2t + 1, y = 3t − 1, − 2 ≤ t ≤ 2
3: Review of Motion Along a Line
3 Homework
1. 2003 AP Calculus AB Exam, Item 25 (no calculator):
A particle moves along the x-axis so that at time t ≥ 0 its position is given by
()
x t = 2t 3 − 21t 2 + 72t − 53. At what time t is the particle at rest?
(A) t = 1 only
(B) t = 3 only
(C) t =
7
only
2
7
2
(E) t = 3 and t = 4
(D) t = 3 and t =
2. 1998 AP Calculus AB Exam, Item 24 (no calculator):
The maximum acceleration attained on the interval 0 ≤ t ≤ 3 by the particle whose
velocity is given by v t = t 3 − 3t 2 + 12t + 4 is
()
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40
3. AP Calculus AB, sample multiple-choice Item 9 (no calculator):
The position of a particle moving along a line is given by
s t = 2t 3 − 24t 2 + 90t + 7 for t ≥ 0 .
()
For what values of t is the speed of the particle increasing?
(A) 3 < t < 4 only
(B) t > 4 only
(C) t > 5 only
(D) 0 < t < 3 and t > 5
(E) 3 < t < 4 and t > 5
4. 2003 AP Calculus AB Exam, Item 76 (calculator):
A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by
v t = 3 + 4.1cos(0.9t ). What is the acceleration of the particle at time t = 4?
()
(A) –2.016
(B) –0.677
(C) 1.633
(D) 1.814
(E) 2.97
5. 2003 AP Calculus AB Exam, Item 91 (calculator):
A particle moves along the x-axis so that at any time t > 0, its acceleration is
given by a t = ln 1 + 2t . If the velocity of the particle is 2 at time t = 1, then the
velocity of the particle at time t = 2 is
()
(
)
(A) 0.462
(B) 1.609
(C) 2.555
(D) 2.886
(E) 3.346
6. AP Calculus AB, sample multiple-choice Item 19 (calculator):
Two particles start at the origin and move along the x-axis. For 0 ≤ t ≤ 10 , their
respective position functions are given by x1 = sin t and x2 = e− 2t − 1. For how
many values of t do the particles have the same velocity?
(A) None
(B) One
(C) Two
(D) Three
(E) Four
7. AP Calculus AB, sample multiple-choice Item 15 (calculator):
(− t ) ( )
()
A particle travels along a straight line with a velocity of v t = 3e 2 sin 2t
meters per second. What is the total distance traveled by the particle during the
time interval 0 ≤ t ≤ 2 seconds?
(A) 0.835
(B) 1.850
(E) 7.025
(D) 2.261
(C) 2.055
Free-Response Questions:
8. 2004 AP Calculus AB Exam, FRQ 3 (calculator):
A particle moves along the y-axis so that its velocity at time t ≥ 0 is given by
()
( )
v t = 1 − tan −1 et . At time t = 0, the particle is at y = –1. (Note: tan −1 x = arctan x.)
(a) Find the acceleration of the particle at time t = 2.
(b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason
for your answer.
(c) Find the time t ≥ 0 at which the particle reaches its highest point. Justify your
answer.
(d) Find the position of the particle at time t = 2. Is the particle moving toward the
origin or away from the origin at time t = 2? Justify your answer.
9. 2006 AP Calculus AB/BC Exams, Item 4 (no calculator):
t (seconds)
0
10
20
30
40
50
60
70
80
v(t) (feet per second)
5
14
22
29
35
40
44
47
49
()
Rocket A has positive velocity v t after being launched upward from an initial
height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for
selected values of t over the interval 0 ≤ t ≤ 80 seconds, as shown in the table
above.
(a) Find the average acceleration of rocket A over the time interval 0 ≤ t ≤ 80
seconds. Indicate units of measure.
(b) Using correct units, explain the meaning of
∫ 10 v (t ) dt
70
in terms of the rocket’s
flight. Use a midpoint Riemann sum with 3 subintervals of equal length to
approximate
∫ 10 v (t ) dt .
70
()
(c) Rocket B is launched upward with an acceleration of a t =
3
feet per
t +1
second. At time t = 0 seconds, the initial height of the rocket is 0 feet, and the
initial velocity is 2 feet per second. Which of the two rockets is traveling faster
at time t = 80 seconds? Explain your answer.
4: Motion Along a Curve — Vectors
Example 1 (no calculator):
A particle moves in the xy-plane so that at any time t, the position of the particle is given
by x t = t 3 + 4t 2 , y t = t 4 − t 3.
