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Mathematics LibreTexts

Functions of Multiple Variables (Exercises)

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    13.1: Functions of Multiple Variables

    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.

    For exercises 5 - 10, find the domain and range of the given function. State the domain in set-builder notation and the range in interval notation.

    5) \( V(x,y)=4x^2+y^2\)

    Answer:
    Domain: \(\{(x, y) | x \in \rm I\!R, y \in \rm I\!R\}\) That is, all points in the \(xy\)-plane
    Range: \( (-\infty, \infty) \)

    6) \( f(x,y)=\sqrt{x^2+y^2−4}\)

    Answer:
    Domain: \( \{(x, y) | x^2+y^2 \ge 4\}\)
    Range: \( [0, \infty) \)

    7) \( f(x,y)=4\ln(y^2−x)\)

    Answer:
    Domain: \( \{(x, y) | x<y^2 \}\)
    Range: \( (-\infty, \infty) \)

    8) \( g(x,y)=\sqrt{16−4x^2−y^2}\)

    Answer:
    Domain: \( \{(x, y) | \frac{x^2}{4} + \frac{y^2}{16} \le 1\}\)
    Range: \( [0, 4] \)

    9) \( z=\arccos(y−x)\)

    Answer:
    Domain: \( \{(x, y) | x - 1 \le y \le x + 1\}\) That is, all points between the graphs of \(y = x -1\) and \(y = x +1 \).
    Range: \( [0, \pi] \)

    10) \( f(x,y)=\dfrac{y+2}{x^2}\)

    Answer:
    Domain: \( \{(x, y) | x\neq 0 \}\)
    Range: \( (-\infty, \infty) \)

    Find the range of the functions.

    11) \( g(x,y)=\sqrt{16−4x^2−y^2}\)

    Answer:
    \( \{z|0≤z≤4\}\)

    12) \( V(x,y)=4x^2+y^2\)

    13) \( z=y^2−x^2\)

    Answer:
    The set \(\rm I\!R\)

    In exercises 14 - 29, find the level curves of each function at the indicated values of \( c\) to visualize the given function. Sketch a contour plot for those exercises where you are asked for more than 3 values of \(c\).

    14) \( z(x,y)=y^2−x^2, \quad c=1\)

    15) \( z(x,y)=y^2−x^2,\quad c=4\)

    Answer:
    \( y^2−x^2=4,\) a hyperbola

    16) \( g(x,y)=x^2+y^2;\quad c=0, 1, 2, 3, 4, 9\)

    17) \( g(x,y)=4−x−y;\quad c=0,1, 2, 3, 4\)

    Answer:
    Level curves are lines with \( y = -x + (4 - c) \).
    For each value of \(c\) these are:
    \( c = 0: \, y = -x + 4\),
    \( c = 1: \, y = -x + 3\),
    \( c = 2: \, y = -x + 2\),
    \( c = 3: \, y = -x + 1\),
    \( c = 4: \, y = -x \).
    The contour plot consists of a series of parallel lines.
    contourplot1.png

    18) \( f(x,y)=xy;c=1;\quad c=−1\)

    19) \( h(x,y)=2x−y;\quad c=-2,0,2\)

    Answer:
    \( 2x−y=0,2x−y=−2,2x−y=2;\) three lines

    20) \( f(x,y)=x^2−y;\quad c=1,2\)

    21) \( g(x,y)=\dfrac{x}{x+y};c=−1,0,1,2\)

    Answer:
    Level curves are lines with the form \( y = x \left( \frac{1-c}{c} \right) \). At \(c = 0\), we solve it directly from the equation \(\dfrac{x}{x+y}=0\) to get \(x = 0\).
    For each value of \(c\) these are:
    \( c = -1: \, y = -2x\),
    \( c = 0: \, x = 0\),
    \( c = 1: \, y = 0\),
    \( c = 2: \, y = -\frac{1}{2}x\).
    contourplot2.png

    22) \( g(x,y)=x^3−y;\quad c=−1,0,2\)

    23) \( g(x,y)=e^{xy};\quad c=\frac{1}{2},3\)

    Answer:
    The level curves have the form, \( y = \frac{\ln c}{x}\).
    For each value of \(c\) these are:
    \( c = \frac{1}{2}: \, y = \frac{\ln \frac{1}{2}}{x}\) that can be rewritten as, \(y = -\frac{\ln 2}{x}\)
    \( c = 3: \, y = \frac{\ln 3}{x}\).

    24) \( f(x,y)=x^2;\quad c=4,9\)

    25) \( f(x,y)=xy−x;\quad c=−2,0,2\)

    Answer:
    Level curves have the form: \( y = \frac{c}{x} + 1\).
    Here \(y = \frac{-2}{x} + 1,\quad y = 1,\quad y = \frac{2}{x} + 1\) or \( xy−x=−2,\,xy−x=0,\,xy−x=2\)

    26) \( h(x,y)=\ln(x^2+y^2);\quad c=−1,0,1\)

    27) \( g(x,y)=\ln(\frac{y}{x^2});\quad c=−2,0,2\)

    Answer:
    The level curves have the form, \( y =e^c x^2\).
    For each value of \(c\) these are:
    \( c = -2: \, y = e^{-2} x^2 \),
    \( c = 0: \, y = x^2 \),
    \( c = 2: \, y = e^{2} x^2 \).

