november1992 said:Homework Statement
Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = [itex]x^{3}[/itex] sin(x) at a = 0.
Homework Equations
[itex]f^{n}[/itex](x) * [itex](x-a)^{n}[/itex] / (n!)
The Attempt at a Solution
I got the question wrong: my answer was 1/3! + 1/5! + 1/7!
Here is the answer below. I was wondering how my teacher got that answer. Do you have to do the product rule to find the derivatives?
A Taylor polynomial is a polynomial function that approximates a given function by using a specific point as a starting point and taking into account the function's derivatives at that point.
A Taylor polynomial is calculated using the Taylor series, which is a formula for finding the coefficients of the polynomial. The formula involves taking the derivatives of the function at the chosen point and plugging them into the formula.
The purpose of a Taylor polynomial is to approximate a function with a simpler polynomial function. This can be useful in situations where the original function is difficult to work with, but the polynomial is easier to manipulate.
The Taylor polynomial of f(x) = x^3sin(x) is useful because it provides a good approximation for the function near the chosen point. It can also be used to calculate values of the function at points that are close to the chosen point.
A Taylor polynomial is a finite polynomial function, while a Taylor series is an infinite sum of terms. The Taylor polynomial only takes into account a finite number of derivatives, while the Taylor series includes all derivatives of the function.