I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Monday, September 28, 2009
Minimize the area between a curve and a horizontal line
Let f be continuous increasing function on the interval [a,b]. Find c so that the area of the region bounded by x=a, x=b, y=f(x), and y=c is a minimum.
Since f is continuous and increasing, it has an inverse function g. By turning the graph sideways, we see that the area A(c) is given by the following equation.
A(c) = int (f(a) to c) [g(x) - a] dx + int (c to f(b)) [b - g(x)] dx.
Using the fundamental theorem of calculus,
A'(c) = 2g(c) - a - b,
so the minimum area occurs when A'(c) = 0, or when g(c) = (a+b)/2, which is equivalent to c = f((a+b)/2).
The answer is c = f((a+b)/2).
ReplyDeleteSince f is continuous and increasing, it has an inverse function g. By turning the graph sideways, we see that the area A(c) is given by the following equation.
A(c) = int (f(a) to c) [g(x) - a] dx + int (c to f(b)) [b - g(x)] dx.
Using the fundamental theorem of calculus,
A'(c) = 2g(c) - a - b,
so the minimum area occurs when A'(c) = 0, or when g(c) = (a+b)/2, which is equivalent to c = f((a+b)/2).