Q1: What is the correct formula for the $k$-th Taylor polynomial $T_k$ of a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with expansion point $\widetilde{x}$.

A1: $$ T_k(h) = \sum_{|\alpha| \leq k} \frac{D^\alpha f(\widetilde{x})}{\alpha!} h^\alpha $$

A2: $$ T_k(h) = \sum_{|\alpha| \leq k} \frac{D^\alpha f(h)}{\alpha!} h^\alpha $$

A3: $$ T_k(h) = \sum_{|\alpha| \leq k} \frac{D^\alpha f(h)}{\alpha!} {\widetilde{x}}^\alpha $$

A4: $$ T_k(h) = \sum_{|\alpha| \leq k+1} \frac{D^\alpha f(\widetilde{x})}{\alpha!} {\widetilde{x}}^\alpha $$

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = x_1 + x_2^2$. What is the second Taylor polynomial $T_2$ of $f$ with expansion point $(0,0)$?

A1: $T_2(h_1, h_2) = h_1 + h_2^2$

A2: $T_2(h_1, h_2) = h_1 + h_2$

A3: $T_2(h_1, h_2) = h_1^2 + h_2$

A4: $T_2(h_1, h_2) = h_1^2 $

A5: $T_2(h_1, h_2) = h_1^3 + h_2^2$

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = x_1 + \exp(x_2)$. What is the first Taylor polynomial $T_1$ of $f$ with expansion point $(0,0)$?

A1: $T_1(h_1, h_2) = 1 + h_1 + h_2 $

A2: $T_1(h_1, h_2) = h_1 + h_2 $

A3: $T_1(h_1, h_2) = h_1 + \exp(h_2) $

A4: $T_1(h_1, h_2) = h_1 $

A5: $T_1(h_1, h_2) = h_2 $

Q4: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2^5$ and $\alpha = (1,2)$. What is $D^\alpha f$?

A1: $$ D^\alpha f(x_1, x_2) = 40 x_2^3 $$

A2: $$ D^\alpha f(x_1, x_2) = x_2^3 $$

A3: $$ D^\alpha f(x_1, x_2) = x_1 $$

A4: $$ D^\alpha f(x_1, x_2) = 40 x_1 $$

A4: $$ D^\alpha f(x_1, x_2) = 4 x_1 x_2^4 $$