()
()
(a) Find the velocity vector when t = 1.
Solution:
()
v t =
()
dx dy
d 3
d 4 3
,
=
t + 4t 2 ,
t −t
dt dt
dt
dt
(
) (
)
= 3t 2 + 8t , 4t 3 − 3t 2
v 1 = 11, 1
(b) Find the acceleration vector when t = 2.
Solution:
()
a t =
()
d ⎛ dx ⎞ d ⎛ dy ⎞
d
d
4t 3 − 3t 2
, ⎜ ⎟ =
3t 2 + 8t ,
⎜
⎟
dt ⎝ dt ⎠ dt ⎝ dt ⎠
dt
dt
(
) (
)
= 6t + 8, 12t 2 − 6t
a 2 = 20, 36
Example 2 (no calculator):
A particle moves in the xy-plane so that at any time t, t ≥ 0, the position of the particle is
given by x t = t 2 + 3t , y t = t 3 − 3t 2 . Find the magnitude of the velocity vector when t = 1.
()
()
Solution:
The magnitude or length of the velocity vector can be found by using the Pythagorean
Theorem, since the horizontal and vertical components make a right triangle, with the
vector itself as the hypotenuse. Therefore its length is given by:
2
Magnitude of velocity vector =
⎛ dx ⎞
⎛ dy ⎞
⎜⎝ dt ⎟⎠ + ⎜⎝ dt ⎟⎠
2
dy
dt
dx
dt
For our problem,
dy d 3
dx d 2
= ⎡⎣t − 3t 2 ⎤⎦ = 3t 2 − 6t .
= ⎡⎣t + 3t ⎤⎦ = 2t + 3 and
dt dt
dt dt
Magnitude of velocity vector =
( 2t + 3) + (3t 2 − 6t )
2
2
=
25 + 9 = 34 .
t =1
Notice that the formula for the magnitude of the velocity vector is the same as the formula for
the speed of the vector, which makes sense since speed is the magnitude of velocity.
Example 3 (no calculator):
A particle moves in the xy-plane so that
x = 3 − 4 cos t and y = 1 − 2 sin t , where 0 ≤ t ≤ 2π .
The path of the particle intersects the x-axis twice. Write an expression that
represents the distance traveled by the particle between the two x-intercepts. Do not evaluate.
Solution:
Figure 15
The path of the particle intersects the x-axis at the points where the y-component is equal
1
to zero. Note that 1 − 2 sin t = 0 when sin t = . For 0 ≤ t ≤ 2π , this will occur when
2
t=
dx d ⎡
dy d
π
5π
=
3 − 4 cos t ⎤ = 4 sin t and
= ⎡1 − 2 sin t ⎤⎦ = − 2 cos t ,
. Since
and t =
⎣
⎦
dt dt
dt dt ⎣
6
6
the distance traveled by the particle is Distance =
5π
6
∫ π 6 ( 4 sin t ) + ( −2 cos t )
2
2
dt .
4 Homework
Use your calculator on problems 10 and 13c only.
dy
.
dx
2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is
3
1. If x = t 2 − 1 and y = et , find
(
)
ln t 2 + 5t , 3t 2 , find its velocity vector at time t = 2.
3. A particle moves in the xy-plane so that at any time t, its coordinates are given by
x = t 5 − 1 and y = 3t 4 − 2t 3. Find its acceleration vector at t = 1.
4. If a particle moves in the xy-plane so that at time t its position vector is
⎛
π⎞
π
sin ⎜ 3t − ⎟ ,3t 2 , find the velocity vector at time t = .
2⎠
2
⎝
5. A particle moves on the curve y = ln x so that its x-component has derivative
()
x′ t = t + 1 for t ≥ 0.At time t = 0, the particle is at the point (1, 0). Find
the position of the particle at time t = 1.
6. A particle moves in the xy-plane in such a way that its velocity vector is
1 + t, t 3 . If the position vector at t = 0 is 5, 0 , find the position of
the particle at t = 2.
= 10. If x = 2 and
7. A particle moves along the curve xy = 10. If x = 2 and
dy
= 3, w
dt
dx
?
dt
8. The position of a particle moving in the xy-plane is given by the parametric
3
equations x = t 3 − t 2 − 18t + 5 and y = t 3 − 6t 2 + 9t + 4. For what value(s) of
2
t is the particle at rest?
what is the value of
9. A curve C is defined by the parametric equations x = t 3 and y = t 2 − 5t + 2.Write
the equation of the line tangent to the graph of C at the point (8, –4).
