What is the Quadratic Maclaurin Polynomial for f(x)=x*sin(x)?

In summary, the conversation is about a person struggling with a Maclaurin Series question involving the function f(x) = x*sin(x). They are trying to find the Quadratic Maclaurin Polynomial and have calculated the first and second derivatives. However, they made a mistake in the second derivative and eventually realized it was due to an error in the derivative. They are seeking help from others to find out where they went wrong.
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Homework Statement




I'm having a bit of trouble with this Maclaurin Series question. It should be simple enough but I can't get the answer which is given as x2. It's been a while since I've done series and my being rusty is a little annoying. Hopefully someone can help :) Consider:

f(x)=x*sin(x)

Question asks to find the Quadratic Maclaurin Polynomial for f(x).
So this is what I did. First I got the nth derivatives:

f1(x) = x*cos(x) + sin(x)
f2(x) = -x*sin(x) + cos(x) + sin(x)

P2(x): 0 + x*0 + (x^2)(0 + 1 + 0)*(1/2!)

= x2/2


I'm sure I'm doing something blatantly wrong here but I can't seem to work it out. Fresh eyes might be useful :)

Thanks a lot.
 
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  • #2
Oh no.. it was the derivative.. lol I've been racking my brain for 20 minutes staring at Taylor's Theorem thinking I made the error there.. ugh well that's embarrassing.
 

1. What is a Maclaurin Polynomial function?

A Maclaurin Polynomial function is a type of polynomial function that is used to approximate more complex functions by using only a few terms. It is named after the Scottish mathematician Colin Maclaurin who first introduced it in the 18th century.

2. How is a Maclaurin Polynomial function different from a traditional polynomial function?

A Maclaurin Polynomial function is a special case of a traditional polynomial function, where the function is centered at x=0. This means that the coefficients of the function are determined by the derivatives of the original function evaluated at x=0.

3. What are the applications of Maclaurin Polynomial functions?

Maclaurin Polynomial functions are commonly used in calculus and other areas of mathematics to approximate more complex functions. They are also useful in physics, engineering, and other fields where precise calculations are needed.

4. How do you find the coefficients of a Maclaurin Polynomial function?

The coefficients of a Maclaurin Polynomial function can be found by taking the derivatives of the original function and evaluating them at x=0. These coefficients are then used to form the polynomial function.

5. What are the limitations of using a Maclaurin Polynomial function?

Maclaurin Polynomial functions can only approximate a function within a certain range and may not be accurate for all values of x. Additionally, the more terms that are included in the function, the more accurate the approximation will be, but the more complex the function becomes.

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