A graph of ƒ and the lines tangent to ƒ at x = 1, 2 and 3 are given. If x₀ = 3, find the values of x₁, x₂, and x₃, that are obtained by applying Newton’s method. <IMAGE>

Verified step by step guidance

1. Understand Newton's Method: Newton's method is an iterative process used to find successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.
2. Identify the initial guess: In this problem, the initial guess x₀ is given as 3. This is the starting point for applying Newton's method.
3. Calculate f(x₀) and f'(x₀): To apply Newton's method, you need the value of the function f at x₀ and the value of its derivative f' at x₀. Use the graph or any given information to determine these values.
4. Apply Newton's formula: Use the values obtained in the previous step to calculate x₁ using the formula x₁ = x₀ - \frac{f(x₀)}{f'(x₀)}. This will give you the next approximation.
5. Repeat the process: Use x₁ as the new approximation and repeat the process to find x₂ and x₃. For each step, calculate f(x_n) and f'(x_n), then apply the formula x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} to find the next approximation.

Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f is the function and f' is its derivative. This method is particularly effective for finding roots of functions and converges quickly under suitable conditions.

Tangent Lines

A tangent line to a function at a given point represents the instantaneous rate of change of the function at that point. It is defined by the slope, which is the derivative of the function at that point. In the context of Newton's Method, the tangent line at the current approximation provides a linear approximation of the function, guiding the next iteration towards the root.

Derivatives

The derivative of a function measures how the function's output changes as its input changes, essentially representing the slope of the function at any given point. It is a fundamental concept in calculus that allows us to analyze the behavior of functions, including identifying local maxima and minima, and is crucial for applying Newton's Method effectively.