Problem 27 - Single Variable Calculus

Question

Estimate the value of the radical accurate to four decimal places by using three iterations of Newton's method to solve the equation \(f(x)=0\) with initial estimate \(x_{0}\). \(\sqrt[4]{20} ; \quad f(x)=x^{4}-20 ; x_{0}=2.1\)

Step-by-Step Solution

Answer

After three iterations of Newton's method with the initial estimate \(x_0 = 2.1\), we have approximated the fourth root of 20 accurate to four decimal places as follows: \(\sqrt[4]{20} \approx x_3 = 2.0241\).

Step 1: Derive Newton's Method Formula

To use Newton's method, we need to find the iterative formula. The general formula for the method is given by: \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\] For the given function \(f(x) = x^4 - 20\), we first need to find its derivative, \(f'(x)\). The derivative of \(f(x)\) is: \[f'(x) = \frac{d}{dx}(x^4 - 20) = 4x^3\] Now we have: \[x_{n+1} = x_n - \frac{x_n^4 - 20}{4x_n^3}\] This is the formula we will use for our Newton's method iterations.

Step 2: Perform the First Iteration (n = 0)

Plug in the given initial estimate, \(x_0 = 2.1\), into the formula: \[x_{1} = 2.1 - \frac{2.1^4 - 20}{4(2.1)^3}\] Compute the value of \(x_1\): \[x_{1} \approx 2.0349\]

Step 3: Perform the Second Iteration (n = 1)

Plug in the updated value of \(x_1 \approx 2.0349\) into the formula: \[x_{2} = 2.0349 - \frac{2.0349^4 - 20}{4(2.0349)^3}\] Compute the value of \(x_2\): \[x_{2} \approx 2.0246\]

Step 4: Perform the Third Iteration (n = 2)

Plug in the updated value of \(x_2 \approx 2.0246\) into the formula: \[x_{3} = 2.0246 - \frac{2.0246^4 - 20}{4(2.0246)^3}\] Compute the value of \(x_3\): \[x_{3} \approx 2.0241\] After three iterations of Newton's method, we have approximated the radical accurate to four decimal places as follows: \[\sqrt[4]{20} \approx x_3 = 2.0241\]

Key Concepts

Numerical Approximation

Numerical approximation is a crucial technique in mathematics and engineering when finding exact solutions analytically is challenging or impossible. It enables us to estimate solutions with a high degree of accuracy through iterative processes.
Newton's method is one of the most popular numerical approximation methods. It allows for the estimation of roots of a function by successively refining guesses until the results are sufficiently precise.
With initial estimates like the one given in the exercise (\(x_0 = 2.1\)), you perform several iterations to improve the approximation of \(\sqrt[4]{20}\). Each iteration brings us closer to the actual root, aiming for the desired level of precision, such as four decimal places in this example. The power of numerical approximation lies in its ability to handle complex calculations that might not be practically solvable through direct algebraic means. It transforms unsolvable analytical circumstances into highly manageable computational tasks.

Derivatives

Derivatives represent a key concept in calculus often connecting to rates of change and slopes of curves. In Newton's method, derivatives are fundamental as they define the slope of the tangent line at any given point on a function.
The formula for Newton's method relies heavily on the derivative of the function to make effective approximations. The derivative used in the exercise is given by \(f'(x) = 4x^3\), which is the slope of the tangent to the curve \(f(x) = x^4 - 20\).

• The knowledge of derivatives allows us to adjust our estimation more accurately with each iteration.
• Derivatives help the method "home in" on the actual root by guiding the approximations towards the point where the tangent crosses the x-axis.

Without calculating derivatives, the efficiency and effectiveness of Newton's method could not be realized, as the derivative provides the necessary direction and rate of adjustment for each step.

Root Finding Algorithms

Root finding algorithms are essential in solving equations where the goal is to find the value of \(x\) such that \(f(x) = 0\). These algorithms vary in complexity and efficiency, each suitable for different types of functions or desired levels of precision.
Newton's method falls under this category and is especially powerful for polynomial equations, like the one we encounter in our exercise. It takes advantage of both the function and its derivative to iteratively close in on a root. Some key characteristics of Newton's method as a root-finding algorithm include:

• Fast convergence near the root, especially when starting close to actual roots.
• Requires calculation of derivatives, making it potent but dependent on differentiable functions.
• Depending on the initial guess, convergence is not guaranteed and may require a sensible choice of starting points.

In practice, Newton's method's efficiency in root finding makes it a popular choice for problems requiring high precision and quick convergence, navigating through complex function behavior effectively.