Understanding Maclaurin Polynomials: Exploring Substitution Techniques

In summary, the conversation is about understanding Maclaurin polynomials and using substitution on known polynomials. The expert points out the mistake of using "R_2(x)" and "R_4(x)" instead of "R_2(x+2)" and "R_2(x^2+ 2)", and confirms that the correct way is shown on the right in the given images.
  • #1
estro
241
0
I'm trying to understand the reminder of Maclaurin polynomials

[PLAIN]http://estro.uuuq.com/0.png
[PLAIN]http://estro.uuuq.com/1.png
[PLAIN]http://estro.uuuq.com/2.png
[PLAIN]http://estro.uuuq.com/3.png
[PLAIN]http://estro.uuuq.com/4.png
Here I show few attempts to use substitution on known polynomials.

Where I'm wrong?
 
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  • #2
The only thing you are doing wrong is that you have "[itex]R_2(x)[/itex]" and "[itex]R_4(x)[/itex]". What you should have is [itex]R_2(x+2)[/itex] and [itex]R_2(x^2+ 2)[/quote] as you do on the right.
 
  • #3
Thank you for your response,
So this is mistake in my book?: [PLAIN]http://estro.uuuq.com/book_wrong.png

So the right way is: [PLAIN]http://estro.uuuq.com/right.png
 
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Related to Understanding Maclaurin Polynomials: Exploring Substitution Techniques

1. What is a Maclaurin polynomial?

A Maclaurin polynomial, also known as a Taylor polynomial centered at 0, is a polynomial function that approximates a given function at a specific point by using its derivatives at that point. It is named after the Scottish mathematician Colin Maclaurin.

2. How is a Maclaurin polynomial calculated?

A Maclaurin polynomial is calculated by using the formula: P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ... + (f^n(0)/n!)x^n. This formula uses the derivatives of the function at 0 to create a polynomial that closely approximates the function at that point.

3. What is the purpose of a Maclaurin polynomial?

The purpose of a Maclaurin polynomial is to approximate a given function at a specific point, which can be useful in situations where the function may be difficult to evaluate or graph. It can also be used to find the value of a function at a point without having to use the function's entire series expansion.

4. How accurate is a Maclaurin polynomial?

The accuracy of a Maclaurin polynomial depends on the degree of the polynomial and the function it is approximating. The higher the degree, the more accurate the approximation will be. However, it is important to note that a Maclaurin polynomial is only an approximation and will never exactly match the original function.

5. What are some real-life applications of Maclaurin polynomials?

Maclaurin polynomials have many real-life applications, including in physics, engineering, and economics. They can be used to approximate complex mathematical models and make predictions about systems that are difficult to analyze directly. They are also used in numerical methods for solving differential equations and in computer graphics to create smooth curves and surfaces.

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