Explain briefly the method of iteration to compute
a real root of the equation ƒ (x) = 0, stating the
condition of convergence of the sequence of
approximations. Use the method of iteration to
find, correct to four significant figures, a real root
of each of the following equations.
(a) ex = 3x
(b) x = 1 / (x+1)2
(c) 1 + x2 = x3
(d) x- sin x = 1/2

### Method of Iteration to Compute a Real Root

The method of iteration, also known as the fixed-point iteration, is a numerical technique to find a root of the equation \( f(x) = 0 \). The procedure involves rewriting the equation in the form \( x = g(x) \) and then using the iterative formula:

\[ x_{n+1} = g(x_n) \]

Here are the key steps and conditions for convergence:

1. **Rewrite the Equation**: Express the equation \( f(x) = 0 \) in the form \( x = g(x) \).

2. **Initial Guess**: Choose an initial guess \( x_0 \).

3. **Iteration Process**: Compute successive approximations using \( x_{n+1} = g(x_n) \).

4. **Convergence Check**: The sequence \( \{x_n\} \) converges to a fixed point \( \alpha \) if \( g(\alpha) = \alpha \).

5. **Condition for Convergence**: The method converges if \( |g'(x)| < 1 \) in the neighborhood of the fixed point.

Now, let's apply the method to the given equations:

#### (a) \( e^x = 3x \)

Rewrite the equation as:

\[ x = \frac{e^x}{3} \]

Choose \( g(x) = \frac{e^x}{3} \).

Let's iterate using an initial guess, \( x_0 = 1 \):

1. \( x_1 = g(x_0) = \frac{e^1}{3} \approx 0.9061 \)
2. \( x_2 = g(x_1) = \frac{e^{0.9061}}{3} \approx 0.9622 \)
3. Continue this process until \( x_n \) converges to four significant figures.

#### (b) \( x = \frac{1}{(x+1)^2} \)

This equation is already in the form \( x = g(x) \) where:

\[ g(x) = \frac{1}{(x+1)^2} \]

Choose an initial guess, \( x_0 = 0.5 \):

1. \( x_1 = g(x_0) = \frac{1}{(0.5+1)^2} = \frac{1}{1.5^2} \approx 0.4444 \)
2. \( x_2 = g(x_1) = \frac{1}{(0.4444+1)^2} \approx 0.4762 \)
3. Continue this process until \( x_n \) converges to four significant figures.

#### (c) \( 1 + x^2 = x^3 \)

Rewrite the equation as:

\[ x = \sqrt[3]{1 + x^2} \]

Choose \( g(x) = \sqrt[3]{1 + x^2} \).

Choose an initial guess, \( x_0 = 1 \):

1. \( x_1 = g(x_0) = \sqrt[3]{1 + 1^2} \approx 1.2599 \)
2. \( x_2 = g(x_1) = \sqrt[3]{1 + (1.2599)^2} \approx 1.2931 \)
3. Continue this process until \( x_n \) converges to four significant figures.

#### (d) \( x - \sin x = \frac{1}{2} \)

Rewrite the equation as:

\[ x = \sin x + \frac{1}{2} \]

Choose \( g(x) = \sin x + \frac{1}{2} \).

Choose an initial guess, \( x_0 = 0.5 \):

1. \( x_1 = g(x_0) = \sin(0.5) + \frac{1}{2} \approx 0.9794 \)
2. \( x_2 = g(x_1) = \sin(0.9794) + \frac{1}{2} \approx 1.3195 \)
3. Continue this process until \( x_n \) converges to four significant figures.

### Summary

- **Iteration Formula**: \( x_{n+1} = g(x_n) \)
- **Convergence Condition**: \( |g'(x)| < 1 \) near the fixed point

Would you like to see the detailed calculations or have any questions?

### Suggested Questions

1. Can you explain the convergence condition in more detail?
2. How do you determine a good initial guess for the iteration?
3. What happens if the sequence does not converge?
4. Can you show the iteration steps for a specific example?
5. How is the derivative \( g'(x) \) used in practice to check convergence?

### Tip

When using the method of iteration, always check the derivative \( g'(x) \) to ensure the convergence condition \( |g'(x)| < 1 \) is satisfied near the expected root.

Explore the method of iteration to compute real roots of equations like ex = 3x, x = 1 / (x+1)^2, 1 + x^2 = x^3, and x - sin(x) = 1/2. Understand the convergence conditions and iterative process involved in finding solutions correct to four significant figures.

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