Differences between revisions 16 and 18 (spanning 2 versions)

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   2     print "on"
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   1   public void main(String[] args]){
   2      System.out.println("Hello world!");
   3   }

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Term: Definition

-  ⇤ ← Revision 16 as of 2008-09-28 21:15:32 → 
  Size: 2701
  Editor: pong
  Comment:
+   ← Revision 18 as of 2008-11-14 13:42:04 → ⇥
  Size: 1312
  Editor: anonymous
  Comment: converted to 1.6 markup
-Deletions are marked like this.
+Additions are marked like this.
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-http://moinmoin.wikiwikiweb.de/ [http://www.python.org/ Python]
+http://moinmoin.wikiwikiweb.de/ [[http://www.python.org/|Python]]
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-http://c2.com/sig/wiki.gif
+{{http://c2.com/sig/wiki.gif}}
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-drawing:mytest
+{{drawing:mytest}}
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-== A graphical illustration of sin(x)/x -> 1 as x-> 0 ==
by Wai Yan Pong

{{{
x=var('x')
@interact
def _(x = slider(-7/10,7/10,1/20,1/2)):
    html('<h3>A graphical illustration of  $lim_{x - > 0} \sin (x) / x = 1$ </h3>')
    html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')
    html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.') 
    html('From the picture, we see that |sin(x)|  $\leq$  |x|  $\leq$  |tan(x)|.')
    html('It follows easily from this that cos(x)  $\leq$  sin(x)/x  $\leq$  1 when x is near 0.')
    html('As  $lim_{x - > 0} \cos (x) = 1$ , we conclude that  $lim_{x - > 0} \sin (x) / x = 1$ .')
    if not (x == 0):
        pretty_print("sin(x)/x = "+str(sin(float(x))/float(x)))
    elif x == 0:
        pretty_print("The limit of sin(x)/x as x tends to 0 is 1.")
    C=circle((0,0),1, rgbcolor='black')
    mvp = (cos(x),sin(x));tpt = (1, tan(x))
    p1 = point(mvp, pointsize=30, rgbcolor='red'); p2 = point((1,0), pointsize=30, rgbcolor='red')
    line1 = line([(0,0),tpt], rgbcolor='black'); line2 = line([(cos(x),0),mvp], rgbcolor='red') 
    line3 = line([(0,0),(1,0)], rgbcolor='black'); line4 = line([(1,0),tpt], rgbcolor='blue')
    result = C+p1+p2+line1+line2+line3+line4
    result.show(aspect_ratio=1, figsize=[3,3], axes=False)
}}}
attachment:sinelimit.png

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