Calculus Maximus WS 7.1: Param & Vector Intro
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Name_________________________________________ Date________________________ Period______
Worksheet 7.1—Intro to Parametric & Vector Calculus
Show all work. No calculator unless explicitly stated.
Short Answer
1. If
2
1xt
and
3
t
ye
, find
dy
dx
.
2. If a particle moves in the
xy
plane so that at any time
0t
, its position vector is
22
ln 5 ,3ttt
,
find its velocity vector at time
2t
.
3. A particle moves in the
xy
plane so that at any time t, its coordinates are given by
5
1xt
,
43
32yt t
. Find its acceleration vector at
1t
.
Calculus Maximus WS 7.1: Param & Vector Intro
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4. If a particle moves in the
xy
plane so that at time t, its position vector is
2
sin 3 ,3
2
tt
, find the
velocity vector at time
2
t
.
5. A particle moves on the curve
lnyx
so that its x-component has velocity
() 1xt t
for
0t
. At
time
0t
, the particle is at the point
1, 0
. Find the position of the particle at time
1t
.
6. A particle moves in the
xy
plane in such a way that its velocity vector is
3
1,tt
. If the position
vector at
0t
is
5, 0
, find the position of the particle at
2t
.
Calculus Maximus WS 7.1: Param & Vector Intro
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7. A particle moves along the curve
10xy
. If
2x
and
3
dy
dt
, what is the value of
dx
dt
?
8. The position of a particle moving in the
xy
plane is given by the parametric equations
32
3
18 5
2
xt t t
and
32
694yt t t
. For what value(s) of t is the particle at rest?
9. A curve C is defined by the parametric equations
3
xt
and
2
52yt t
. Write an equation of the
line tangent to the graph of C at the point
8, 4
.
Calculus Maximus WS 7.1: Param & Vector Intro
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10. (Calculator Permitted) A particle moves in the
xy
plane so that the position of the particle is given by
() 5 3sinxt t t
and
() 8 1 cosyt t t
. Find the velocity vector at the time when the particle’s
horizontal position is
25x
.
Free Response:
11. The position of a particle at any time
0t
is given by
2
() 3xt t
and
3
2
()
3
yt t
.
(a) Find the magnitude of the velocity vector at time
5t
.
(b) Find the total distance traveled by the particle from
0t
to
5t
.
(c) Find
dy
dx
as a function of x.
Calculus Maximus WS 7.1: Param & Vector Intro
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12. Point
,Pxy
moves in the
xy
plane in such away that
1
1
dx
dt t
and
2
dy
t
dt
for
0t
.
(a) Find the coordinates of P in terms of t when
1t
,
ln 2x
, and
0y
.
(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
1t
.
Calculus Maximus WS 7.1: Param & Vector Intro
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13. Consider the curve C given by the parametric equations
23cosxt
and
32sinyt
, for
22
t
.
(a) Find
dy
dx
as a function of t.
(b) Find an equation of the tangent line at the point where
4
t
.
(c) (Calculator Permitted) The curve C intersects the y-axis twice. Approximate the length of the
curve between the two y-intercepts.
Calculus Maximus WS 7.1: Param & Vector Intro
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Multiple Choice:
14. A parametric curve is defined by
sinxt
and
cscyt
for
0
2
t
. This curve is
(A) increasing & concave up (B) increasing & concave down (C) decreasing & concave up
(D) decreasing & concave down (E) decreasing with a point of inflection
15. The parametric curve defined by
lnxt
,
yt
for
0t
is identical to the graph of the function
(A)
lnyx
for all real x (B)
lnyx
for
0x
(C)
x
ye
for all real x
(D)
x
ye
for
0x
(E)
ln
x
ye
for
0x
16. The position of a particle in the
xy
plane is given by
2
1xt
and
ln 2 3yt
for all
0t
. The
acceleration vector of the particle is
(A)
2
2,
23
t
t
(B)
2
4
2,
23
t
t
(C)
2
4
2,
23t
(D)
2
2
2,
23t
(E)
2
4
2,
23t
