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The location
P(t)
of an object moving in the
xy
-plane at time
t
seconds is given by the equations
P(t)=(x(t),y(t))
, where
x(t)=a +
10
t
and
y(t)=b +
6
t
,
a,b
are constants and distances are measured in units
of meters. The equations
x(t), y(t)
describe
linear parametrized motion
; see section 10.1 of the textbook
for review.
(a) The location of the object at time
t=1
is (
$$
a
+10
,
$$
b
+6
)
(b) The average rate of change of
x(t)
between 1 and 2 seconds is

(c) The instantaneous rate of change of
x(t)
at time
t=1
is
.

(d) What is the instantaneous horizontal velocity of the object at time
t
?

(e) The average rate of change of
y(t)
between 1 and 2 seconds is
$$6

(f) The instantaneous rate of change of
y(t)
at time
t=1
is
.

(g) What is the instantaneous vertical velocity of the object at time
t
?

(h) The line along which the object is moving in the plane has the equation:
y=
$$35
x
+
$$
b
−35·
a
.
(i) Let
d(t)
be the distance the object has traveled after
t
seconds. The formula for
d(t)
$$√102+62·
t
.
(j) The instantaneous rate of change of
d(t)
at time
t
is
is

. This is called the
speed
along the line of motion.
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If a rock is thrown upward on the planet Mars with a velocity of
12
m/s, its height (in meters) after
seconds is given by
H
=
12
t
− 1.86
t
2
(a) Find the velocity of the rock after
one
second.
t
.