Calculate 2nd order Taylor polynomial of a given function

[mahler1]

Homework Statement .

Let $f:\mathbb R^2 \to \mathbb R$ / $f \in C^2$, $f(0,0)=0$ and $\nabla f(0,0)=(0,1)$ $Df(0,0)=\begin{pmatrix} 1 & 1\\ 1 & 2\\ \end{pmatrix}$
Let $g:\mathbb R^2 \to \mathbb R$ / $g \in C^2$ and

$g(x,y)=\int_0^{f(x,y)} e^{t^2}dt$

Calculate the 2nd order Taylor polynomial of $g$ at $(0,0)$


The attempt at a solution.

If $P_{(0,0)}$ is the 2nd order polynomial of $g$ at the origin, then $P_(0,0)$ is

$P_{(0,0)}=g(0,0)+<\nabla g(0,0),(x,y)>+(x,y)H_{(g)(0,0)}{(x,y)}^T$

$H_{(g)(0,0)}$ denotes the Hessian matrix of $g$ at $(0,0)$.

I got stuck at the very beginning of the exercise, I have basic doubts: I don't know how to calculate the first and second partial derivatives of $g$, that is the whole point of the exercise, but I still don't have any idea what to do. I suppose I must use the fundamental theorem of calculus and the chain rule at some point.

I would appreciate if someone could show me how to calculate the partial derivatives of $g$ with an example.

 

[Ray Vickson]

Homework Statement .

Let $f:\mathbb R^2 \to \mathbb R$ / $f \in C^2$, $f(0,0)=0$ and $\nabla f(0,0)=(0,1)$ $Df(0,0)=\begin{pmatrix} 1 & 1\\ 1 & 2\\ \end{pmatrix}$
Let $g:\mathbb R^2 \to \mathbb R$ / $g \in C^2$ and

$g(x,y)=\int_0^{f(x,y)} e^{t^2}dt$

Calculate the 2nd order Taylor polynomial of $g$ at $(0,0)$


The attempt at a solution.

If $P_{(0,0)}$ is the 2nd order polynomial of $g$ at the origin, then $P_(0,0)$ is

$P_{(0,0)}=g(0,0)+<\nabla g(0,0),(x,y)>+(x,y)H_{(g)(0,0)}{(x,y)}^T$

$H_{(g)(0,0)}$ denotes the Hessian matrix of $g$ at $(0,0)$.

I got stuck at the very beginning of the exercise, I have basic doubts: I don't know how to calculate the first and second partial derivatives of $g$, that is the whole point of the exercise, but I still don't have any idea what to do. I suppose I must use the fundamental theorem of calculus and the chain rule at some point.

I would appreciate if someone could show me how to calculate the partial derivatives of $g$ with an example.

You say you "suppose I must use the fundamental theorem of calculus and the chain rule..." So just do it! If something is stopping you from doing it you need to tell us exactly where you are stuck.