11.1: Functions of Several Variables (Exercises)
- Page ID
- 104877
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, evaluate each function at the indicated values.
1) \( W(x,y)=4x^2+y^2.\) Find \( W(2,−1), W(−3,6)\).
- Answer
- \( W(2,−1) = 17,\quad W(−3,6) = 72\)
2) \( W(x,y)=4x^2+y^2\). Find \( W(2+h,3+h).\)
3) The volume of a right circular cylinder is calculated by a function of two variables, \( V(x,y)=πx^2y,\) where \( x\) is the radius of the right circular cylinder and \( y\) represents the height of the cylinder. Evaluate \( V(2,5)\) and explain what this means.
- Answer
- \( V(2,5) = 20π\,\text{units}^3\) This is the volume when the radius is \( 2\) and the height is \( 5\).
4) An oxygen tank is constructed of a right cylinder of height \( y\) and radius \( x\) with two hemispheres of radius \( x\) mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, \( x\) and \( y\), find \( V(10,2)\), and explain what this means.
5) A thin plate made of iron is located in the \(xy\)-plane The temperature \( T\) in degrees Celsius at a point \( P(x,y)\) is inversely proportional to the square of its distance from the origin. Express \( T\) as a function of \( x\) and \( y\).
- Answer
- \( T(x,y)=\dfrac{k}{x^2+y^2}\)
6) Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point \( P(1,2)\) is \( 50°C.\) Use this constant to determine the temperature at point \( Q(3,4).\)
For exercises 7 - 18, find the domain and range of the given function.
7) \( V(x,y)=4x^2+y^2\)
- Answer
- Domain: \(\big\{(x, y) \, | \, x \in \rm I\!R, y \in \rm I\!R\big\}\) That is, all points in the \(xy\)-plane
Range: \( [0, \infty) \)
8) \( f(x,y)=\sqrt{x^2+y^2−4}\)
- Answer
- Domain: \( \big\{(x, y) \, | \, x^2+y^2 \ge 4\big\}\)
Range: \( [0, \infty) \)
9) \( f(x,y)=4\ln(y^2−x)\)
- Answer
- Domain: \( \left\{(x, y) \mid x \lt y^2 \right\} \) Range: \( (-\infty, \infty) \)
10) \( g(x,y)=\sqrt{16−4x^2−y^2}\)
- Answer
- Domain: \( \big\{(x, y) \, | \, \dfrac{x^2}{4} + \dfrac{y^2}{16} \le 1\big\}\)
Range: \( [0, 4] \)
11) \( z=\arccos(y−x)\)
- Answer
- Domain: \( \big\{(x, y) \, | \, x - 1 \le y \le x + 1\big\}\) That is, all points between the graphs of \(y = x -1\) and \(y = x +1 \).
Range: \( [0, \pi] \)
12) \( f(x,y)=\dfrac{y+2}{x^2}\)
- Answer
- Domain: \( \big\{(x, y) \, | \, x\neq 0 \big\}\)
Range: \( (-\infty, \infty) \)
13) \( g(x,y)=\sqrt{16−4x^2−y^2}\)
- Answer
- \( \big\{z \, | \, 0≤z≤4\big\}\) or in interval notation: \([0,4]\)
14) \( V(x,y)=4x^2+y^2\)
15) \( z=y^2−x^2\)
- Answer
- The set \(\rm I\!R\)
16) \( z=\sqrt{100−4x^2−25y^2}\)
- Answer
- Domain: \( \big\{(x, y) \, | \, \dfrac{x^2}{25}+\dfrac{y^2}{4}≤1\big\}\)
Range: \( [0, 10] \)
17) \( z=\ln(x−y^2)\)
18) \( f(x,y)=\cos\sqrt{x^2+y^2}\)