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Title: Application of the Inverse Function Theorem
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Series: Multivariable Calculus
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YouTube-Title: Multivariable Calculus 24 | Application of the Inverse Function Theorem
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Subtitle on GitHub: mc24_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: The following equation $\sin(x^3) + \cos(y^2 + \sin(z)) = 1$ should be given in an explicit form $z(x,y)$ around the point $(0,0,1)$. What should be the chosen function $f$ to apply the inverse function theorem?
A1: $$f: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \text{ with } f(x,y,z) = \begin{pmatrix} x \ y \ \sin(x^3) + \cos(y^2 + \sin(z)) \end{pmatrix}$$
A2: $$f: \mathbb{R}^3 \rightarrow \mathbb{R} \text{ with } f(x,y,z) = \sin(x^3) + \cos(y^2 + \sin(z)) - 1$$
A3: $$f: \mathbb{R}^3 \rightarrow \mathbb{R} \text{ with } f(x,y,z) = \begin{pmatrix} x \ y \ z \end{pmatrix}$$
Q2: The following equation $\sin(x^3) + \cos(y^2 + \sin(z)) = 1$ should be given in an explicit form $z(x,y)$ around the point $(0,0,1)$. We choose the following function $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ with $$f(x,y,z) = \begin{pmatrix} x \ y \ \sin(x^3) + \cos(y^2 + \sin(z)) \end{pmatrix}$$ what is the Jacobian determinant at $(0,0,1)$
A1: $1$
A2: $0$
A3: $-\cos(1)$
A4: $-\cos(1) \sin(\sin(1))$
A5: $\sin(1)+\cos(1) \sin(-\sin(1))$
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Last update: 2024-10
