# Multivariable function questions and answers

Recent questions in Multivariable functions

### I have the multivariable function $$\displaystyle{\log{{\left({y}^{{2}}+{4}{x}^{{2}}-{4}\right)}}}$$ and I have found the maximal domain to be $$\displaystyle{x}^{{2}}+{\frac{{{y}^{{2}}}}{{{4}}}}{>}{1}$$

Sam Longoria 2022-01-06 Answered

### let x be a differentiable function $$\displaystyle{x}:{\left[{a},{b}\right]}\rightarrow{R}$$ which satisfies: $$\displaystyle{\frac{{{\left.{d}{x}\right.}{\left({t}\right)}}}{{{\left.{d}{t}\right.}}}}={f{{\left({t},{x}{\left({t}\right)}\right)}}}$$ $$\displaystyle{x}{\left({a}\right)}={x}_{{a}}$$ In particular I am trying to understand what $$\displaystyle{f{{\left({t},{x}{\left({t}\right)}\right)}}}$$ means, I understand this represents a multivariable function with parameters $$\displaystyle{t}$$ and $$\displaystyle{x}{\left({t}\right)}$$, but I can't think of what this would mean in the given context?

Walter Clyburn 2022-01-05 Answered

### Differentiation of multivariable function proof $$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{v}{\left({x}\right)}}}^{{{u}{\left({x}\right)}}}}{f{{\left({t},{x}\right)}}}{\left.{d}{t}\right.}={u}'{\left({x}\right)}{f{{\left({u}{\left({x}\right)},{x}\right)}}}-{v}'{\left({x}\right)}{f{{\left({v}{\left({x}\right)},{x}\right)}}}+{\int_{{{v}{\left({x}\right)}}}^{{{u}{\left({x}\right)}}}}{\frac{{\partial}}{{\partial{x}}}}{f{{\left({t},{x}\right)}}}{\left.{d}{t}\right.}$$

Kathleen Rausch 2022-01-05 Answered

### Sketch the level curves for $$\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{2}}-{y}$$ together with the gradient $$\displaystyle\nabla{f}$$ at a few typical points. Write an equation for the tangent line of the level curve $$\displaystyle{f{{\left({x},{y}\right)}}}={1}$$ at the point $$\displaystyle{\left(\sqrt{{{2}}},{1}\right)}$$.

Stefan Hendricks 2022-01-04 Answered

### Derivative of a multivariable function evaluate the derivative of a function $$\displaystyle\mathbb{R}\rightarrow\mathbb{R}$$ defined as $$\displaystyle{g{{\left({t}\right)}}}={f{{\left({x}+{t}{\left({y}−{x}\right)}\right)}}}$$ where $$\displaystyle{f}:\mathbb{R}^{{n}}\rightarrow\mathbb{R}$$ is a multivariable function and $$\displaystyle{x},{y}\in\mathbb{R}^{{n}}$$. Prove that $$\displaystyle{g}′{\left({t}\right)}={\left({y}−{x}\right)}^{{T}}\nabla{f{{\left({x}+{t}{\left({y}−{x}\right)}\right)}}}$$

Donald Johnson 2022-01-04 Answered

### How do you define iterations of multivariable functions? To be clear(example): If $$\displaystyle{f}:\mathbb{R}^{{2}}\rightarrow\mathbb{R}$$ How do you define $$\displaystyle{f}\circ{f},\text{or}{f}\circ\ldots\circ{f}?$$?

Teddy Dillard 2022-01-04 Answered

Sho 2021-12-13