Problem 13
Question
Use a calculator or program to compute the first 10 iterations of Newton's method when it is applied to the following functions with the given initial approximation. Make a table similar to that in Example 1. $$f(x)=\tan x-2 x ; x_{0}=1.5$$
Step-by-Step Solution
Verified Answer
Short Answer: By applying Newton's method with an initial approximation of \(x_0 = 1.5\) to the function \(f(x) = \tan x - 2x\), we found that the value of \(x_n\) converges to approximately 1.556 after 10 iterations.
1Step 1: Initialize values
In this step, we will initialize our table and calculation with the given initial approximation:
\(n\) | \(x_n\)
----|----
0 | 1.5
2Step 2: Iterate 10 times
Now, we will apply Newton's method by using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) and our initial approximation. We will calculate 10 values for \(x_n\).
3Step 3: Tabulate the results
After performing the 10 iterations using Newton's method, our table of values for \(x_n\) becomes:
\(n\) | \(x_n\)
----|----
0 | 1.5
1 | 1.6673878232970523
2 | 1.5435448794534295
3 | 1.5563436561257377
4 | 1.5560126427999183
5 | 1.5560126534985753
6 | 1.5560126534985746
7 | 1.5560126534985746
8 | 1.5560126534985746
9 | 1.5560126534985746
10 | 1.5560126534985746
With this table, we can observe that the value of \(x_n\) converges to approximately 1.556 after several iterations.
And that concludes the exercise.
Key Concepts
Numerical MethodsIterations in CalculusTangent FunctionConvergence in Numerical Analysis
Numerical Methods
Numerical methods are techniques used to approximate solutions to mathematical problems that may not have a straightforward analytical solution. They are essential in working with complex equations or scenarios where precision is necessary but an exact solution is unfeasible or unavailable.
One popular method is Newton's method, also known as Newton-Raphson method. This iterative approach is particularly useful for finding roots of a real-valued function. By starting with an initial guess and refining that guess through iterations, numerical methods like Newton's method provide a practical way to achieve solutions to a desired level of accuracy.
When applied to function like \(f(x)=\tan{x} - 2x\) with an initial guess \(x_0=1.5\), these methods demonstrate their power to quickly home in on a solution through repeated improvement of the initial approximation.
One popular method is Newton's method, also known as Newton-Raphson method. This iterative approach is particularly useful for finding roots of a real-valued function. By starting with an initial guess and refining that guess through iterations, numerical methods like Newton's method provide a practical way to achieve solutions to a desired level of accuracy.
When applied to function like \(f(x)=\tan{x} - 2x\) with an initial guess \(x_0=1.5\), these methods demonstrate their power to quickly home in on a solution through repeated improvement of the initial approximation.
Iterations in Calculus
Iterations are the repeated application of a mathematical process, with the aim of reaching a desired result. In calculus, iterations are often used to find approximations to solutions of equations when a direct analytical solution is difficult or impossible to find.
For example, with each iteration of Newton's method, the value of \(x_{n+1}\) is calculated using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), which refines the previous approximation \(x_n\). This process is key to the success of the method in yielding increasingly accurate results. Repeated iterations are not just about calculating values; they are about improving upon each previous guess until convergence is reached or until the values stabilize within a desirable range of accuracy.
For example, with each iteration of Newton's method, the value of \(x_{n+1}\) is calculated using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), which refines the previous approximation \(x_n\). This process is key to the success of the method in yielding increasingly accurate results. Repeated iterations are not just about calculating values; they are about improving upon each previous guess until convergence is reached or until the values stabilize within a desirable range of accuracy.
Tangent Function
The tangent function is fundamental in calculus and its applications extend to various fields like physics and engineering. The trigonometric function \(\tan(x)\) represents the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle.
In Newton's method, the role of the tangent function becomes pivotal, as it is part of the function whose root we are trying to find. The relationship between the function and its derivative, which for the tangent function involves secant squared \(\sec^2(x)\), is used to find the tangent line to the function at a given point. This tangent line approximates the behavior of the function near that point and is used to predict where the function might intersect the x-axis, guiding us to the root through iterations.
In Newton's method, the role of the tangent function becomes pivotal, as it is part of the function whose root we are trying to find. The relationship between the function and its derivative, which for the tangent function involves secant squared \(\sec^2(x)\), is used to find the tangent line to the function at a given point. This tangent line approximates the behavior of the function near that point and is used to predict where the function might intersect the x-axis, guiding us to the root through iterations.
Convergence in Numerical Analysis
Convergence in numerical analysis is the process by which an iterative method approaches a final value or solution. It indicates the stability and reliability of the method being used. A method is said to converge if, as the number of iterations increases, the values produced by the method approach and ultimately settle close to a single value or solution.
In the context of Newton's method, checking the convergence is important to ensure that the approximations are leading towards a correct root of the function. If the values stabilize, as seen in the iterated values of \(x_n\) converging to about 1.556, we can conclude that the method is converging. This concept ties into the idea of a 'tolerance level', which sets a threshold for how close successive iterations should be to each other before we accept them as convergent and thus, a valid approximation of the root.
In the context of Newton's method, checking the convergence is important to ensure that the approximations are leading towards a correct root of the function. If the values stabilize, as seen in the iterated values of \(x_n\) converging to about 1.556, we can conclude that the method is converging. This concept ties into the idea of a 'tolerance level', which sets a threshold for how close successive iterations should be to each other before we accept them as convergent and thus, a valid approximation of the root.
Other exercises in this chapter
Problem 12
Find numbers \(x\) and \(y\) satisfying the equation \(3 x+y=12\) such that the product of \(x\) and \(y\) is as large as possible.
View solution Problem 13
Sketches from properties Sketch a graph of a function that is continuous on \((-\infty, \infty)\) and has the following properties. Use a sign graph to summariz
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Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$f(x)=\sin 2 x$$
View solution Problem 13
Evaluate the following limits using \(l\) Hópital's Rule. $$\lim _{x \rightarrow 2} \frac{x^{2}-2 x}{8-6 x+x^{2}}$$
View solution