"); 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  x * cos[(x)/ (x-2)]=0

exp7.jpg for x*cos[(x)/(x-2)]

   The graph of this equation is given in the figure. 

   Let a = 1 and b = 1.5
 

Iteration
No.
a
b
c
f(a) * f(c)
1
1
1.5
1.133
0.159 (+ve)
2
1.133
1.5
1.194
0.032 (+ve)
3
1.194
1.5
1.214
3.192E-3 (+ve)
4
1.214
1.5
1.22
2.586E-4(+ve)
5
1.22
1.5
1.222
1.646E-5 (+ve)
6
1.222
1.5
1.222
3.811E-9(+ve)

   
   So one of the roots of x * cos[(x)/ (x-2)]=0 is approximately 1.222.


 
 

2. Find the root of  x2 = (exp(-2x) - 1) / x

exp8.jpg for x^2=(exp[-2x]-1)/x

   The graph of this equation is given in the figure. 

   Let a = -0.5 and b = 0.5 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
-0.5
0.5
0.209
-0.646 (-ve)
2
-0.5
0.208
0.0952
-0.3211 (-ve)
3
-0.5
0.0952
0.0438
-0.1547 (-ve)
4
-0.5
0.0438
0.0201
-0.0727 (-ve)
5
-0.5
0.0201
9.212E-3
-0.0336 (-ve)
6
-0.5
9.212E-3
4.218E-3
-0.015 (-ve)
7
-0.5
4.218E-3
1.931E-3
-7.1E-3 (-ve)
8
-0.5
1.931E-3
8.83E-4
-3.2E-3 (-ve)
   So one of the roots of x2 = (exp(-2x) - 1) / x is approximately 8.83E-4.


 
 
 

3. Find the root of  exp(x2-1)+10sin(2x)-5 = 0

exp9.jpg for exp[x^2-1]+10sin2x-5

   The graph of this equation is given in the figure. 

   Let a = 0 and b = 0.5 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
0
0.5
0.272
-2.637 (-ve)
2
0
0.272
0.242
-0.210 (-ve)
3
0
0.242
0.24
-0.014 (-ve)
4
0
0.24
0.24
-2.51E-3(-ve)
   So one of the roots of exp[x2-1]+10sin(2x)-5 = 0 is approximately 0.24.

 


4. Find the root of   exp(x)-3x2=0

exp10.jpg for exp[x]-3x^2

   The graph of this equation is given in the figure. 

   Let a = 3 and b = 4 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
3
4
3.512
24.137 (+ve)
2
3.512
4
3.681
3.375 (+ve)
3
3.681
4
3.722
0.211 (+ve)
4
3.722
4
3.731
9.8E-3 (+ve)
5
3.731
4
3.733
3.49E-4 (+ve)
6
3.733
4
3.733
1.733*10-3 (+ve)
   So one of the roots of exp(x)-3x2=0 is approximately 3.733. 

 


5. Find the root of    tan(x)-x-1 = 0

exp11.jpg for tan[x]-x-1

   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
0.576
0.8836 (+ve)
2
0.576
1.5
0.644
0.8274 (+ve)
3
0.644
1.5
0.705
0.762 (+ve)
4
0.705
1.5
0.76
0.692 (+ve)
5
0.76
1.5
0.808
0.616 (+ve)
6
0.808
1.5
0.851
0.541 (+ve)
. . . . .
33
1.128
1.5
1.129
1.859E-4 (+ve)
34
1.129
1.5
1.129
2.947E-6 (+ve)
   So one of the roots of tan(x)-x-1 = 0 is approximately 1.129.

 


6. Find the root of  sin(2x)-exp(x-1) =0

exp12.jpg for sin[2x]-exp[x-1]

   The graph of this equation is given in the figure. 

   Let a = 0 and b = 0.5 
 

Iteration
No.
a
b
c
f(a) * f(c)
1
0
0.5
0.305
-0.027 (-ve)
2
0
0.305
0.254
-4.497E-3(-ve)
3
0
0.254
0.246
-6.384E-4 (-ve)
4
0
0.246
0.245
-9.782E-5 (-ve)
5
0
0.245
0.245
-3.144E-5 (-ve)
   So one of the roots of sin(2x)-exp(x-1) = 0 is approximately 0.245.

 

Problems to Work-Out:
 
7. Find the root of  cos(x) - x * exp(x) = 0 [Graph]
 
8. Find the root of  x4-x-10 = 0 [Graph]
 
9. Find the root of  x - exp(-x) = 0 [Graph]
 
10. Find the root of  exp(-x) * (x2+5x+2) + 1= 0 [Graph]
 
11. Find the root of  x-sin(x) - (1/2) = 0 [Graph]
 
12. Find the root of  exp(-x) = 3log(x) [Graph]

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