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Discussion 3b

Course: MATH 231, Fall 2008
School: University of Illinois,...
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230/249 Math Fall 2006 Recitation 5 (Week 3b) R. Ghrist D. Beck Name: _______________ More Taylor Expansions 1. Warmup Expand about x=0 to order x2 (&quot;second order&quot;) 1 1 1 n a) (1 + x ) = 1 + nx + n(n - 1)x 2 + O x 3 b) = 1 - nx + n(n + 1)x 2 + O x 3 n 2! 2! (1 + x ) 2 Examples - expand about x=0 to order x 1 2 d) (1 - x ) = 1 - 2 x + x 2 c) (1 - x ) = 1 - x ( ) ( ) e) What is going on...

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230/249 Math Fall 2006 Recitation 5 (Week 3b) R. Ghrist D. Beck Name: _______________ More Taylor Expansions 1. Warmup Expand about x=0 to order x2 ("second order") 1 1 1 n a) (1 + x ) = 1 + nx + n(n - 1)x 2 + O x 3 b) = 1 - nx + n(n + 1)x 2 + O x 3 n 2! 2! (1 + x ) 2 Examples - expand about x=0 to order x 1 2 d) (1 - x ) = 1 - 2 x + x 2 c) (1 - x ) = 1 - x ( ) ( ) e) What is going on here?? The series is truncated (higher order terms are all zero), reproducing the simple forms as we would expect. f) (1 + x ) 1 3/ 2 = 1- 3 135 2 x+ x + O x3 2 222 ( ) 2. A capstan is a disk-shaped piece of equipment on a sailboat. It is used to help sailors hold one end of a rope (Fsailor) when something, e.g. a sail, is pulling on the other with great force (Fsail). The way we analyze the problem is to break up the part of the rope in contact with the capstan into n pieces. Each small section of rope subtends an angle of /n radians. It turns out that the tension in each section of rope is always a fixed fraction percentage or (proportional to the length of the small section) larger than the last piece, e.g. F1 = Fsailor 1 + , n F2 = F1 1 + , n M Fsail = Fn -1 1 + n = Fsailor 1 + n n Fsail Fsailor where is a constant (coefficient of friction). When we take the limit of n , we get an interesting alternate definition for the exponential function Math 230/249 Fall 2006 Recitation 5 (Week 3b) x e x = lim1 + , so n n = Fsailor e , Fsailor = Fsail e - n R. Ghrist D. Beck Fsail We wish to analyze this expression for ex using our results from part 1. First, write the Taylor series ...

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University of Illinois, Urbana Champaign - MATH - 231
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