Recent questions in Multivariable calculus

Multivariable calculusAnswered question

omgespit9q 2022-10-14

In a function like $f(x,y)={x}^{2}+y$, 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 $$y$$?

Say for $$1$$ for $$x$$ and $$99$$ for $$y$$?

Multivariable calculusAnswered question

Oscar Burton 2022-10-14

Limit of multivariable functions

limit of $\sqrt{{x}^{2}+{y}^{2}}\mathrm{sin}(1/y)$ as $(x,y)\to (0,0)$

limit of $2xy/({x}^{2}-y+5xy)$ as $(x,y)\to (0,0)$

limit of $\sqrt{{x}^{2}+{y}^{2}}\mathrm{sin}(1/y)$ as $(x,y)\to (0,0)$

limit of $2xy/({x}^{2}-y+5xy)$ as $(x,y)\to (0,0)$

Multivariable calculusAnswered question

Amina Richards 2022-10-13

How do you optimize $f(x,y)=2{x}^{2}+3{y}^{2}-4x-5$ subject to ${x}^{2}+{y}^{2}=81$?

Multivariable calculusAnswered question

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 $f$:

$$f(x,y)={e}^{y}\mathrm{sin}(x)$$

If we're given a $\overrightarrow{v}$ say, [1, 2] would the cross section of the graph of the plane formed by the $z$-axis and $\overrightarrow{v}$ inherently be closed form as $f$ is on the axes?

But is the same true for the diagonal cross sections? Say for example we have the function $f$:

$$f(x,y)={e}^{y}\mathrm{sin}(x)$$

If we're given a $\overrightarrow{v}$ say, [1, 2] would the cross section of the graph of the plane formed by the $z$-axis and $\overrightarrow{v}$ inherently be closed form as $f$ is on the axes?

Multivariable calculusAnswered question

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 $$\sum _{k=0}^{m}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{f}^{(n-k)}(x){g}^{(k)}(x)$$ where ${g}^{(k)}$ 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 $(1,2)$ in ${\mathbb{R}}^{2}.$.

What is a meaning of $\beta \le \alpha ?$?

Since the formula is for multivariable, I suppose that 𝛽 and 𝛼 are vectors, like $(1,2)$ in ${\mathbb{R}}^{2}.$.

Multivariable calculusAnswered question

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.

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.

Multivariable calculusAnswered question

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?

Multivariable calculusAnswered question

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 $\varphi (t)=f(x+tv)$, $v\in {\mathbb{R}}^{d}$. What are the steps to calculating ${\varphi}^{\prime}(t),{\varphi}^{\u2033}(t),{\varphi}^{\u2034}(t)$, etc?

Multivariable calculusAnswered question

kenedirkitch 2022-09-27

Use implicit differentiation to find

∂*z*/∂*x* and ∂*z*/∂*y*.

*x^*2 + 2*y^*2 + 5*z^*2 = 9

Multivariable calculusOpen question

kenedirkitch 2022-09-27

Use implicit differentiation to find

∂*z*/∂*x* and ∂*z*/∂*y*.

*x^*2 + 2*y^*2 + 5*z^2* = 9

Multivariable calculusAnswered question

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?

Multivariable calculusAnswered question

Vrbljanovwu 2022-09-27

What is the maximum possible area of the rectangle that is to be inscribed in a semicircle of radius 8?

Multivariable calculusAnswered question

Stacy Barr 2022-09-27

Following functions:

$f({x}_{1},{x}_{2})=\left[\begin{array}{c}{x}_{1}{x}_{2}^{2}+{x}_{1}^{3}{x}_{2}\\ {x}_{1}^{2}{x}_{2}+{x}_{1}+{x}_{2}^{3}\end{array}\right]$

$g(u)=\left[\begin{array}{c}{e}^{u}\\ {u}^{2}+u\end{array}\right]$

Is it possible to take a derivative of $f(g)$ or $g(f)$. If not - why?

$f({x}_{1},{x}_{2})=\left[\begin{array}{c}{x}_{1}{x}_{2}^{2}+{x}_{1}^{3}{x}_{2}\\ {x}_{1}^{2}{x}_{2}+{x}_{1}+{x}_{2}^{3}\end{array}\right]$

$g(u)=\left[\begin{array}{c}{e}^{u}\\ {u}^{2}+u\end{array}\right]$

Is it possible to take a derivative of $f(g)$ or $g(f)$. If not - why?

Multivariable calculusAnswered question

Will Underwood 2022-09-26

Consider a function $f:{Y}_{n}\to {X}_{{n}^{2}}$

For any $(a,b)\in {Y}_{n}$, let $f((a,b))=f(a,b)=an+b.$ 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 $f(a,b)=an+b$ is a bijection, but not certain how to find its specific inverse. It would be important to note that ${Y}_{n}={X}_{n}\times {X}_{n}$ where ${X}_{n}=\{0,1,2,...,n-1\}$.

