Q1: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be differentiable. Which of the following statements is correct?

A1: $$ \Biggl \langle \mathrm{grad} f (\mathbf{x} ), \begin{pmatrix} 0 \ 1 \end{pmatrix} \Biggr \rangle = \frac{ \partial f }{ \partial x_2 }(\mathbf{x})$$

A2: $$ \Biggl \langle \mathrm{grad} f(\mathbf{x}), \begin{pmatrix} 0 \ 1 \end{pmatrix} \Biggr \rangle = \frac{ \partial f }{ \partial x_1 }(\mathbf{x})$$

A3: $$ \Biggl \langle \mathrm{grad} f(\mathbf{x}), \begin{pmatrix} 1 \ 1 \end{pmatrix} \Biggr \rangle = \frac{ \partial f }{ \partial x_1 }(\mathbf{x})$$

A4: $$ \Biggl \langle \mathrm{grad} f(\mathbf{x}), \begin{pmatrix} 1 \ 1 \end{pmatrix} \Biggr \rangle = \frac{ \partial f }{ \partial x_2 }(\mathbf{x})$$

Q2: Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be differentiable. For which $\mathbf{v} \in \mathbb{R}^n$ with $| \mathbf{v} | = 1$ is the directional derivative $ \partial_{\mathbf{v}} f (\mathbf{x}) $ maximal?

A1: For $ \mathbf{v} $ in the direction of $ \mathrm{grad} f(\mathbf{x}) $.

A2: For $ \mathbf{v} $ in the direction of $\mathbf{e}_1$.

A3: For $ \mathbf{v} $ in the direction of $\mathbf{e}_n$.

A4: For $ \mathbf{v} $ orthogonal to the direction of $ \mathrm{grad} f(\mathbf{x}) $.

Q3: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be differentiable and given by $f(x_1, x_2) = 2 x_1^2 + 3 x_2^2$. What is the gradient at point $(1,1)$?

A1: $ \binom{4}{6} $

A2: $ \binom{2}{3} $

A3: $ \binom{1}{1} $

A4: $ 2$

A5: $ 4$

A6: $ 0 $