Select all that apply

The Secant Method
1. requires only one initial guess: x subscript 0
2. requires f（x） evaluations only
3 (uses a finite-difference approximation for the derivative
4. requires f(x) and f' (x) evaluations
5. requires two initial guesses: x subscript 0 and x subscript 1

The Secant Method is a numerical method for finding the roots of a nonlinear equation \( f(x) = 0 \). Let's analyze the given statements:

1. **Requires only one initial guess: \( x_0 \)**: 
   - Incorrect. The Secant Method requires **two initial guesses**.

2. **Requires \( f(x) \) evaluations only**:
   - Correct. The Secant Method does not require the derivative of \( f(x) \), only the function \( f(x) \).

3. **Uses a finite-difference approximation for the derivative**:
   - Correct. The Secant Method approximates the derivative using a finite difference between two points.

4. **Requires \( f(x) \) and \( f'(x) \) evaluations**:
   - Incorrect. The method does not require evaluating the derivative \( f'(x) \).

5. **Requires two initial guesses: \( x_0 \) and \( x_1 \)**:
   - Correct. The Secant Method starts with two initial approximations \( x_0 \) and \( x_1 \).

### The correct options are: 
**2, 3, 5**

Would you like more details or have any questions? 

Here are 5 related questions to explore this topic further:
1. How does the Secant Method differ from Newton's method?
2. Why does the Secant Method require two initial guesses?
3. Can the Secant Method converge faster than Newton’s method?
4. In what situations would you prefer using the Secant Method over other root-finding methods?
5. What are the limitations of the Secant Method in terms of convergence?

**Tip:** The Secant Method is useful when the derivative of the function is difficult to compute.

The Secant Method
1. requires only one initial guess: x_0
2. requires f(x) evaluations only
3. uses a finite-difference approximation for the derivative
4. requires f(x) and f'(x) evaluations
5. requires two initial guesses: x_0 and x_1

Explore the Secant Method, a numerical technique for solving nonlinear equations. Learn about the core concepts, including the use of finite differences and two initial guesses. Understand how it differs from Newton's method and its practical applications. Ideal for students studying numerical methods or root-finding algorithms.

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