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| Solution to the Curvature Problem |
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Since this is called a curvature problem, we will start by writing the formula for (what else?) the curvature
of the graph of y = f(x) :
Since we are assuming that the curve is concave upward, y'' is nonegative. Thus we can get rid of the absolute value signs and write
Now if we knew
and y' at P, we could solve for y'' and be done. But finding y' at P is easy; it's just the slope of the blue tangent line at P. Using the points P and Q to compute the slope, we get
To find
,
recall that
where
is the radius of curvature of the graph. Thus
is the radius of the red circle of curvature in the picture (the distance between C and P):
Then
Finally, we can solve the curvature equation for y'' and finish the problem:
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