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Title: Taylor’s Theorem - Examples
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 17 | Taylor’s Theorem - Examples
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Bright video: https://youtu.be/rDHrX87iwHM
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Dark video: https://youtu.be/0gADi9YZG_A
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mc17_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What is the correct formula for the Hessian of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$?
A1: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_1^2 }(x) & \frac{ \partial^2 f }{\partial x_1 \partial x_2 } (x)\ \frac{ \partial^2 f }{\partial x_2 \partial x_1 }(x) & \frac{ \partial^2 f }{\partial x_2^2 } (x) \end{pmatrix} $$
A2: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_1^2 }(x) & \frac{ \partial^2 f }{\partial x_2 \partial x_2 }(x) \ \frac{ \partial^2 f }{\partial x_2 \partial x_2 }(x) & \frac{ \partial^2 f }{\partial x_2^2 }(x) \end{pmatrix} $$
A3: $$ H_f(x) = \begin{pmatrix} \frac{ \partial f }{\partial x_1 }(x) & \frac{ \partial f }{\partial x_2 }(x) \ \frac{ \partial f }{\partial x_2 }(x) & \frac{ \partial f }{\partial x_1 } (x) \end{pmatrix} $$
A4: $$ H_f(x) = \begin{pmatrix} \frac{ \partial^2 f }{\partial x_2^2 }(x) & \frac{ \partial^2 f }{\partial x_1 \partial x_2 } (x) \ \frac{ \partial^2 f }{\partial x_2 \partial x_1 } (x) & \frac{ \partial^2 f }{\partial x_1^2 } (x) \end{pmatrix} $$
Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x_1, x_2) = x_1 + x_2^2 + x_1^4$. 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 $f(x_1, x_2) = \cos(x_1 x_2)$. What is the first Taylor polynomial $T_1$ of $f$ with expansion point $(0,0)$?
A1: $T_1(h_1, h_2) = 1$
A2: $T_1(h_1, h_2) = 1+ h_1 + h_2 $
A3: $T_1(h_1, h_2) = h_1 + \cos(h_2) $
A4: $T_1(h_1, h_2) = 1+ h_1 $
A5: $T_1(h_1, h_2) = h_2 $
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Last update: 2024-10