# Become a Multivariable Calculus Pro!

Recent questions in Multivariable calculus
Frances Pham 2022-11-09

## Find a function $f\left(x,y\right)$ that is a solution of the following integral equation.${\int }_{0}^{y}{\int }_{0}^{x}f\left({x}^{\prime },{y}^{\prime }\right)d{x}^{\prime }d{y}^{\prime }=\left(x-y{\right)}^{2}$How can we find the $f\left(x,y\right)$?

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 $g$?

Widersinnby7 2022-11-06

## Let $f\left(x,y\right)=\frac{y}{1+xy}-\frac{1-y\mathrm{sin}\left(\frac{\pi x}{y}\right)}{\mathrm{arctan}\left(x\right)}$Compute the limit:$g\left(x\right)=\underset{y\to \mathrm{\infty }}{lim}f\left(x,y\right)$Wouldn't the value of the limit also depend on $x$?

Uriel Hartman 2022-11-05

## How do you find the dimensions of a rectangle whose area is 100 square meters and whose perimeter is a minimum?

klasyvea 2022-11-03

## Function $f:{\mathbb{R}}^{4}\to \mathbb{R}$, and we denote the four variables $x,y,z,w$, are the following statements equivalent?i) $f$ is continuousii) $f{|}_{x},f{|}_{y},f{|}_{z},f{|}_{w}$ are each continuousHere, $f{|}_{x}$ stands for the function attained by fixing the variables $w,y,z$.

Alexia Avila 2022-11-03

## A soft drink can is h centimeters tall and has a radius of r cm. The cost of material in the can is 0.0015 cents per cm^2 and the soda itself costs 0.002 cents per cm^3. The cans are currently 10 cm tall and have a radius of 2 cm. Use calculus to estimate the effect on costs of increasing the radius by 0.5 cm and decreasing the height by 0.7 cm.

Emmanuel Giles 2022-11-02

## How to find the extreme values on a specific restriction ,i.e, $f\left(x,y,z\right)={x}^{2}+{y}^{2}-z$ on the restriction $2x-3y+z-6=0$.

Emmy Swanson 2022-10-31

## Derivative of multivariable function $f:{\mathbb{R}}^{n}\setminus \left\{0\right\}\to {\mathbb{R}}^{n},f\left(x\right)=g\left(‖x‖\right)x$I used the product and chain rule.${J}_{f}\left(x\right)=x\cdot {g}^{\prime }\left(‖x‖\right)\cdot \frac{1}{‖x‖}\cdot x+g\left(r\right){I}_{n}$The solution says it's ${J}_{f}\left(x\right)={g}^{\prime }\left(r\right)\frac{1}{‖x‖}x{x}^{T}+g\left(r\right){I}_{n}$why do we have $x{x}^{T}$? Why the transpose?

caschaillo7 2022-10-29

## How to prove that $f$ is discontinuous at origin using epsilon delta method?$f\left(x,y\right)=\left\{\begin{array}{ll}\frac{{x}^{3}+{y}^{3}}{x-y}& x\ne y\\ 0& x=y\end{array}$

George Morales 2022-10-28

## determine if the following functions are differentiable at $\left(x,y\right)=\left(0,0\right)$.$f\left(x,y\right)=\sqrt{|xy|}$$g\left(x,y\right)={e}^{{|x|}^{3}y}$

Stephany Wilkins 2022-10-27

## How do you find two positive numbers whose sum is 300 and whose product is a maximum?

snaketao0g 2022-10-27

## Given a multivariable function $f:{\mathbb{R}}^{n}\to \mathbb{R}$ which is both convex and concave, a new function is constructed: $h\left(x\right)=f\left(x\right)-f\left(0\right)$.Prove the following:1. $g\left(tx\right)=t\cdot g\left(x\right),\mathrm{\forall }t\ge 0$2. $g\left(x+y\right)=g\left(x\right)+g\left(y\right),\mathrm{\forall }x,y\in {\mathbb{R}}^{n}$In addition, based on those two thing I need to prove that $f$ is linear $f\left(x\right)={a}^{T}x+b$.

Elise Kelley 2022-10-25

## Few functions and I have to study the following aspects:Continuity in the point (0,0)If the derivative exists at (0,0)Continuity of the partial derivatives at (0,0)Directional derivatives at (0,0)One of the functions is, for example:$f\left(x,y\right)=\left\{\begin{array}{ll}\frac{{x}^{2}{y}^{2}}{\sqrt{\left(}{x}^{2}+{y}^{2}\right)},& \text{if if (x,y) not (0,0)}\\ 0,& \text{if (x,y) = (0,0)}\end{array}$

Aldo Ashley 2022-10-24

## How to integrate this multivariable function when one of the variable in the function is in the form of limits:$\int {\left[f\left(x,y\right)\right]}_{y=0}^{y=b}\phantom{\rule{thinmathspace}{0ex}}dx$by ignoring the limits:$\int f\left(x,y\right)\phantom{\rule{thinmathspace}{0ex}}dx$If yes, how can it be justified?

Paloma Sanford 2022-10-23

## Let $V={R}^{d}$ and $\left({x}_{i},{y}_{i}{\right)}_{1\le i\le n}\in \left(V×\left\{-1,1\right\}{\right)}^{n}$Let $C=\left\{\left(w,b\right)\in V×R:1-{y}_{i}\left({w}^{t}{x}_{i}-b\right)\le 0,\mathrm{\forall }i\in \left[1,n\right]\right\}$Show that $mi{n}_{\left(w,b\right)\in C}||w|{|}^{2}$ has a solution $\left(w,b\right)$What I tried:$||w|{|}^{2}$ is clearly continuous, I proved that C is closed. If C is bounded, then C is compact and the solution exists (In this case, I don't know how to prove that C is bounded). If C is not bounded, I don't know how to conclude either.

4enevi 2022-10-23

## Suppose a multivariable function $f:{\mathbb{R}}^{n}\to \mathbb{R}$ is concave and sufficiently smooth. We have:$‖f\left(\mathbf{x}\right)-f\left({\mathbf{x}}_{0}\right)‖\le M‖\mathbf{x}-{\mathbf{x}}_{0}‖$ for some positive constant $M$.If the $f$ is univariate, we know that $M$ is the absolute value of the slope of the tangent line at ${\mathbf{x}}_{0}$. But what is it for multivariable case, is there a special name in math given to it?

Keyla Koch 2022-10-21