dannyboi2006tk
2022-09-29
Answered

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?

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Allvin03

Answered 2022-09-30
Author has **7** answers

Well, consider the function $s:\mathbb{R}\to {\mathbb{R}}^{d}$ such that ${s}^{\prime}(t)=v$. Then 𝑠′(𝑡)=𝑣. Since $\varphi =f\circ s$, you may apply the multivariable chain rule to find

asked 2021-09-08

There are 100 two-bedroom apartments in the apartment building Lynbrook West.. The montly profit (in dollars) realized from renting out x apartments is given by the following function.

$P\left(x\right)=-12{x}^{2}+2136x-41000$

How many units should be rented out in order to optimize the monthly rental profit?

What is the maximum monthly profit realizable?

asked 2021-02-05

Use polar coordinates to find the limit. [Hint: Let $x=r\mathrm{cos}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}y=r\mathrm{sin}$ , and note that (x, y) (0, 0) implies r 0.]
$\underset{(x,y)\to (0,0)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}$

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An organization that tracks crime statistics reports that cars of make A model U, make A model V, and make B model W are the cars most frequently reported stolen, while cars of make C model X, make D model Y, and make E model Z are stolen least often. Is it reasonable to say that theres

asked 2021-12-13

Consider the following statement of masses,

If both masses start from rest,

A) find a1? B) find a2 ? C) How fast is m1 moving after sliding the 2.00 m shown?

*Assume both masses are simultaneously released from the rest*

asked 2021-11-16

Stock analysis. The price-earning ratios of 100 randomly selected stocks from the New York Stock

$$\overline{)\begin{array}{cc}Interval& Frequency\\ -0.5-4.5& 7\\ 4.5-9.5& 54\\ 9.5-14.5& 22\\ 14.5-19.5& 10\\ 19.5-24.5& 2\\ 24.5-29.5& 3\\ 29.5-34.5& 2\end{array}}$$

a. Find the mean of the price-earning ratios.

a. Find the mean of the price-earning ratios.

asked 2021-09-10

A real estate office handles a 60-unit apartment complex. When the rent is $530 per month, all units are occupied. For each $40 increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $65 per month for service and repairs. What rent should be charged to obtain a maximum profit?

asked 2022-01-22

A quick question; is it possible to say in a way analogous to the single variable case that a multivariable function is "asymptotically equivalent" to a second multivariable function? For example, consider the function of $n}_{1},{n}_{2}\in \mathbb{R$ given by

$\text{Var}\left(\hat{\mu}\right)=\frac{{\sigma}^{2}({n}_{1}+2{n}_{2})}{{({n}_{1}+{n}_{2})}^{2}}.$

where$\sigma}^{2$ is a constant.

Can we say that$\text{Var}\left(\hat{\mu}\right)\approx \frac{1}{{n}_{1}+{n}_{2}}$ and then conclude that $\text{Var}\left(\hat{\mu}\right)\to 0$ as $n}_{1}\to \mathrm{\infty$ and $n}_{2}\to \mathrm{\infty$ ?

$\underset{(x,y)\to (\mathrm{\infty},\mathrm{\infty})}{lim}\frac{x+2y}{{(x+y)}^{2}}$

does not exist. Am I wrong to think of$\text{Var}\left(\hat{\mu}\right)$ as a function of two variables?

where

Can we say that

does not exist. Am I wrong to think of