ws15 - WORKSHEET 15 - Fall 1995 1. The Intermediate-Value...

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WORKSHEET 15 - Fall 1995 1. The Intermediate-Value Theorem Let f be continuous throughout the closed interval [ a, b ]. Let m be any number between f ( a )and f ( b ). a) Draw several pictures of functions satisfying this hypotheses. Make sure you include both the case that f ( a ) f ( b )andthat f ( a ) f ( b ). b) In each of your pictures, pick a value for m and draw the line y = m . How many times does this line intersect the graph of f in each case. Can you draw an example where they don’t intersect? c) Give an argument that given the hypotheses above, there must be at least one c in [ a, b ] such that f ( c )= m . 2. Use the Intermediate Value Theorem to show that there is a number c such that 4 c =2 c . 3. A set in the plane bounded by a curve is convex if for any two points P and Q in the set, the line segment joining them also lies in the set. Q Q P P non-convex convex a) Let L be a line in the plane and let K be a convex set. Show that there is a line parallel to L that cuts K into two pieces of equal area.
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.

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