Let's solve this step by step: ### Step 1: Fill in the blanks #### 1. If \( f \) is a function **differentiable** on \((a, b)\) and **continuous** on \([a, b]\), where \( c \in [a, b] \) and \( f(c) = 0 \), then Newton's Method uses **tangent** lines to approximate \( c \). #### 2. First, estimate an initial \( x_1 \) close to \( c \). Then, determine \( x_2 \) using \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \). Calculate a third estimate using \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \). Continue this process until \( |x_{n+1} - x_n| \) is within the desired accuracy, and let \( x_{n+1} \) be the final approximation of \( c \). --- ### **Final Answer (with explanations):** If \( f \) is a function **differentiable** on \( (a, b) \) and **continuous** on \( [a, b] \), where \( c \in [a, b] \) and \( f(c) = 0 \), then Newton's Method uses **tangent** lines to approximate \( c \). First, estimate an initial \( x_1 \) close to \( c \). Then, determine \( x_2 \) using \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \). Calculate a third estimate using \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \). Continue this process until \( |x_{n+1} - x_n| \) is within the desired accuracy, and let \( x_{n+1} \) be the final approximation of \( c \). --- #### **Explanation:** - **Differentiable**: Newton's Method requires the function to have a derivative. - **Continuous**: The function must be continuous so the root exists in the interval. - **Tangent lines**: The method uses tangent lines to find better approximations for the root. - **Iteration formula**: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). - **Stopping condition**: When the difference between successive approximations is small enough, we stop and take the latest value as the approximation for the root.

Certainly! Let's carefully analyze each part of the statement, filling in the blanks with detailed explanations, and then summarize the final answer. --- ### Step-by-step Explanation of Each Blank: **1. "If \( f \) is a function **differentiable** on \((a, b)\) and **continuous** on \([a, b]\), where \( c \in [a, b] \) and \( f(c) = 0 \), then Newton's Method uses **tangent** lines to approximate \( c \)."** - **Why "differentiable" and "continuous"?** Newton's method relies on the derivative \( f'(x) \) to construct tangent lines. For the tangent line at a point \( x \) to be well-defined and useful, \( f \) must be differentiable at that point, and the derivative must be continuous in the interval to ensure convergence. - **"uses tangent lines"** Newton's method approximates the root by using the tangent line at a current guess and finding where this tangent intersects the x-axis, thus providing a better approximation. --- **2. "First, estimate an initial \( x_1 \) close to \( c \). Then, determine \( x_2 \) using \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \). Calculate a third estimate using \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \). Continue this process until \( |x_{n+1} - x_n| \) is within the desired accuracy, and let \( x_{n+1} \) be the final approximation of \( c \)."** - **"Estimate an initial \( x_1 \) close to \( c \)":** Choose a starting point near the actual root because the convergence of Newton's method depends on the initial guess being sufficiently close to the actual root. - **"Determine \( x_2 \) using \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \)":** This is the core iterative formula. It computes a better approximation \( x_2 \) by moving from \( x_1 \) along the tangent line at \( x_1 \). - **"Calculate a third estimate using \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \)":** Repeat the process to refine the approximation further, using the previous estimate \( x_2 \). - **"Continue this process until \( |x_{n+1} - x_n| \) is within the desired accuracy"**: The iterations stop when the change between successive approximations is smaller than a predetermined tolerance (indicating convergence). - **"and let \( x_{n+1} \) be the final approximation of \( c \)"**: Once the process converges, \( x_{n+1} \) is taken as the best estimate of the root. --- ### **Final Completed Explanation and Answer:** **Complete Version:** **"If \( f \) is a function _differentiable_ on \((a, b)\) and _continuous_ on \([a, b]\), where \( c \in [a, b] \) and \( f(c) = 0 \), then Newton's Method uses _tangent_ lines to approximate \( c \). First, estimate an initial \( x_1 \) close to \( c \). Then, determine \( x_2 \) using \( x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \). Calculate a third estimate using \( x_3 = x_2 - \frac{f(x_2)}{f'(x_2)} \). Continue this process until \( |x_{n+1} - x_n| \) is within the desired accuracy, and let \( x_{n+1} \) be the final approximation of \( c \)."** --- ### **Summary of Key Points:** - **Function Conditions:** \( f \) must be differentiable in the interval and continuous to ensure the tangent line approach is valid. - **Initial Guess:** Starting near the actual root improves convergence. - **Iteration Formula:** \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] It uses the tangent line at \( (x_n, f(x_n)) \) to find a better approximation. - **Stopping Criterion:** When the difference between successive approximations is sufficiently small, stop iterating. --- I hope this detailed explanation clarifies each part of the process!