 # Become a Multivariable Calculus Pro!

Recent questions in Multivariable calculus omgespit9q 2022-10-14

## In a function like $f\left(x,y\right)={x}^{2}+y$, are $x$ and $y$ independent of each other and are we allowed to pick values for each deliberately?Say for $1$ for $x$ and $99$ for $y$? Oscar Burton 2022-10-14

## Limit of multivariable functionslimit of $\sqrt{{x}^{2}+{y}^{2}}\mathrm{sin}\left(1/y\right)$ as $\left(x,y\right)\to \left(0,0\right)$limit of $2xy/\left({x}^{2}-y+5xy\right)$ as $\left(x,y\right)\to \left(0,0\right)$ Amina Richards 2022-10-13

## How do you optimize $f\left(x,y\right)=2{x}^{2}+3{y}^{2}-4x-5$ subject to ${x}^{2}+{y}^{2}=81$? Emmy Swanson 2022-10-13

## If we have some closed form multivariable function say 2-in, 1-out, the cross sections of the graph parallel to the $xz$ and $yz$ plane have equations that are inherently closed form.But is the same true for the diagonal cross sections? Say for example we have the function $f$:$f\left(x,y\right)={e}^{y}\mathrm{sin}\left(x\right)$If we're given a $\stackrel{\to }{v}$ say, [1, 2] would the cross section of the graph of the plane formed by the $z$-axis and $\stackrel{\to }{v}$ inherently be closed form as $f$ is on the axes? Aidyn Crosby 2022-10-08

## Fix a natural number $n.$ Let $f,g:\mathbb{R}\to \mathbb{R}$ be $n$ times differentiable functions. General Leibniz rule states that $n$th derivative of the product $fg$ is given by $\sum _{k=0}^{m}\left(\genfrac{}{}{0}{}{n}{k}\right){f}^{\left(n-k\right)}\left(x\right){g}^{\left(k\right)}\left(x\right)$ where ${g}^{\left(k\right)}$ means that $g$ is differentiated $k$ times.What is a meaning of $\beta \le \alpha ?$?Since the formula is for multivariable, I suppose that 𝛽 and 𝛼 are vectors, like $\left(1,2\right)$ in ${\mathbb{R}}^{2}.$. Austin Rangel 2022-10-06

## We have just started studying functions of several variables and their derivatives and our professor suggested the following problem as food for thought.Two squares, both with length l=1 intersect in a rectangle that has an area equal with 1/8 . Find the minimum and maximum distance between the centers of the squares. Damon Vazquez 2022-10-06

## How do you find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum? dannyboi2006tk 2022-09-29

## What is the explicit procedure for differentiating multivariable functions with respect to a scalar? For a very simple example, I have a function $f:{\mathbb{R}}^{d}\to \mathbb{R}$ and $\varphi \left(t\right)=f\left(x+tv\right)$, $v\in {\mathbb{R}}^{d}$. What are the steps to calculating ${\varphi }^{\prime }\left(t\right),{\varphi }^{″}\left(t\right),{\varphi }^{‴}\left(t\right)$, etc? kenedirkitch 2022-09-27

## Use implicit differentiation to find∂z/∂x and ∂z/∂y.x^2 + 2y^2 + 5z^2 = 9 kenedirkitch 2022-09-27

## Use implicit differentiation to find∂z/∂x and ∂z/∂y.x^2 + 2y^2 + 5z^2 = 9 Nathanael Perkins 2022-09-27

## How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm? Vrbljanovwu 2022-09-27

## What is the maximum possible area of the rectangle that is to be inscribed in a semicircle of radius 8? Stacy Barr 2022-09-27

## Following functions:$f\left({x}_{1},{x}_{2}\right)=\left[\begin{array}{c}{x}_{1}{x}_{2}^{2}+{x}_{1}^{3}{x}_{2}\\ {x}_{1}^{2}{x}_{2}+{x}_{1}+{x}_{2}^{3}\end{array}\right]$$g\left(u\right)=\left[\begin{array}{c}{e}^{u}\\ {u}^{2}+u\end{array}\right]$Is it possible to take a derivative of $f\left(g\right)$ or $g\left(f\right)$. If not - why? Will Underwood 2022-09-26

## Consider a function $f:{Y}_{n}\to {X}_{{n}^{2}}$For any $\left(a,b\right)\in {Y}_{n}$, let $f\left(\left(a,b\right)\right)=f\left(a,b\right)=an+b.$ Prove that $f$ is a bijection and find its inverse ${f}^{-1}$.How to find an inverse of this multivariable function. I was already able to prove that $f\left(a,b\right)=an+b$ is a bijection, but not certain how to find its specific inverse. It would be important to note that ${Y}_{n}={X}_{n}×{X}_{n}$ where ${X}_{n}=\left\{0,1,2,...,n-1\right\}$. besnuffelfo 2022-09-26

## Confusion in integrating multivariable functionIntegrate ${\int }_{C}\frac{-x}{{x}^{2}+{y}^{2}}dx+\frac{y}{{x}^{2}+{y}^{2}}dy$$C:x=\mathrm{cos}t,y=\mathrm{sin}t,$ $\phantom{\rule{1em}{0ex}}0\le t\le \frac{\pi }{2}$In this case, It's incorrect to integrate it as $\frac{-1}{2}\mathrm{ln}\left({x}^{2}+{y}^{2}\right){|}_{a}^{b}+\frac{1}{2}\mathrm{ln}\left({x}^{2}+{y}^{2}\right){|}_{c}^{d}$. But,${\int }_{0}^{1}{\int }_{0}^{1}\frac{1}{1-xy}dxdy={\int }_{0}^{1}{\int }_{0}^{1}\frac{-1}{y}\frac{-y}{1-xy}dxdy={\int }_{0}^{1}\frac{-1}{y}\left[\mathrm{ln}\left(1-xy{\right]}_{0}^{1}\right)dy$When integrands are multivariable functions for both cases, why does only the bottom case work? Addyson Bright 2022-09-24

## Consider a NLP $min\left\{f\left(x\right):g\left(x\right)\le 0\right\}$. There are no equality constraints. The problem is feasible for small steps $t>0$. I have to prove that $g\left(x+td\right)\le 0$ if $g\left(x\right)<0$, where $t$ is the step length and $d$ is the direction of the line search (gradient descent).I was thinking that since $t$ is positive and the direction $d$ can not be negative (not too sure about this fact), hence their multiplication is positive. The only way for $g\left(x+td\right)$ to be 0 or negative is for $g\left(x\right)$ to be negative. 2k1ablakrh0 2022-09-24

## How to apply the chain rule to a double partial derivative of a multivariable function?$\begin{array}{rl}f\left(x,y\right)& ={e}^{xy}\\ g\left(x,y\right)& =f\left(\mathrm{sin}\left({x}^{2}+y\right),{x}^{3}+2y+1\right)\end{array}$Let’s compute $\frac{{\mathrm{\partial }}^{2}g}{\mathrm{\partial }{x}^{2}}\left(0,0\right)$ koraby2bc 2022-09-23

## What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches? ct1a2n4k 2022-09-23 Thordiswl 2022-09-23