Q1: Consider the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$. What is the correct definition for the notion: $f$ is partially differentiable with respect to $x_3$ at $\tilde{x}$?

A1: $$ \lim_{h \rightarrow 0} \frac{f(\tilde{x})}{\tilde{x}} $$ exists.

A2: $$ \lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2, \tilde{x}_3)}{h} $$ exists.

A3: $$ \lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2, \tilde{x}_3 + h) - f(\tilde{x}) }{h} $$ exists.

A4: $$ \lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2+h, \tilde{x}_3 + h) - f(\tilde{x}) }{h} $$ exists.

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $$ f(x_1, x_2) = x_1 \cdot x_2 $$ What is $\frac{\partial f}{ \partial x_2}(0,1)$?

A1: $0$

A2: $1$

A3: $2$

A4: $3$

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $$ f(x_1, x_2) = x_1^2 \cdot x_2 $$ What is $\frac{\partial f}{ \partial x_1}(\tilde{x})$?

A1: $2 \tilde{x}_1 \tilde{x}_2$

A2: $2 x_1 \tilde{x}_2$

A3: $2 x_1 x_2$

A4: $2 \tilde{x}_1$