Q1: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a totally differentiable function. What is the definition of the gradient of $f$ at the position $ x \in \mathbb{R}^n$?

A1: $ \begin{pmatrix} \frac{\partial f}{\partial x_1}(x) \ \vdots \ \frac{\partial f}{\partial x_n}(x) \end{pmatrix} $

A2: $ \begin{pmatrix} f(x_1) \ \vdots \ f(x_n) \end{pmatrix} $

A3: $ \begin{pmatrix} \frac{\partial f}{\partial x_n}(x) \ \vdots \\frac{\partial f}{\partial x_1}(x) \end{pmatrix} $

Q2: Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a totally differentiable function given by $$ f(x_1, x_2, x_3) = x_1^3 + x_2 + x_3^5 $$ What is $\mathrm{grad}f(x_1,x_2,x_3)$?

A1: $ \begin{pmatrix} 3x_1^2 \ 1 \ 5 x_3^4 \end{pmatrix} $

A2: $ \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} $

A3: $ \begin{pmatrix} 5 x_3^4 \ x_2 \ x_1 \end{pmatrix} $

A4: $ 3x_1^2 + 1 + 5 x_3^4 $

A5: $ \begin{pmatrix} 1 \ x_2 \ 1 \end{pmatrix} $

A6: $ \begin{pmatrix} x_1^2 \ x_2 \ x_3^5 \end{pmatrix} $

Q3: Let $\gamma: \mathbb{R} \rightarrow \mathbb{R}^3$ and $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be two totally differentiable functions. What is the derivative $\frac{d}{dt}( f(\gamma(t)))$?

A1: $0$

A2: $1$

A3: $ \langle \mathrm{grad}f (\gamma(t)) , \gamma^\prime(t) \rangle $

A4: $\begin{pmatrix} 0 & 4\ 1 & 0 \end{pmatrix}$

A5: $ \mathrm{grad}f (\gamma(t)) - \gamma^\prime(t)$