"); 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(); }  
 

5. Find the root of    x - sin(x) - (1/2)= 0

exp5.jpg for x-sin[x]-1/2

   The graph of this equation is given in the figure. 

   Let a = 1 and b = 2 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
1
2
1.5
-8.554*10-4 (-ve)
2
1
1.5
1.25
0.068 (+ve)
3
1.25
1.5
1.375
0.021 (+ve)
4
1.375
1.5
1.437
5.679*10-3 (+ve)
5
1.437
1.5
1.469
1.42*10-3 (+ve)
6
1.469
1.5
1.485
3.042*10-4 (+ve)
7
1.485
1.5
1.493
5.023*10-5 (+ve)
8
1.493
1.5
1.497
2.947*10-6 (+ve)

   So one of the roots of x-sin(x)-(1/2) = 0 is approximately 1.497.
 


6. Find the root of  exp(-x)=3log(x)

exp6.jpg for exp[-x]=3log[x]

   The graph of this equation is given in the figure. 

   Let a = 0.5 and b = 1.5 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
0.5
1.5
1
0.555 (+ve)
2
1.00
1.5
1.25
-1.554*10-3 (-ve)
3
1.00
1.25
1.125
0.063 (+ve)
4
1.125
1.25
1.187
0.014 (+ve)

   So one of the roots of exp(-x)=3log(x) is approximately 1.187.
 

Problems to Work-Out:
 
7. Find the root of x * cos[(x)/ (x-2)] = 0  [Graph]
 
8. Find the root of x2 = (exp(-2x) - 1) / x [Graph]
 
9. Find the root of exp(x2-1)+10sin2x-5 = 0 [Graph]
 
10. Find the root of  exp(x)-3x2=0 [Graph]
 
11. Find the root of tan(x)-x-1 = 0 [Graph]
 
12. Find the root of sin(2x)-exp(x-1) = 0 [Graph]