10. A particle moves in the xy-plane so that the position of the particle is given by
()
() (
)(
)
x t = 5t + 3sin t and y t = 8 − t 1 − cos t . Find the velocity vector at the time
when the particle’s horizontal position is x = 25.
()
()
11. The position of a particle at any time t ≥ 0 is given by x t = t 2 − 3 and y t =
(a) Find the magnitude of the velocity vector at time t = 5.
2 3
t .
3
(b) Find the total distance traveled by the particle from t = 0 to t = 5.
(c) Find
dy
as a function of x.
dx
dx
1
12. Point P x, y moves in the xy-plane in such a way that
=
and
dt t + 1
dy
nd
= 2t for t ≥ 0.
dt
(
)
dy
Find
the value
coordinates
of P in terms of t given that, when t = 1, x = ln 2 and y = 0.
= 3, (a)
what
is the
o
dt
(b) Write an equation expressing y in terms of x.
(c) Find the average rate of change of y with respect to x as t varies from 0 to 4.
(d) Find the instantaneous rate of change of y with respect to x when t = 1.
13. Consider the curve C given by the parametric equations x = 2 − 3cos t and
π
π
y = 3 + 2sin t , for − ≤ t ≤ .
2
2
dy
as a function of t.
(a) Find
dx
π
(b) Find the equation of the tangent line at the point where t = .
4
(c) The curve C intersects the y-axis twice. Approximate the length of the curve
between the two y-intercepts.
5: Motion Along a Curve — Vectors (continued)
Example (calculator):
( ( ) ( ))
An object moving along a curve in the xy-plane has position x t , y t at time t
dx
dy
= cos t 2 . At time t = 2, the object is at the position (1, 4).
= sin t 3 ,
with
dt
dt
(a) Find the acceleration vector for the particle at t = 2.
( )
( )
(b)Write the equation of the tangent line to the curve at the point where t = 2.
(c) Find the speed of the vector at t = 2.
(d)Find the position of the particle at time t = 1.
Solution:
dx
dy
and
(a) Students should use their calculators to numerically differentiate both
dt
dt
when t = 2 to get a 2 = − 1.476, 3.027 .
()
dy cos 4
=
or − 0.661 , so the tangent line equation is
dx sin 8
cos 4
y−4=
x − 1 or y − 4 = − 0.661 x − 1 .
sin 8
Notice that it is fine to leave the slope as the exact value or to write it as a decimal
correct to three decimal places.
(b)When t = 2,
(
)
2
(
2
)
⎛ dx ⎞
⎛ dy ⎞
2
(c) Speed = ⎜ ⎟ + ⎜ ⎟ = sin 8 + cos 4
⎝ dt ⎠
⎝ dt ⎠
(
) (
)
2
or 1.186
Notice that it is fine to leave the speed as the exact value or to write it as a
decimal correct to three decimal places.
(d)Students should apply the Fundamental Theorem of Calculus to find the x and y
components of the position.
() ( )
2
()
()
()
2
()
x 1 = x 2 − ∫ x ′ t dt
y 1 = y 2 − ∫ y ′ t dt
2
= 4 − ∫ cos t 2 dt
1
( )
= 1 − ∫ sin t 3 dt
1
= 0.782
2
1
1
( )
= 4.443
Therefore the position at time t = 1 is (0.782, 4.443).
5 Homework
Use your calculator on problems 7–11 only.
dy
in terms of t.
dx
2. Write an integral expression to represent the length of the path described by the
π
parametric equations x = cos3 t and y = sin 2 t for 0 ≤ t ≤ .
2
3. For what value(s) of t does the curve given by the parametric equations
x = t 3 − t 2 − 1 and y = t 4 + 2t 2 − 8t have a vertical tangent?
( )
1. If x = e2t and y = sin 3t , find
4. For any time t ≥ 0, if the position of a particle in the xy-plane is given by
x = t 2 + 1 and y = ln 2t + 3 , find the acceleration vector.
(
)
5. Find the equation of the tangent line to the curve given by the parametric
equations x t = 3t 2 − 4t + 2 and y t = t 3 − 4t at the point on the curve
where t = 1.
()
()
()
6. If x t = et + 1 and y = 2e2t are the equations of the path of a particle moving in
the xy-plane, write an equation for the path of the particle in terms of x and y.
7. A particle moves in the xy-plane so that its position at any time t is given by
x = cos 5t and y = t 3. What is the speed of the particle when t = 2?
