Math 1A Quiz 3
February 15th, 2008
Name
SID
1. (Version 1:) Find a degree3 polynomial
p
(
x
) such that the graph of
p
(
x
) has a horizontal tangent line at the point (0
,
0) and a tangent line
with slope 1 at the point (1
,
3).
(Version 2:) Find a degree3 polynomial
p
(
x
) such that the graph of
p
(
x
) has a horizontal tangent line at the point (0
,
0) and a tangent line
with slope 1 at the point (2
,
3).
Solution to Version 2:
Let
p
(
x
) =
Ax
3
+
Bx
2
+
Cx
+
D
. We want
to figure out what
A, B, C,
and
D
should be in order to have
p
satisfy the given conditions. Note that
p
(
x
) = 3
Ax
2
+ 2
Bx
+
C
.
The given conditions tell us four things: the graph of
p
passes
through the points (0
,
0) and (2
,
3) – so
p
(0) = 0 and
p
(2) = 3 –
and the slope of the tangent to the graph of
p
is 0 for
x
= 0 and
1 for
x
= 2, i.e.
p
(0) = 0 and
p
(2) = 1. Each of these conditions
gives us an equation involving A, B, C, and D:
0 =
p
(0) = 0
A
+ 0
B
+ 0
C
+
D
=
D
0 =
p
(0) = 0
·
3
A
+ 0
·
2
B
+
C
=
C
3 =
p
(2) = 8
A
+ 4
B
+ 2
C
+
D
1 =
p
(2) = 4
·
3
A
+ 2
·
2
B
+
C
Using the first two equations (i.e.
C
= 0 =
D
), we reduce the
second two equations to
3 = 8
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 Spring '08
 WILKENING
 Math, Calculus, Derivative, Slope, lim, horizontal tangent line

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