In textbooks and online tutorials I see that the remainder is calculated by using a new unknown variable on the same interval. For example we take the Taylor polynomial $T_n(a)$ but find the remainder $R(x)$ with a new variable $z$ inside it. See this tutorial for an example.
Maybe I'm dense but I haven't seen an explanation for this other variable being used. Or at least not one I can understand. I originally thought the remainder was the difference between the value of $f(a)$ and $T_n(a)$ but it appears instead that it is a function like the derivative that can find the error at any point. But I'm not even sure if that's the case.
What is the purpose of $z$ in that example? And why do we find the remainder at an arbitrary value instead of the value we are estimating?
I'm in Calc II if that's relevant. Thanks.