What’s the derivative of these equations in Taylor series?

Question

I am struggling understand the linear approximation and Taylor.series. Could you give me a hint what are the derivatives of these functions?

$$a_2(x_1-x_0)^2 + a_3(x_1-x_0)^3?$$ If it’s stated that $x_1=x_0$.

Answer 1

Peano's theorem - Let $f : (a,\,b) \to \mathbb{R}$ and $x_0 \in (a,\,b)$. If $f$ is $n$ times derivable at $x_0$, the Taylor polynomial of order $n$ and centered at $x_0$, $$T_n(x) := \sum_{k = 0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$$ is the only polynomial of degree $\le n$ such that $$f(x) = T_n(x) + o\left((x-x_0)^n\right) \quad \text{for} \; x \to x_0$$ and also such that $$f^{(k)}(x_0) = T_n^{(k)}(x_0) \quad \text{for} \; k = 0,\,1,\,\dots,\,n.$$

---

Application example - Let $f : (-3,\,3) \to \mathbb{R}$ of law $f(x) := e^x$ and $x_0 = 0$, then it follows that:

$\hspace{3cm}$

Now the "meaning" of "Taylor polynomials" should be obvious.