"); 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]) = 0exp1.jpg for (cos[x])-(x * exp[x])

   The graph of this equation is given in the figure. 

   Let the initial guess x0 be 2.0
 
 
 
 
 
 

i
0
1
2
3
4
5
6
7
xi
2
1.34157
0.8477
0.58756
0.52158
0.51777
0.51776
0.51776

 
   So the iterative process converges at  0.51776


2. Find the root of  x4-x-10 = 0exp2.jpg for x^4-x-10

   The graph of this equation is given in the figure. 

   Let the initial guess x0 be 2.0
 
 
 
 
 
 

i
0
1
2
3
4
xi
2
1.87097
1.85578
1.85558
1.85558

 
   So the iterative process converges at  1.85558. 

 


3. Find the root of  x-exp[-x] = 0exp3.jpg for x-exp[-x]

   The graph of this equation is given in the figure.l

   Let the initial guess x0 be 2.0
 
 
 
 
 
 

i
0
1
2
3
4
5
xi
2
0.35761
0.55871
0.56713
0.56714
0.56714

 
   So one of the roots of  x-exp[-x] = 0 is approximately 0.56714. 


4. Find the root of   (exp[-x] * (x2+5x+2)) + 1= 0exp4.jpg for exp[-x]*(x^2-5x+2)+1

   The graph of this equation is given in the figure. 

   let the initial guess x0 be -2.0
 
 
 
 
 
 

i
0
1
2
3
 4 5
6
xi
-2
-1.22707
-0.77562
-0.60291
 -0.57955 -0.57916
-0.57916

 
   So one of the roots of (exp[-x] * (x2+5x+2)) + 1= 0 is approximately -0.57916. 

 


5. Find the root of    x-sin[x]-(1/2)= 0exp5.jpg for x-sin[x]-1/2

   The graph of this equation is given in the figure. 

   let the initial guess x0 be 2.0
 
 
 
 
 
 

i
0
1
2
3
4
xi
2
1.58228
1.50087
1.49731
1.4973

 
   So one of the roots of x-sin[x]-(1/2) = 0 is approximately 1.4973. 


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 the initial guess x0 be 2.0
 
 
 
 
 
 

i
0
1
2
3
4
5
xi
2
1.02418
1.22523
1.24663
1.24682
1.24682

 
   So one of the roots of exp[-x]=3log[x] is approximately 1.24682. 

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]