Give a geometric explanation of Newton's method. Choose the correct answer. A. Newton's method generates a sequence of \( y \)-intercepts, for specific lines tangent to the graph of \( f(x) \), which ideally converge to a root of \( f(x) \). B. Newton's method generates a sequence of \( x \)-intercepts, for specific secant lines of the graph of \( f(x) \), which ideally converge to a root of \( f(x) \). C. Newton's method generates the \( x \)-intercept of a line tangent to the graph of \( f(x) \) to approximate a root of \( f(x) \). D. Newton's method generates a sequence of \( x \)-intercepts, for specific lines tangent to the graph of \( f(x) \), which ideally converge to a root of \( f(x) \).

The correct geometric explanation of Newton's method is: **D. Newton's method generates a sequence of \( x \)-intercepts, for specific lines tangent to the graph of \( f(x) \), which ideally converge to a root of \( f(x) \).** **Explanation:** Newton's method is an iterative technique used to approximate the roots of a real-valued function. Geometrically, the method works as follows: 1. **Start with an initial guess** \( x_0 \) for the root of the function \( f(x) \). 2. **Draw the tangent line** to the graph of \( f(x) \) at the point \( (x_0, f(x_0)) \). 3. **Find the \( x \)-intercept** of this tangent line. This intercept, denoted as \( x_1 \), becomes the next approximation. 4. **Repeat the process** using \( x_1 \) to find \( x_2 \), and so on. Each step involves generating a new \( x \)-intercept from the tangent line at the current approximation, ideally bringing the sequence \( \{x_n\} \) closer to an actual root of \( f(x) \). **Option D** accurately captures this iterative process by emphasizing the sequence of \( x \)-intercepts derived from tangent lines converging to a root. **Answer:** Option D