MATH 100 Study Guide - Midterm Guide: Intermediate Value Theorem, Cartesian Coordinate System
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MATH 100 Full Course Notes
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Problems: prove that there exists a vertical line that divides the following irregular shape in half. Solution: place this shape within the cartesian plane, where x and y are both positive. De ne x = a as the left endpoint of the shape and x = b as the right endpoint of the shape. De ne a(x) as the percentage of area to the left of the vertical line at x. We know that this function is continuous since the area of the shape is changing constantly, without any sudden jumps. Now, we note that a(a) = 0 since x = a is the left endpoint. Also, we know that a(b) = 1 since x = b is the right endpoint. Since a(x) is continous, we can apply the intermediate value theorem to it. Since a(a) = 0 and a(b) = 1, there must exist c [a, b] such that a(c) = 0. 5.
