 # Expert Help with Multivariable Function Questions

Recent questions in Multivariable functions zookeeper1930r8k 2023-03-29

## In a regression analysis, the variable that is being predicted is the "dependent variable."a. Intervening variable b. Dependent variable c. None d. Independent variable? Arjun Patterson 2023-03-21

## What is ${R}^{*}$ in math? Bradyn Cherry 2022-12-26

## Repeated addition is called ? A)Subtraction B)Multiplication C)Division Exceplyclene72 2022-12-24

## Multiplicative inverse of 1/7 is _? Leandro Acosta 2022-12-18

## Use Lagrange multipliers to find the point on a surface that is closest to a plane.Find the point on $z=1-2{x}^{2}-{y}^{2}$ closest to $2x+3y+z=12$ using Lagrange multipliers.I recognize $z+2{x}^{2}+{y}^{2}=1$ as my constraint but am unable to recognize the distance squared I am trying to minimize in terms of 3 variables. May someone help please. Will Osborn 2022-12-04

## Just find the curve of intersection between ${x}^{2}+{y}^{2}+{z}^{2}=1$ and $x+y+z=0$ Jaden Bird 2022-11-30

## Which equation illustrates the identity property of multiplication? A$\left(a+\mathrm{bi}\right)×c=\left(\mathrm{ac}+\mathrm{bci}\right)$ B$\left(a+\mathrm{bi}\right)×0=0$ C$\left(a+\mathrm{bi}\right)×\left(c+\mathrm{di}\right)=\left(c+\mathrm{di}\right)×\left(a+\mathrm{bi}\right)$ D$\left(a+\mathrm{bi}\right)×1=\left(a+\mathrm{bi}\right)$ lascieflYr 2022-11-30

## The significance of partial derivative notationIf some function like $f$ depends on just one variable like $x$, we denote its derivative with respect to the variable by:$\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)$Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_{i}}f\left({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n}\right)$Now my question is what is the significance of this notation? I mean what will be wrong if we show "Partial derivative" of $f$ with respect to ${x}_{i}$ like this? :$\frac{\mathrm{d}}{\mathrm{d}{x}_{i}}f\left({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n}\right)$Does the symbol $\mathrm{\partial }$ have a significant meaning? Kierra Griffith 2022-11-22

## The function $f\left(x,y,z\right)$ is a differentiable function at $\left(0,0,0\right)$ such that ${f}_{y}\left(0,0,0\right)={f}_{x}\left(0,0,0\right)=0$ and $f\left({t}^{2},2{t}^{2},3{t}^{2}\right)=4{t}^{2}$ for every $t>0$. Define $u=\left(6/11,2/11,9/11\right)$, with the given about. Is it possible to calculate ${f}_{u}\left(1,2,3\right)$ or ${f}_{u}\left(0,0,0\right)$, or ${f}_{z}\left(0,0,0\right)$? jorgejasso85xvx 2022-11-19

## Given topological spaces ${X}_{1},{X}_{2},\dots ,{X}_{n},Y$, consider a multivariable function $f:\prod _{i=1}^{n}{X}_{i}\to Y$ such that for any $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in \prod _{i=1}^{n}{X}_{i}$, the functions in the family $\left\{x↦f\left({x}_{1},\dots ,{x}_{i-1},x,{x}_{i+1},\dots ,{x}_{n}\right){\right\}}_{i=1}^{n}$ are all continuous. Must $f$ itself be continuous? Jenny Roberson 2022-11-18

## Let $x$ be an independent variable. Does the differential dx depend on $x$?(from the definition of differential for variables & multivariable functions) Alberto Calhoun 2022-11-18

## Let $f:M\left(n,\mathbb{R}\right)\to M\left(n,\mathbb{R}\right)$ and let $f\left(A\right)=A{A}^{t}$. Then find derivative of $f$, denoted by $df$ .So, Derivative of $f\left(df\right)$ if exists, will satisfy $limH\to 0\frac{||f\left(A+H\right)-f\left(A\right)-df\left(H\right)||}{||H||}=0$. Nicholas Hunter 2022-11-17

## if $F\left(x,y\right)$ and $y=f\left(x\right)$,$\frac{dy}{dx}=-\frac{\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(F\right)}{\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(F\right)}$1) $F\left(x,y\right)$ 𝑎𝑛𝑑 $y=f\left(x\right)$ so his means that the function $F$ is a function of one variable which is $x$2) while we were computing 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 we treated $y$ and $x$ as two independent variables although that $y$ changes as $x$ changes but while doing the 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 w.r.t $x$ we treated $y$ and $x$ as two independent varaibles and considered $y$ as a constant Laila Murphy 2022-11-17

## Let $f:{\mathbb{R}}^{2}\to \mathbb{R}$ be defined asthen check whether its differentiable and also whether its partial derivatives ie ${f}_{x},{f}_{y}$ are continuous at $\left(0,0\right)$. I dont know how to check the differentiability of a multivariable function as I am just beginning to learn it. For continuity of partial derivative I just checked for ${f}_{x}$ as function is symmetric in $y$ and $x$. So ${f}_{x}$ turns out to be${f}_{x}\left(x,y\right)=2x\mathrm{cos}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)+\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{sin}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)$which is definitely not $0$ as $\left(x,y\right)\to \left(0,0\right)$. Same can be stated for ${f}_{y}$. But how to proceed with the first part? Sophie Marks 2022-11-16

## How "messy" can a multivariable function be?$f\left(x,y\right)=\frac{2xy}{{x}^{2}+{y}^{2}}\phantom{\rule{1em}{0ex}}f\left(0,0\right)=0$ Alberto Calhoun 2022-11-13

## Many mathematical texts define a multivariable function $f$ in the following way$f:=f\left(x,y\right)$However, if we focus on the fact that a function is really a binary relation on two sets, (say the real numbers), the definition would be as follows$f:{\mathbb{R}}^{2}\to \mathbb{R}$This seems to imply that the domain of the function is a set of ordered pairs of the form $\left(x,y\right)$.The set $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\left(f\right)\subset {\mathbb{R}}^{2}×\mathbb{R}$, would then comprise ordered pairs of the form$\left\{\left(\left({x}_{0},{y}_{0}\right),a\right),\left(\left({x}_{1},{y}_{1}\right),b\right),\dots \right\}$In line with this, does it not follow that the correction notation for $f$ should be be$f:=f\left(\left(x,y\right)\right)$ Siemensueqw 2022-11-11

## What is the definition of a path along a multivariable function?$\begin{array}{r}f\left(x,y\right)=\left\{\begin{array}{ll}\frac{{x}^{2\alpha }}{{x}^{2}+{y}^{2}},& \text{if (}x\text{,}y\text{)}\ne \text{(0,0)}\\ 0& \text{if (}x\text{,}y\text{) = (0,0)}\end{array}\end{array}$Long story short, we ended up switching to polar coordinates and simplifying down to:$\underset{r\to 0}{lim}{r}^{2\alpha -2}\cdot {\mathrm{cos}}^{2\alpha }\theta$Besides the fact that we can now analyze different cases based on the value of $\alpha$, we came to the conclusion that since the point is approached at by infinitely different "paths" (depending on the value of $\theta$), the limit doesn't even exist in the first place. Jaiden Elliott 2022-11-10

## Why is continuous andseem to be discontinuity? Widersinnby7 2022-11-09
## How is ⟨f,g⟩ where $⟨f,g⟩\text{where}f\left(x,y\right),g\left(x,y\right)$ defined, where $⟨\cdot ,\cdot ⟩$ means the inner product of $f$ and $g$? Frances Pham 2022-11-09