Questions: 3. Draw a picture to explain the formula for successive approximations.

Solution Steps

The question asks for a drawing to explain a formula, which is not something that can be directly solved using Python code. However, I can provide a high-level approach to understanding successive approximations and how you might visualize it.

1. Successive approximations involve iteratively improving an estimate to get closer to the desired solution.
2. Start with an initial guess.
3. Apply the formula to get a new approximation.
4. Repeat the process until the approximation is sufficiently accurate.

For example, if we are using the Newton-Raphson method to find the root of a function, the formula is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

To visualize this:

1. Plot the function \( f(x) \).
2. Start with an initial guess \( x_0 \).
3. Draw the tangent line at \( x_0 \) and find where it intersects the x-axis to get \( x_1 \).
4. Repeat the process to get \( x_2, x_3, \ldots \) until convergence.

Since this is a conceptual explanation, no Python code is needed for this part.### Step 1: Understanding Successive Approximations

Successive approximations, also known as the method of iterations, is a technique used to find solutions to equations or systems of equations. The basic idea is to start with an initial guess and then iteratively improve that guess using a specific formula until the solution converges to a desired level of accuracy.

The general formula for successive approximations can be written as: \[ x_{n+1} = f(x_n) \] where \( x_n \) is the current approximation and \( x_{n+1} \) is the next approximation obtained by applying the function \( f \) to \( x_n \).

To visualize the process of successive approximations, consider the following steps:

1. Initial Guess: Start with an initial guess \( x_0 \).
2. Iteration: Apply the function \( f \) to the current guess to get the next approximation.
3. Convergence: Repeat the iteration until the approximations converge to the actual solution.

Let's consider an example where we want to find the fixed point of the function \( f(x) = \cos(x) \). The fixed point is the value of \( x \) such that \( f(x) = x \).

1. Initial Guess: Start with \( x_0 = 0.5 \).
2. First Iteration: \( x_1 = \cos(0.5) \approx 0.8776 \).
3. Second Iteration: \( x_2 = \cos(0.8776) \approx 0.6390 \).
4. Third Iteration: \( x_3 = \cos(0.6390) \approx 0.8027 \).

Final Answer