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 March 15th, 2011, 09:11 AM #1 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Combination of knowledge! [color=#000000]1. Let A(1,0), B(0,1) and ?(1,1) three points on the 2D plane. For a random point M, which lies on the line segment AB without it being point A or B, prove that (? is the rate of direction and O is the beginning of the axis) 2. If a function f is continuous on the interval [0,1] and differentiable on (0,1) with f(0)=0 and f(1)=1 prove that there exist and with such that .[/color] March 17th, 2011, 09:14 PM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Combination of knowledge! Intermediate value theorem, mean value theorem? No-one else going to touch this? ("? is the rate of direction" means "? is the slope.") March 24th, 2011, 04:53 PM #3 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Combination of knowledge! [color=#000000]Aswoods is right, noone is going to touch this?[/color] March 25th, 2011, 02:54 AM #4 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Combination of knowledge! 1. We have |OB|=1, |A?|=1, |OA|=1 and |B?|=1 (and data provided by ZardoZ) The line segment is described by y=1-x Let M have co�rdinates (x,y) such that it has co�rdinates (x,1-x). Let's find slope It is for Now, we need slope as well. It is for Now, for 2. Not sure for this one. Maybe, split in 3 cases. C1: f'(0)<1. Then there exists f'(x)>1 (for otherwise, )? such that C2: f'(0)=1. Then there exists f'(x)=1 such that C3: f'(0)>1. Then there exists f'(x)=1 (for otherwise, )?such that Maybe, some-one is inspired now. March 25th, 2011, 06:01 AM #5 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Combination of knowledge! Let g(x) = 1-x. Then f(0)=0, g(0)=1, but f(1)=1, g(1)=0. Use the intermediate value theorem to prove that the curves intersect for some x=a, then apply the mean value theorem to f over the intervals [0,a] and [a,1]. March 26th, 2011, 10:22 AM #6 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Combination of knowledge! [color=#000000]Well done![/color] Tags combination, knowledge Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Emma.k Math Books 0 March 23rd, 2014 04:17 AM eindoofus Calculus 4 December 8th, 2010 05:38 PM Abishai100 Applied Math 0 April 17th, 2009 07:41 AM bsjams New Users 1 March 22nd, 2008 10:21 PM panmaled Applied Math 3 December 30th, 2007 06:15 AM

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