• Title: Taylor’s Theorem

• Series: Multivariable Calculus

• YouTube-Title: Multivariable Calculus 16 | Taylor’s Theorem

• Bright video: https://youtu.be/Nhj8Gz_lo5g

• Dark video: https://youtu.be/ponHSUQs1O0

• Subtitle on GitHub: mc16_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

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

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