Recent questions in Multivariable calculus

Multivariable calculusAnswered question

Frances Pham 2022-11-09

Find a function $f(x,y)$ that is a solution of the following integral equation.

${\int}_{0}^{y}{\int}_{0}^{x}f({x}^{\prime},{y}^{\prime})d{x}^{\prime}d{y}^{\prime}=(x-y{)}^{2}$

How can we find the $f(x,y)$?

${\int}_{0}^{y}{\int}_{0}^{x}f({x}^{\prime},{y}^{\prime})d{x}^{\prime}d{y}^{\prime}=(x-y{)}^{2}$

How can we find the $f(x,y)$?

Multivariable calculusAnswered question

Widersinnby7 2022-11-09

How is ⟨f,g⟩ where $$\u27e8f,g\u27e9\text{where}f(x,y),g(x,y)$$ defined, where $\u27e8\cdot ,\cdot \u27e9$ means the inner product of $f$ and $g$?

Multivariable calculusAnswered question

Widersinnby7 2022-11-06

Let $f(x,y)=\frac{y}{1+xy}-\frac{1-y\mathrm{sin}(\frac{\pi x}{y})}{\mathrm{arctan}(x)}$

Compute the limit:

$g(x)=\underset{y\to \mathrm{\infty}}{lim}f(x,y)$

Wouldn't the value of the limit also depend on $x$?

Compute the limit:

$g(x)=\underset{y\to \mathrm{\infty}}{lim}f(x,y)$

Wouldn't the value of the limit also depend on $x$?

Multivariable calculusAnswered question

Uriel Hartman 2022-11-05

Multivariable function $$F(t)=f(x(t),y(t))$$, and we find derivative of $$F$$ with respect to $$t....$$ in our derivative the first member is $$\frac{df}{dx}\ast \frac{dx}{dt}+.....$$) Why $$\frac{df}{dx}$$ depends on $$x$$ and $$y$$.

Multivariable calculusAnswered question

Adison Rogers 2022-11-04

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

Multivariable calculusAnswered question

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 continuous

ii) $f{|}_{x},f{|}_{y},f{|}_{z},f{|}_{w}$ are each continuous

Here, $f{|}_{x}$ stands for the function attained by fixing the variables $w,y,z$.

i) $f$ is continuous

ii) $f{|}_{x},f{|}_{y},f{|}_{z},f{|}_{w}$ are each continuous

Here, $f{|}_{x}$ stands for the function attained by fixing the variables $w,y,z$.

Multivariable calculusAnswered question

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.

Multivariable calculusAnswered question

Emmanuel Giles 2022-11-02

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

Multivariable calculusAnswered question

Emmy Swanson 2022-10-31

Derivative of multivariable function $f:{\mathbb{R}}^{n}\setminus \{0\}\to {\mathbb{R}}^{n},f(x)=g(\Vert x\Vert )x$

I used the product and chain rule.

$${J}_{f}(x)=x\cdot {g}^{\prime}(\Vert x\Vert )\cdot \frac{1}{\Vert x\Vert}\cdot x+g(r){I}_{n}$$

The solution says it's ${J}_{f}(x)={g}^{\prime}(r)\frac{1}{\Vert x\Vert}x{x}^{T}+g(r){I}_{n}$

why do we have $x{x}^{T}$? Why the transpose?

I used the product and chain rule.

$${J}_{f}(x)=x\cdot {g}^{\prime}(\Vert x\Vert )\cdot \frac{1}{\Vert x\Vert}\cdot x+g(r){I}_{n}$$

The solution says it's ${J}_{f}(x)={g}^{\prime}(r)\frac{1}{\Vert x\Vert}x{x}^{T}+g(r){I}_{n}$

why do we have $x{x}^{T}$? Why the transpose?

