Math 184 Group Work 1 Solutions
1. A car is traveling at night along a highway shaped like the parabola
y
=
x
2
.
There is a statue located 3 km east and 5 km north of the origin. Will the car’s
headlights ever illuminate the statue?
If so, where will the car be when this
happens?
Solution:
The headlights of the car will always follow the line tangent to
y
=
f
(
x
) =
x
2
at th location of the car. We are then looking for a tangent line
to
y
=
x
2
which passes through the point (3
,
5).
What we need to determine is exactly which tangent line this is. To characterize
a tangent line, it’s easiest to just give a variable name to the
x
coordinate of
the point where the line is tangent to
y
=
x
2
. We call this value
a
, and so we
are describing the unknown tangent line as the one which is tangent to
y
=
x
2
at
x
=
a
. We will then attempt to solve for
a
.
From class (or the textbook), we know that the equation for this tangent line
to
y
=
x
2
at
x
=
a
is
y

f
(
a
) =
f
′
(
a
)(
x

a
). However, we also know that the
point (3
,
5) is on this line, and so the equation must remain valid if we plug in
x
= 3 and
y
= 5:
5

f
(
a
) =
f
′
(
a
)(3

a
)
Also, we know that
f
(
a
) =
a
2
and
f
′
(
a
) = 2
a
. Therefore,
5

a
2
= 2
a
(3

a
)
5

a
2
= 6
a

2
a
2
a
2

6
a
+ 5 = 0
(
a

5)(
a

1) = 0, so
a
= 1 or
a
= 5.
So, the only tangent lines to
y
=
x
2
which pass through (3
,
5) are tangent to
y
=
x
2
at either
x
= 1 or
x
= 5.
However, notice that when the car has an
x
coordinate of 5, it is at the point (5
,f
(5)) = (5
,
25), and even though the
point (3
,
5) is on the tangent line to
y
=
x
2
at (5
,
25), the headlights of the car
will not illuminate the statue at (3
,
5). This is because the statue is behind
(to
the left of) the car at this time. So, the only point at which the car’s headlights
will illuminate the statue is when it has an
x
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 Spring '09
 James

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