Alexander's question via email about Newton's Method

In summary, Alexander is asking for the application of Newton's Method to find an approximate solution for the equation $\displaystyle \mathrm{e}^{1.2\,x} = 1.5 + 2.5\cos^2{\left( x \right) } $ with an initial estimate of $\displaystyle x_0 = 1 $. Prove It provides the necessary steps for solving the equation using Newton's Method and obtains an approximate solution of $\displaystyle x_3 = 0.81797 $ after three iterations.
  • #1
Prove It
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Alexander asks:

Apply three iterations of Newton's Method to find an approximate solution of the equation

$\displaystyle \mathrm{e}^{1.2\,x} = 1.5 + 2.5\cos^2{\left( x \right) } $

if your initial estimate is $\displaystyle x_0 = 1 $.

What solution do you get?
 

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@Prove It answers:

Newton's Method solves an equation of the form $\displaystyle f\left( x \right) = 0 $, so we need to rewrite the equation as

$\displaystyle \mathrm{e}^{1.2\,x} - 1.5 - 2.5\cos^2{\left( x \right) } = 0 $

Thus $\displaystyle f\left( x \right) = \mathrm{e}^{1.2\,x} - 1.5 - 2.5\cos^2{\left( x \right) }$.

Newton's Method is: $\displaystyle x_{n+1} = x_n - \frac{f\left( x_n \right) }{f'\left( x_n \right) } $

We will need the derivative, $\displaystyle f'\left( x \right) = 1.2\,\mathrm{e}^{1.2\,x} + 5\sin{\left( x \right) }\cos{\left( x \right) } $.I have used my CAS to do this problem:

View attachment 9644

View attachment 9645

So after three iterations the root is approximately $\displaystyle x_3 = 0.81797 $.
 
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FAQ: Alexander's question via email about Newton's Method

What is Newton's Method?

Newton's Method is an algorithm used to find the roots of a function. It is commonly used to solve equations that cannot be solved analytically.

How does Newton's Method work?

Newton's Method starts with an initial guess for the root of a function. It then uses the derivative of the function to iteratively refine the guess until it reaches a value that is close enough to the actual root.

What are the advantages of using Newton's Method?

Newton's Method is a fast and efficient way to find roots of a function. It also has a high convergence rate, meaning that it quickly approaches the actual root with each iteration.

What are the limitations of Newton's Method?

Newton's Method can only find real roots of a function. It also requires an initial guess, which can be difficult to determine for some functions. Additionally, it may not converge if the initial guess is too far from the actual root.

How is Newton's Method used in real-world applications?

Newton's Method is commonly used in fields such as engineering, physics, and economics to solve complex equations and systems of equations. It is also used in computer science for numerical analysis and optimization problems.

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