"); grf.document.close(); } function expcls(){ grf.window.close(); } function tip(){ popup=window.open('','','toolbar=no,width=350,height=100,location=top'); popup.document.open(); popup.document.writeln("the intermediate value theorem for continuous functions "); popup.document.writeln("For any continuous function f (x) in the interval [a,b] which satisfies f (a) * f (b) < 0 must have a zero of  'f ' in the interval [a,b] ."); popup.document.writeln(""); popup.document.close(); } function tipclose(){ popup.window.close(); } function hlts(){ hlt=window.open('','','toolbar=no,width=350,height=250,location=top'); hlt.document.open(); hlt.document.writeln("highlights"); hlt.document.writeln("AdvantagesDisadvantages."); hlt.document.writeln(""); hlt.document.close(); } function hltsclose(){ hlt.window.close(); }  
 
1. Find the root of cos(x) - x * exp(x) = 0

exp1.jpg for (cos[x])-(x * exp[x])

   The graph of this equation is given in the figure. 

   Let a = 0 and b = 1 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
0
1
0.5
0.053 (+ve)
2
0.5
1
0.75
-0.046 (-ve)
3
0.5
0.75
0.625
-0.357 (-ve)
4
0.5
0.625
0.562
-7.52 * 10-3(-ve)
5
0.5
0.562
0.531
-2.168 * 10-3 (-ve)
6
0.5
0.531
0.516
3.648 * 10-4 (+ve)
7
0.516
0.531
0.524
-9.371 * 10-5(-ve)
8
0.516
0.524
0.520
-3.649 * 10-5(-ve)
9
0.516
0.520
0.518
-3.941 * 10-6(-ve)
10
0.516
0.518
0.517
1.229 * 10-5(+ve)

   So one of the roots of cos(x) - x * exp(x) = 0 is approximately 0.517.