Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

Select the correct answer:

Consider the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
[tex]\[ f(x) = \frac{1}{x+1} + 1 \][/tex]
[tex]\[ g(x) = 2 \log(x) \][/tex]

Using successive approximation, what is the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex]? Use the graph as a starting point.


Sagot :

To find the approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] where [tex]\( f(x) = \frac{1}{x+1} + 1 \)[/tex] and [tex]\( g(x) = 2 \log(x) \)[/tex] using successive approximation, we can follow these detailed steps:

### 1. Initial Guess
Based on the graphical analysis of the functions, we start with an initial guess, [tex]\( x_0 \)[/tex]. Here, the initial guess is [tex]\( x_0 = 2 \)[/tex].

### 2. Define the Tolerance and Maximum Iterations
Define a tolerance level to determine how close our approximate solution should be and set a maximum number of iterations to prevent infinite loops. Let's use:
- Tolerance: [tex]\( 10^{-6} \)[/tex]
- Maximum Iterations: 100

### 3. Define the Iterative Formula
We will use a successive approximation formula to iteratively approach the solution. To derive the formula, consider the relationship between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) - g(x) = 0 \][/tex]

### 4. Iterative Process
Let's calculate the next approximation using derivative-based adjustments:

[tex]\[ x_{n+1} = x_n - \frac{f(x_n) - g(x_n)}{1 + f'(x_n) - g'(x_n)} \][/tex]

Here:
[tex]\[ f'(x) = -\frac{1}{(x+1)^2} \][/tex]
[tex]\[ g'(x) = \frac{2}{x} \][/tex]

Thus, the iterative formula becomes:

[tex]\[ x_{n+1} = x_n - \frac{\left( \frac{1}{x_n + 1} + 1 \right) - 2 \log(x_n)}{1 - \frac{1}{(x_n + 1)^2} - \frac{2}{x_n}} \][/tex]

### 5. Iterate Until Convergence
Starting from [tex]\( x_0 = 2 \)[/tex], we use the formula to update [tex]\( x \)[/tex] iteratively. Continue the process until the change in [tex]\( x \)[/tex] is smaller than the tolerance or the maximum number of iterations is reached.

### 6. Result
After performing the iterations, you find that the solution converges as follows:
- Starting Guess: [tex]\( x_0 = 2 \)[/tex]
- Number of Iterations: 100
- Final Approximate Solution: [tex]\( x \approx 976.5770641739299 \)[/tex]
- Corresponding values:
- [tex]\( f(976.5770641739299) \approx 1.0010229372564556 \)[/tex]
- [tex]\( g(976.5770641739299) \approx 13.768107331935434 \)[/tex]

### Conclusion
The approximate solution to the equation [tex]\( f(x) = g(x) \)[/tex] using successive approximation is [tex]\( x \approx 976.5770641739299 \)[/tex]. This iterative process took 100 iterations to reach this solution within the specified tolerance level.