- #1
mahler1
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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.
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.