Calculating directional derivatives and Taylor series

1. Consider the function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}$$ given by $$\begin{equation*} f(x,y) := e^{x^2+y^2}-1. \end{equation*}$$
Find the directional derivative of $$f$$ in the direction of $$\mathbf{u} = (u_1,u_2)$$ at the point $$(0,0)$$.

To get the directional derivative of $$f$$ in the direction of $$\mathbf{u} = (u_1,u_2)$$ we need to normalise $$\mathbf{u}$$ and take the dot product with $$\nabla{f}$$. So $$||\mathbf{u}|| = \sqrt{u_1^2+u_2^2}$$, so $$\hat{\mathbf{u}} = \frac{1}{\sqrt{u_1^2+u_2^2}}(u_1,u_2) = \left(\frac{u_1}{\sqrt{u_1^2+u_2^2}},\frac{u_2}{\sqrt{u_1^2+u_2^2}}\right)$$. Hence, the directional derivative of $$f$$ at the point $$(0,0)$$ in the direction of $$\mathbf{u} = (u_1,u_2)$$ is $$\begin{equation*} \hat{\mathbf{u}}\cdot \nabla{f} = \begin{bmatrix} \frac{u_1}{\sqrt{u_1^2+u_2^2}} & \frac{u_2}{\sqrt{u_1^2+u_2^2}} \end{bmatrix}\cdot \begin{bmatrix} 0 \\ 0 \end{bmatrix} = 0. \end{equation*}$$

1. The error function $$\text{erf} : \mathbb{R}\mapsto \mathbb{R}$$ is defined by $$\begin{equation*} \text{erf}(x) := \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} \; dt. \end{equation*}$$

Explain why erf is infinitely differentiable and find the quadratic Taylor polynomial of erf about $$a = 0$$.
Is question 1 good and for question 2 do we show there is an nth derivative for erf(x) and not too sure about the Taylor polynomial.

• Is there any particular connection between your $1$ and $2$? If not, is there any particular reason you're asking about both of them in one question instead of $2$ questions? – John Omielan Aug 2 '19 at 1:53
• No reason. Should I have asked in different threads??? I’m still relatively new to this!!!! – squenshl Aug 2 '19 at 2:12
• In general, you should not combine unrelated questions into one. There are various reasons for this such as some people may be able to answer only one question while others may answer another, people reading or searching questions may waste time reading something unrelated, etc. However, your questions here are relatively simple & short, so it's not a big deal, at least to me. Nonetheless, in the future, please ask them as separate questions. Thanks. – John Omielan Aug 2 '19 at 2:34

2. Yes, to explain why a function $$\ f\$$ is infinitely differentiable, you need to explain why it has an $$\ n^\mathrm{th}\$$ derivative for every positive integer $$\ n\$$.
3. Presumably the "quadratic Taylor polynomial" of a function $$\ f\$$ means the quadratic polynomial formed by the first three terms of its Taylor series: $$\ f\left(0\right) + f'\hskip{-0.2em}\left(0\right)x +f''\hskip{-0.2em}\left(0\right)x^2\$$, so you'd have to determine what that is for $$\ f=\mathrm{erf}\$$.