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 ...