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Math 124 Exam 4 Nov. 30, 2006
SOLUTIONS
Directions
1. SHOW YOUR WORK and be thorough in your solutions. Partial credit will only be given for work
shown.
2. Any numerical answers should be left in exact form, i.e, no decimal approximations.
3. You may use calculators.
Good luck!
1)
Find parametric equations for a line through the points
A
= (

1
,
1) and
B
= (2
,
3).
Solution:
We use the parametric equations
x
=
x
0
+
at
y
=
y
0
+
bt
a
= 2

(

1) = 3 and
b
= 3

1 = 2. Hence
x
=

1 + 3
t, y
= 1 + 2
t
2)
A ladybug moves on the
xy
plane according to the equations
x
= 2
t
(
t

6) and
y
= 1

t
, where
t
is time in minutes.
(a) Does the lady bug ever stop moving? If so, when and at what
x, y
coordinates?
(b) Is the ladybug ever moving straight up or down? If so, what and at what
x, y
coordinates?
(c) Find the speed of the ladybug when it’s at coordinates (

10
,
0).
Solutions:
(a) We compute
dx
dt
= 2(
t

6) + 2
t
= 4
t

12 and
dy
dt
=

1. In order for the bug to stop, both derivatives
must be 0. However,
dy
dt
=

1, hence the bug never stops.
(b) Setting
dx
dt
= 0, we see that
t
= 3. This means the bug is not moving horizontally at coordinates
(

18
,

2). The bug is never moving only horizontally, since the
y
derivative is never 0.
(c) The
t
value at (

10
,
0) is
t
= 1. The derivatives at
t
= 1 are
dx
dt
=

8 and
dy
dt
=

1. Hence,
Speed =
s
±
dx
dt
²
2
+
±
dy
dt
²
2
=
p
(

8)
2
+ (

1)
2
=
√
65
3)
At time
t
, in seconds, the velocity,
v
, in miles per hour, of a car is given by
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This note was uploaded on 06/16/2008 for the course MATH 124 taught by Professor Kennedy during the Spring '08 term at University of Arizona Tucson.
 Spring '08
 KENNEDY
 Calculus, Approximation

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