What tis the complete domain D and range R of the following multivariable functions: w(x,y)=sqrt(y-4x^2)

Question
Multivariable functions
asked 2020-10-28
What tis the complete domain D and range R of the following multivariable functions: \(\displaystyle{w}{\left({x},{y}\right)}=\sqrt{{{y}-{4}{x}^{{2}}}}\)

Answers (1)

2020-10-29
\(\displaystyle{w}{\left({x},{y}\right)}=\sqrt{{{y}-{4}{x}^{{2}}}}\)
\(\displaystyle{y}−{4}{x}^{{2}}\ge{0}\)
\(\displaystyle−{4}{x}^{{2}}\ge−{y}\)
\(\displaystyle−{1}\times{\left(−{4}{x}^{{2}}\right)}\le−{1}×{\left(−{y}\right)}\)
\(\displaystyle{4}{x}^{{2}}\le{y}\)
\(\displaystyle{x}^{{2}}{>}{0}\)
\(\displaystyle\Rightarrow{4}{x}^{{2}}{>}{0}\)
\(\displaystyle\Rightarrow{y}\ge{0}\)
\(\displaystyle{D}=\mathbb{R}^{{2}}\) such that \(\displaystyle{y}\ge{0}{\quad\text{and}\quad}{4}{x}^{{2}}\le{y}.\)
\(\displaystyle\sqrt{{{y}-{4}{x}^{{2}}}}\ge{0}\)
So, \(\displaystyle{w}\ge{0}\)
\(\displaystyle{R}\in\mathbb{R}\) such \(\displaystyle{w}{\left({x},{y}\right)}\ge{0}\)
0

Relevant Questions

asked 2020-10-18
What tis the complete domain D and range R of the following multivariable functions: \(\displaystyle{w}{\left({x},{y}\right)}=\frac{{1}}{{{x}{\left({y}-{1}\right)}}}\)
asked 2021-01-31
What tis the complete domain D and range R of the following multivariable functions:
\(\displaystyle{w}={2}{\sin{{x}}}{y}\)
asked 2020-11-02
a) Find the function's domain .
b) Find the function's range.
c) Find the boundary of the function's domain.
d) Determine if the domain is an open region, a closed region , or neither.
e) Decide if the domain is bounded or unbounded.
for: \(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{{\left({16}-{x}^{{2}}-{y}^{{2}}\right)}^{{\frac{{1}}{{2}}}}}}\)
asked 2021-01-05
Let \(\displaystyle{z}={e}^{{{x}^{{2}}+{3}{y}}}+{x}^{{3}}{y}^{{2}}\), where \(\displaystyle{x}={t}{\cos{{r}}}{\quad\text{and}\quad}{y}={r}{t}^{{4}}\). Use the chain rule fot multivariable functions (Cals III Chain Rule) to find \(\displaystyle\frac{{\partial{z}}}{{\partial{r}}}\) and \(\displaystyle\frac{{\partial{z}}}{\partial}{t}{)}\). Give your answers in terms of r and t only. Be sure to show all of your work.
asked 2021-03-02
COnsider the multivariable function \(\displaystyle{g{{\left({x},{y}\right)}}}={x}^{{2}}-{3}{y}^{{4}}{x}^{{2}}+{\sin{{\left({x}{y}\right)}}}\). Find the following partial derivatives: \(\displaystyle{g}_{{x}}.{g}_{{y}},{g}_{{{x}{y}}},{g{{\left(\times\right)}}},{g{{\left({y}{y}\right)}}}\).
asked 2020-12-17
A surface is represented by the following multivariable function,
\(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}\)
a) Calculate \(\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}\)
b) Calculate coordinates of stationary points.
c) Classify all stationary points.
asked 2021-02-19
Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: \(\displaystyle{a}{\left({x},{y}\right)}=\frac{{{8}{x}-{y}}}{{{e}^{{{x}^{{2}}+{y}^{{2}}}}}}\)
asked 2021-01-06
Let \(\displaystyle{z}{\left({x},{y}\right)}={e}^{{{3}{x}{y}}},{x}{\left({p},{q}\right)}=\frac{{p}}{{q}}{\quad\text{and}\quad}{y}{\left({p},{q}\right)}=\frac{{q}}{{p}}\) are functions. Use multivariable chain rule of partial derivatives to find
(i) \(\displaystyle\frac{{\partial{z}}}{{\partial{p}}}\)
(ii) \(\displaystyle\frac{{\partial{z}}}{{\partial{q}}}\).
asked 2020-11-27
Comsider this, multivariable fucntion
\(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}\)
a)What is the value of f(-1,3)?
b)Find all x-values such that f(x,x)=0
asked 2021-02-09
Consider this multivariable function. f(x,y)=xy+2x+y−36
a) What is the value of f(2,−3)?
b) Find all x-values such that f (x,x) = 0
...