I am given the following problem:
The order $p$ of a root $x$ of a function $f$ is the order of the smallest non vanishing derivative at the point $x$. For example, if $f(x) = x^2$, then the order of the root at $x = 0$ is 2. Find the order of the root at $x = 0$ of the function \begin{equation} f(x) = e^{sin(x)}-sin(e^x-1)-1 \end{equation}
I have the following questions regarding the above.
- What exactly is the meaning of "an order of a root". Is it the same meaning as "multiplicity of a root"? That is, an order of a root of 2 means that there are two roots with the same value as in the example given.
- In very simple terms, given the above, how do I find the order of the root of the equation given? (Note that the question is asked in the context of using a computer algebra system, so its not required that it be calculated by hand).
Yes, what you should be computing is the multiplicity of the root. In this case the answer is $4$, because the Taylor series of your function centered at $0$ is$$\frac{x^4}{12}+\frac{x^5}8+\cdots$$