Differences between revisions 16 and 18 (spanning 2 versions)
 ⇤ ← 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: localhost Comment: converted to 1.6 markup Deletions are marked like this. Additions are marked like this. Line 38: Line 38: http://moinmoin.wikiwikiweb.de/ [http://www.python.org/ Python] http://moinmoin.wikiwikiweb.de/ [[http://www.python.org/|Python]] Line 43: Line 43: http://c2.com/sig/wiki.gif {{http://c2.com/sig/wiki.gif}} Line 59: Line 59: drawing:mytest {{drawing:mytest}} Line 65: Line 65: == A graphical illustration of sin(x)/x -> 1 as x-> 0 ==by Wai Yan Pong{{{x=var('x')@interactdef _(x = slider(-7/10,7/10,1/20,1/2)):    html('

A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$

')    html('Below is the unit circle, so the length of the red line is |sin(x)|')    html('and the length of the blue line is |tan(x)| where x is the length of the arc.')     html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')    html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 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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   1 def syntax(highlight):
2     print "on"
3     return None


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


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