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Title: Directional Derivative
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 10 | Directional Derivative
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Bright video: https://youtu.be/OxVYmBZqeBU
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Dark video: https://youtu.be/9TOfdo4yem8
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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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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mc10_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: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. What is the definition of the directional derivative of $f$ along the vector $\mathbf{v}$ at the point $\tilde{\mathbf{x}}$.
A1: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) - f( \tilde{\mathbf{x}} )}{h} $$
A2: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) + f( \tilde{\mathbf{x}} )}{h} $$
A3: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) - f( \tilde{\mathbf{x}} + h )}{h} $$
Q2: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. What is an alternative definition of the directional derivative of $f$ along the vector $\mathbf{v}$ at the point $\tilde{\mathbf{x}}$.
A1: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} + t \mathbf{v} )|_{\mid t=0} $$
A2: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - t \mathbf{v} )|_{\mid t=1} $$
A3: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - \mathbf{v} )|_{\mid t=0} $$
A4: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - t^2 \mathbf{v} )|_{\mid t=1} $$
Q3: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be totally differentiable and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. How can the directional derivative be calculated?
A1: $$ \langle \mathrm{grad} f (\tilde{\mathbf{x}}), \mathbf{v} \rangle $$
A2: $$ \langle \mathrm{grad} f (\mathbf{v}), \tilde{\mathbf{x}} \rangle $$
A3: $$ - \langle \mathrm{grad} f (\mathbf{v}), \tilde{\mathbf{x}} \rangle $$
A4: $$ \langle \mathrm{grad} f (\tilde{\mathbf{x}}) - \mathbf{v}, \tilde{\mathbf{x}} \rangle $$
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Last update: 2024-10
