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1. Find the root of (cos[x])-(x * exp[x]) = 0fpi1.jpg for (cos[x])-(x * exp[x])

   Consider g(x) = cos[x]/exp[x]

   The graph of g(x) and x are given in the figure.

   let the initial guess x0 be 2.0
 
 
 
 

i
0
1
2
3
4
5
6
7
8
 9
10
11
12
13
...
31
32
xi
1
0.199
0.803
0.311
0.698
0.381
0.634
0.427
0.594
 0.458
0.567
0.478
0.551
0.491
...
0.518
0.518

 
 
   That is for g(x) = cos[x]/exp[x] the itirative process is converged to 0.518.


 

2. Find the root of  x4-x-10 = 0fpi2.jpg for x^4-x-10

   Consider g(x) = (x + 10)1/4

   The graph of g(x) and x are given in the figure.

   let the initial guess x0 be 4.0 
 
 
 
 
 
 

i
0
1
2
3
4
5
6
xi
4.0
1.93434
1.85866
1.8557
1.85559
1.85558
1.85558
   That is for g(x) = (x + 10)1/4  the itirative process is converged to 1.85558.


 
 
 

3. Find the root of  x-exp[-x] = 0fpi3.jpg for x-exp[-x]

   Consider g(x) = exp[-x]

   The graph of g(x) and x are given in the figure..

   let the initial guess x0 be 3.0 
 
 
 
 
 
 

i
0
1
2
3
4
5
 6 7 8 9
10
11
12
13
14
xi
3.0
0.05
0.951
0.386
0.68
0.507
 0.602 0.548 0.578 0.561
0.571
0.565
0.568
0.567
0.567

 
   That is for g(x) = exp[-x]  the itirative process is converged to 0.567. 

 


4. Find the root of   (exp[-x] * (x2+5x+2)) + 1= 0fpi4.jpg for exp[-x]*(x^2-5x+2)+1

   Consider g(x) = (exp[x]+x^3+4x^2+2x+2)/(x^2+3x-3)

   The graph of g(x) and x are given in the figure.

   let the initial guess x0 be -2 
 
 
 
 
 
 

i
0
1
2
3
4
5
6
xi
-2
-1.227
-0.776
-0.603
-0.58
-0.579
-0.579

   That is for g(x) = (exp[x]+x^3+4x^2+2x+2)/(x^2+3x-3)  the itirative process is converged to -0.579.


 

5. Find the root of    x-sin[x]-(1/2)= 0fpi5.jpg for x-sin[x]-1/2

   Consider g(x) = sin[x]+(1/2)

   The graph of g(x) and x are given in the figure.

   let the initial guess x0 be 2 
 
 
 
 
 
 

i
0
1
2
3
4
5
xi
2
1.409
1.487
1.496
1.497
1.497

 
   That is for g(x) = sin[x]+(1/2) the itirative process is converged to 1.497. 

 


6. Find the root of  exp[-x]=3log[x]fpi6.jpg for exp[-x]=3log[x]

   Consider g(x) = exp[(exp[-x]/3)]

   The graph of g(x) and x are given in the figure.

   let the initial guess x0 be 2 
 
 
 
 
 
 

i
0
1
2
3
4
5
xi
2
1.046
1.124
1.114
1.116
1.115
   That is for g(x) = exp[(exp[-x]/3)]  the itirative process is converged to 1.115.

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]