    28) \( z=f(x,y)=\sqrt{x^2+y^2},\quad c=3\)

    29) \( f(x,y)=\dfrac{y+2}{x^2},\quad c=\)any constant

    Answer:
    The level curves are parabolas of the form \( y=cx^2−2\).
    contourplot3.png

    In exercises 30-32, find the vertical traces of the functions at the indicated values of \( x\) and \( y\), and plot the traces.

    30) \( z=4−x−y, \quad x=2\)

    31) \( f(x,y)=3x+y^3, \quad x=1\)

    Answer:

    \( z=3+y^3,\) a curve in the \(zy\)-plane with rulings parallel to the \(x\)-axis

    32) \( z=\cos\sqrt{x^2+y^2}, \quad x=1\)

    In exercises 33 - 38, find the domain and range of each function.

    33) \( z=\sqrt{100−4x^2−25y^2}\)

    Answer:
    Domain: \( \{(x, y) | \frac{x^2}{25}+\frac{y^2}{4}≤1\}\)
    Range: \( [0, 10] \)

    34) \( z=\ln(x−y^2)\)

    35) \( f(x,y,z)=\dfrac{1}{\sqrt{36−4x^2−9y^2−z^2}}\)

    Answer:
    Domain: \( \{(x, y, z) | \frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}<1\}\)
    Range: \( [\frac{1}{6}, \infty) \)

    36) \( f(x,y,z)=\sqrt{49−x^2−y^2−z^2}\)

    37) \( f(x,y,z)=\sqrt[3]{16−x^2−y^2−z^2}\)

    Answer:
    Domain: All points in \( xyz\)-space
    Range: \( (-\infty, \infty) \)

    38) \( f(x,y)=\cos\sqrt{x^2+y^2}\)

    In exercises 39 - 40, plot a graph of the function.

    39) \( z=f(x,y)=\sqrt{x^2+y^2}\)

    Answer:

    40) \( z=x^2+y^2\)

    41) Use technology to graph \( z=x^2y.\)

    Answer:

    In exercises 42 - 46, sketch the function by finding its level curves. Verify the graph using technology, such as CalcPlot3D.

    42) \( f(x,y)=\sqrt{4−x^2−y^2}\)

    43) \( f(x,y)=2−\sqrt{x^2+y^2}\)

    Answer:

    44) \( z=1+e^{−x^2−y^2}\)

    45) \( z=\cos\sqrt{x^2+y^2}\)

    Answer:

    46) \( z=y^2−x^2\)

    47) Describe the contour lines for several values of \( c\) for \( z=x^2+y^2−2x−2y.\)

    Answer:
    The contour lines are concentric circles centered at the point, \( (1, 1) \).
    You can see this by completing the square after setting this function equal to \(c\).
    That is, we write \( x^2-2x+1+y^2−2y+1 = c + 2 \) which can be rewritten as, \( (x - 1)^2 + (y - 1)^2 = c + 2 \).
    This gives us circles centered at the point, \( (1, 1) \), each with a radius of \( \sqrt{c+2} \).

    In exercises, 48 - 52, find the level surface for the given value of \(c\) for each function of three variables and describe it.

    48) \( w(x,y,z)=x−2y+z,\quad c=4\)

    49) \( w(x,y,z)=x^2+y^2+z^2,\quad c=9\)

    Answer:
    \( x^2+y^2+z^2=9\), a sphere of radius \( 3\)

    50) \( w(x,y,z)=x^2+y^2−z^2,\quad c=−4\)

    51) \( w(x,y,z)=x^2+y^2−z^2,\quad c=4\)

    Answer:
    \( x^2+y^2−z^2=4,\) a hyperboloid of one sheet

    52) \( w(x,y,z)=9x^2−4y^2+36z^2,\quad c=0\)

    In exercises 53 - 55, find an equation of the level curve of \( f\) that contains the point \( P\).

    53) \( f(x,y)=1−4x^2−y^2,\quad P(0,1)\)

    Answer:
    \( 4x^2+y^2=1,\)

    54) \( g(x,y)=y^2\arctan x,\quad P(1,2)\)

    55) \( g(x,y)=e^{xy}(x^2+y^2),\quad P(1,0)\)

    Answer:
    \( 1=e^{xy}(x^2+y^2)\)

    56) The strength \( E\) of an electric field at point \( (x,y,z)\) resulting from an infinitely long charged wire lying along the \(y\)-axis is given by \( E(x,y,z)=k/\sqrt{x^2+y^2}\), where \( k\) is a positive constant. For simplicity, let \( k=1\) and find the equations of the level surfaces for \( E=10\) and \( E=100.\)

    57) 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:
    \( T(x,y)=\frac{k}{x^2+y^2}\)

    58) 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 \( Q(3,4).\)

    59) Refer to the preceding problem. Find the level curves for \( T=40°C\) and \( T=100°C,\) and describe what the level curves represent.

    Answer:
    \( x^2+y^2=\frac{k}{40}, \quad x^2+y^2=\frac{k}{100}\). The level curves represent circles of radii \( \sqrt{10k}/20\) and \( \sqrt{k}/10\)

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

    • Paul Seeburger (Monroe Community College) edited the LaTeX and added contour plots to answers for problems 17, 21 and 29.