( )
8. The position of a particle at time t ≥ 0 is given by the parametric equations
()
x t
(t − 2)
=
3
()
+ 4 and y t = t 2 − 4t + 4 .
3
(a) Find the magnitude of the velocity vector at t = 1.
(b)Find the total distance traveled by the particle from t = 0 to t = 1.
(c) When is the particle at rest? What is its position at that time?
( ( ) ( ))
9. An object moving along a curve in the xy-plane has position x t , y t at
dx
dy
= 1 + tan t 2 and
= 3e t . Find the acceleration
time t ≥ 0 with
dt
dt
vector and the speed of the object when t = 5.
( )
10. A particle moves in the xy-plane so that the position of the particle is given by
()
()
x t = t + cos t and y t = 3t + 2 sin t , 0 ≤ t ≤ π . Find the velocity vector when
the particle’s vertical position is y = 5.
( ( ) ( ))
11. An object moving along a curve in the xy-plane has position x t , y t at
dx
dy
time t with
= cos t 2 for 0 ≤ t ≤ 4. At time t = 1, the
= 2 sin t 3 and
dt
dt
object is at the position (3, 4).
( )
( )
(a) Write an equation for the line tangent to the curve at (3, 4).
(b)Find the speed of the object at time t = 2.
(c) Find the total distance traveled by the object over the time interval 0 ≤ t ≤ 1.
(d)Find the position of the object at time t = 2.
6 Homework
Use your calculator only on problems 3–7.
()
()
1. The position of a particle at any time t ≥ 0 is given by x t = t 2 − 2, y t =
(a) Find the magnitude of the velocity vector at t = 2.
2 3
t .
3
(b) Set up an integral expression to find the total distance traveled by the particle
from t = 0 to t = 4.
dy
as a function of x.
(c) Find
dx
(d) At what time t is the particle on the y-axis? Find the acceleration vector at this time.
( ( ) ( ))
2. An object moving along a curve in the xy-plane has position x t , y t at time t
⎞
⎛ 1
with the velocity vector v t = ⎜
, 2t ⎟ . At time t = 1, the object is at (ln 2, 4).
⎝ t +1 ⎠
4.
(a) Find the position vector.
()
5.
(b) Write an equation for the line tangent to the curve when t = 1.
(c) Find the magnitude of the velocity vector when t = 1.
( ( ) ( )) have a
(d) At what time t > 0 does the line tangent to the particle at x t , y t
slope of 12?
( ( ) ( ))
3. A particle moving along a curve in the xy-plane has position x t , y t , with
x t = 2t + 3sin t and y t = t 2 + 2cos t, where 0 ≤ t ≤ 10.. Find the velocity vector
at the time when the particle’s vertical position is y = 7.
()
()
( ( ) ( ))
4. A particle moving along a curve in the xy-plane has position x t , y t at time t
dy
dx
= 1 + sin t 3 . The derivative
with
is not explicitly given. For any time t, t
dt
dt
( )
( ( ) ( )) has a slope of t + 3. Find the
≥ 0, the line tangent to the curve at x t , y t
acceleration vector of the object at time t = 2.
( ( ) ( ))
5. An object moving along a curve in the xy-plane has position x t , y t at time
dx
dy
t with
= cos et and
= sin et for 0 ≤ t ≤ 2. At time t = 1, the object is at
dt
dt
the point (3, 2).
( )
( )
(a) Find the equation of the tangent line to the curve at the point where t = 1.
(b) Find the speed of the object at t = 1.
(c) Find the total distance traveled by the object over the time interval 0 ≤ t ≤ 2.
(d) Find the position of the object at time t = 2.
( ( ) ( ))
6. A particle moving along a curve in the xy-plane has position x t , y t at time
dx
dy
t with
= sin t 3 − t and
= cos t 3 − t . At time t = 3, the particle is at the
dt
dt
point (1, 4).
(
)
(
)
(a) Find the acceleration vector for the particle at t = 3.
(b) Find the equation of the tangent line to the curve at the point where t = 3.
(c) Find the magnitude of the velocity vector at t = 3.
(d) Find the position of the particle at time t = 2.
( ( ) ( ))
7. An object moving along a curve in the xy-plane has position x t , y t at time t
dx
dy
= 2 + sin et . The derivative
with
is not explicitly given. At t = 3, the
dt
dt
object is at the point (4, 5).
( )
(a) Find the y-coordinate of the position at time t = 1.
dy
dx
is −1.8 . Find the value of
when t = 3.
dx
dt
(c) Find the speed of the object at time t = 3.
(b)At time t = 3, the value of