For any $(a,b)\in {Y}_{n}$, let $f((a,b))=f(a,b)=an+b.$ 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 $f(a,b)=an+b$ is a bijection, but not certain how to find its specific inverse. It would be important to note that ${Y}_{n}={X}_{n}\times {X}_{n}$ where ${X}_{n}=\{0,1,2,...,n-1\}$.

Multivariable calculusAnswered question

besnuffelfo 2022-09-26

Confusion in integrating multivariable function

Integrate ${\int}_{C}\frac{-x}{{x}^{2}+{y}^{2}}dx+\frac{y}{{x}^{2}+{y}^{2}}dy$

$$C:x=\mathrm{cos}t,y=\mathrm{sin}t,$$ $\phantom{\rule{1em}{0ex}}0\le t\le \frac{\pi}{2}$

In this case, It's incorrect to integrate it as $\frac{-1}{2}\mathrm{ln}({x}^{2}+{y}^{2}){|}_{a}^{b}+\frac{1}{2}\mathrm{ln}({x}^{2}+{y}^{2}){|}_{c}^{d}$. But,

${\int}_{0}^{1}{\int}_{0}^{1}\frac{1}{1-xy}dxdy={\int}_{0}^{1}{\int}_{0}^{1}\frac{-1}{y}\frac{-y}{1-xy}dxdy={\int}_{0}^{1}\frac{-1}{y}[\mathrm{ln}(1-xy{]}_{0}^{1})dy$

When integrands are multivariable functions for both cases, why does only the bottom case work?

Integrate ${\int}_{C}\frac{-x}{{x}^{2}+{y}^{2}}dx+\frac{y}{{x}^{2}+{y}^{2}}dy$

$$C:x=\mathrm{cos}t,y=\mathrm{sin}t,$$ $\phantom{\rule{1em}{0ex}}0\le t\le \frac{\pi}{2}$

In this case, It's incorrect to integrate it as $\frac{-1}{2}\mathrm{ln}({x}^{2}+{y}^{2}){|}_{a}^{b}+\frac{1}{2}\mathrm{ln}({x}^{2}+{y}^{2}){|}_{c}^{d}$. But,

${\int}_{0}^{1}{\int}_{0}^{1}\frac{1}{1-xy}dxdy={\int}_{0}^{1}{\int}_{0}^{1}\frac{-1}{y}\frac{-y}{1-xy}dxdy={\int}_{0}^{1}\frac{-1}{y}[\mathrm{ln}(1-xy{]}_{0}^{1})dy$

When integrands are multivariable functions for both cases, why does only the bottom case work?

Multivariable calculusAnswered question

Addyson Bright 2022-09-24

Consider a NLP $min\{f(x):g(x)\le 0\}$. There are no equality constraints. The problem is feasible for small steps $t>0$. I have to prove that $g(x+td)\le 0$ if $g(x)<0$, 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 $g(x+td)$ to be 0 or negative is for $g(x)$ to be negative.

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 $g(x+td)$ to be 0 or negative is for $g(x)$ to be negative.

Multivariable calculusAnswered question

2k1ablakrh0 2022-09-24

How to apply the chain rule to a double partial derivative of a multivariable function?

$$\begin{array}{rl}f(x,y)& ={e}^{xy}\\ g(x,y)& =f(\mathrm{sin}({x}^{2}+y),{x}^{3}+2y+1)\end{array}$$

Let’s compute $\frac{{\mathrm{\partial}}^{2}g}{\mathrm{\partial}{x}^{2}}(0,0)$

$$\begin{array}{rl}f(x,y)& ={e}^{xy}\\ g(x,y)& =f(\mathrm{sin}({x}^{2}+y),{x}^{3}+2y+1)\end{array}$$

Let’s compute $\frac{{\mathrm{\partial}}^{2}g}{\mathrm{\partial}{x}^{2}}(0,0)$

Multivariable calculusAnswered question

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?

Multivariable calculusAnswered question

ct1a2n4k 2022-09-23

How do you find the length and width of a rectangle whose area is 900 square meters and whose perimeter is a minimum?

Multivariable calculusAnswered question

Thordiswl 2022-09-23

1- What is Optimization? How many methods are there to calculate it?

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

As you start exploring calculus and analysis, you will encounter multivariable calculus equations that are self-explanatory as well because all of them will contain at least two questions related to each variable being involved. See our multivariable calculus examples to receive more help and information regarding how these are used. The answers to solving these multivariable calculus questions should be based on finding the deterministic behavior. These are used in engineering and those fields where the parametric equations solver will provide you an optimal control of time dynamic systems. As an interesting subject, applying at least one equation in practice will keep you inspired!