Multivariable calculusAnswered question

caschaillo7 2022-10-29

How to prove that $$f$$ is discontinuous at origin using epsilon delta method?

$$f(x,y)=\{\begin{array}{ll}\frac{{x}^{3}+{y}^{3}}{x-y}& x\ne y\\ 0& x=y\end{array}$$

$$f(x,y)=\{\begin{array}{ll}\frac{{x}^{3}+{y}^{3}}{x-y}& x\ne y\\ 0& x=y\end{array}$$

Multivariable calculusAnswered question

George Morales 2022-10-28

determine if the following functions are differentiable at $(x,y)=(0,0)$.

$$f(x,y)=\sqrt{|xy|}$$

$$g(x,y)={e}^{{|x|}^{3}y}$$

$$f(x,y)=\sqrt{|xy|}$$

$$g(x,y)={e}^{{|x|}^{3}y}$$

Multivariable calculusAnswered question

Stephany Wilkins 2022-10-27

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

Multivariable calculusAnswered question

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: $h(x)=f(x)-f(0)$.

Prove the following:

1. $g(tx)=t\cdot g(x),\mathrm{\forall}t\ge 0$

2. $g(x+y)=g(x)+g(y),\mathrm{\forall}x,y\in {\mathbb{R}}^{n}$

In addition, based on those two thing I need to prove that $f$ is linear $f(x)={a}^{T}x+b$.

Prove the following:

1. $g(tx)=t\cdot g(x),\mathrm{\forall}t\ge 0$

2. $g(x+y)=g(x)+g(y),\mathrm{\forall}x,y\in {\mathbb{R}}^{n}$

In addition, based on those two thing I need to prove that $f$ is linear $f(x)={a}^{T}x+b$.

Multivariable calculusAnswered question

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:

$$f(x,y)=\{\begin{array}{ll}\frac{{x}^{2}{y}^{2}}{\sqrt{(}{x}^{2}+{y}^{2})},& \text{if if (x,y) not (0,0)}\\ 0,& \text{if (x,y) = (0,0)}\end{array}$$

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:

$$f(x,y)=\{\begin{array}{ll}\frac{{x}^{2}{y}^{2}}{\sqrt{(}{x}^{2}+{y}^{2})},& \text{if if (x,y) not (0,0)}\\ 0,& \text{if (x,y) = (0,0)}\end{array}$$

Multivariable calculusAnswered question

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:

$$\int {[f(x,y)]}_{y=0}^{y=b}\phantom{\rule{thinmathspace}{0ex}}dx$$

by ignoring the limits:

$$\int f(x,y)\phantom{\rule{thinmathspace}{0ex}}dx$$

If yes, how can it be justified?

$$\int {[f(x,y)]}_{y=0}^{y=b}\phantom{\rule{thinmathspace}{0ex}}dx$$

by ignoring the limits:

$$\int f(x,y)\phantom{\rule{thinmathspace}{0ex}}dx$$

If yes, how can it be justified?

Multivariable calculusAnswered question

Paloma Sanford 2022-10-23

Let $V={R}^{d}$ and $({x}_{i},{y}_{i}{)}_{1\le i\le n}\in (V\times \{-1,1\}{)}^{n}$

Let $C=\{(w,b)\in V\times R:1-{y}_{i}({w}^{t}{x}_{i}-b)\le 0,\mathrm{\forall}i\in [1,n]\}$

Show that $mi{n}_{(w,b)\in C}||w|{|}^{2}$ has a solution $(w,b)$

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.

Let $C=\{(w,b)\in V\times R:1-{y}_{i}({w}^{t}{x}_{i}-b)\le 0,\mathrm{\forall}i\in [1,n]\}$

Show that $mi{n}_{(w,b)\in C}||w|{|}^{2}$ has a solution $(w,b)$

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.

Multivariable calculusAnswered question

4enevi 2022-10-23

Suppose a multivariable function $f:{\mathbb{R}}^{n}\to \mathbb{R}$ is concave and sufficiently smooth. We have:

$\Vert f(\mathbf{x})-f({\mathbf{x}}_{0})\Vert \le M\Vert \mathbf{x}-{\mathbf{x}}_{0}\Vert $ 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?

$\Vert f(\mathbf{x})-f({\mathbf{x}}_{0})\Vert \le M\Vert \mathbf{x}-{\mathbf{x}}_{0}\Vert $ 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?

Multivariable calculusAnswered question

Keyla Koch 2022-10-21

How do you find the largest possible area for a rectangle inscribed in a circle of radius 4?

Multivariable calculusAnswered question

Ignacio Riggs 2022-10-19

Range of multivariable function $z={e}^{x+y}\mathrm{arctan}\left(\frac{y}{x}\right)$

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!