9.1: Functions of Several Variables
- Page ID
- 144341
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Find the domain and range of \(f(x, y) = 4x^2 + y^2\).
- Find the domain and range of \(g(x, y) = \sqrt{x^2 + y^2 - 4}\).
- Find the domain and range of \(h(x, y) = 4\ln(y^2 - x)\).
- Find the domain and range of \(z = \dfrac{3x^2}{x - y}\).
- Find the domain and range of \(F(x, y) = \tan(x + y)\).
- Find the domain and range of \(\beta(x, y) = \sqrt{16 - 4x^2 - y^2}\).
- Plot \(z(x, y) = x + y\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(f(x, y) = x^2 + y^2\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(g(x, y) = x^2 - y^2\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(h(x, y) = x^2 + y\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(w(x, y) = \sqrt{x^2 + y^2}\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(z(x, y) = \sqrt{9 - x^2 - y^2}\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(f(x, y) = \ln(x^2 + y^2)\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(g(x, y) =\dfrac{1}{1+x^2 + y^2}\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Plot \(h(x, y) =\ln\left(\dfrac{x}{y^2}\right)\) by finding level curves/traces of the function. Confirm your result with a 3D graphing calculator.
- Find the level surfaces at \(w(x, y, z) = 1, 4, 9\) for the three-variable function \(w(x, y, z) = x^2 + y^2 + z^2\).
- Find the level surfaces at \(w(x, y, z) = -1, 0, 1\) for the three-variable function \(w(x, y, z) = x - 2y + z\).
- Find the level surfaces at \(w(x, y, z) = -4, -1, 0, 1, 4\) for the three-variable function \(w(x, y, z) = x^2 - y^2 + z^2\).