Danielle M. Henak
Calculus II
September 26, 2007
David Sharpe
Manuscript 1
Considering that on the parabola y=x
2
, there are two points A and B which can be connected by a
line. If the x value for A is c and the x value for B is d, the slope of the line connecting these two points
would be equal to
=
ΔyΔx d

 = 
+
 = +
2 c2d c d cd cd c d c
. Because this line must run through the middle of the
parabola, there must be some point on the parabola that has a tangent line equal to this slope. Therefore,
=
= +
dydx 2x d c
= +
x d c2
for the x value of a point C. Thus, there are the points A(c,c
2
), B (d,d
2
) and C (
+
d c2
,
14d
+
+
2 cd 14c2
).
The area bounded by the line AB and the curve of the function y=x
2
, can be found by taking the
area of the trapezoid created by the lines x=c, x=d, AB, and the xaxis subtracted by the area under the
y=x
2
curve. In a mathematical equation that would look like this, considering that the equation for the area
of a trapezoid equals
(
12h b
+
)
1 b2
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 Spring '08
 Sharpe
 Calculus, Slope, Equals sign, Area= 12dcc2+d2x33cd Area=

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