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Functions of Multiple Variables (Exercises)

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For the following exercises, evaluate each function at the indicated values.

1) W(x,y)=4x2+y2. Find W(2,1),W(3,6).

Answer
W(2,1)=17,W(3,6)=72

2) W(x,y)=4x2+y2. 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)=πx2y, 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)=20πunits3 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)=4x2+y2

Answer
Domain: {(x,y)|xIR,yIR} That is, all points in the xy-plane
Range: [0,)

6) f(x,y)=x2+y24

Answer
Domain: {(x,y)|x2+y24}
Range: [0,)

7) f(x,y)=4ln(y2x)

Answer
Domain: {(x,y)|x<y2}
Range: (,)

8) g(x,y)=164x2y2

Answer
Domain: {(x,y)|x24+y2161}
Range: [0,4]

9) z=arccos(yx)

Answer
Domain: {(x,y)|x1yx+1} That is, all points between the graphs of y=x1 and y=x+1.
Range: [0,π]

10) f(x,y)=y+2x2

Answer
Domain: {(x,y)|x0}
Range: (,)

Find the range of the functions.

11) g(x,y)=164x2y2

Answer
{z|0z4} or in interval notation: [0,4]

12) V(x,y)=4x2+y2

13) z=y2x2

Answer
The set IR

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)=y2x2,c=1

15) z(x,y)=y2x2,c=4

Answer
y2x2=4, a hyperbola

16) g(x,y)=x2+y2;c=0,1,2,3,4,9

17) g(x,y)=4xy;c=0,1,2,3,4

Answer
Level curves are lines with y=x+(4c).
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.
Contour plot of the function g(x,y)=4−x−y

18) f(x,y)=xy;c=1;c=1

19) h(x,y)=2xy;c=2,0,2

Answer
2xy=0,2xy=2,2xy=2; three lines

20) f(x,y)=x2y;c=1,2

21) g(x,y)=xx+y;c=1,0,1,2

Answer
Level curves are lines with the form y=x(1cc). At c=0, we solve it directly from the equation xx+y=0 to get x=0.
For each value of c these are:
c=1:y=2x,
c=0:x=0, with y0,
c=1:y=0, with x0,
c=2:y=12x.
Contour Plot of the function g(x, y) = x/(x+y)

22) g(x,y)=x3y;c=1,0,2

23) g(x,y)=exy;c=12,3

Answer
The level curves have the form, y=lncx.
For each value of c these are:
c=12:y=ln12x that can be rewritten as, y=ln2x

c=3:y=ln3x.

24) f(x,y)=x2;c=4,9

25) f(x,y)=xyx;c=2,0,2

Answer
Level curves have the form: y=cx+1.
Here y=2x+1,y=1,y=2x+1 or xyx=2,xyx=0,xyx=2

26) h(x,y)=ln(x2+y2);c=1,0,1

27) g(x,y)=ln(yx2);c=2,0,2

Answer
The level curves have the form, y=ecx2.
For each value of c these are:
c=2:y=e2x2,
c=0:y=x2,
c=2:y=e2x2.

28) z=f(x,y)=x2+y2,c=3

29) f(x,y)=y+2x2,c= any constant

Answer
The level curves are parabolas of the form y=cx22, with x0.
Contour plot of the function f(x, y) = (y + 2)/x^2

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=4xy,x=2

31) f(x,y)=3x+y3,x=1

Answer

z=3+y3, a curve in the zy-plane with rulings parallel to the x-axis

Surface plot of the function f(x,y)=3x+y^3

32) z=cosx2+y2,x=1

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

33) z=1004x225y2

Answer
Domain: {(x,y)|x225+y241}
Range: [0,10]

34) z=ln(xy2)

35) f(x,y,z)=1364x29y2z2

Answer
Domain: {(x,y,z)|x29+y24+z236<1}
Range: [16,)

36) f(x,y,z)=49x2y2z2

37) f(x,y,z)=316x2y2z2

Answer
Domain: All points in xyz-space
Range: (,316]

38) f(x,y)=cosx2+y2

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

39) z=f(x,y)=x2+y2

Answer

Surface plot of the surface f(x,y)=sqrt(x^2+y^2)

40) z=x2+y2

41) Use technology to graph z=x2y.

Answer

Surface plot of the function z=x^2y

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

42) f(x,y)=4x2y2

43) f(x,y)=2x2+y2

Answer

Surface plot of the function f(x,y)=2−sqrt(x^2+y^2)

44) z=1+ex2y2

45) z=cosx2+y2

Answer

Surface plot of the function z=cos(sqrt(x^2+y^2))

46) z=y2x2

47) Describe the contour lines for several values of c for z=x2+y22x2y.

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 x22x+1+y22y+1=c+2 which can be rewritten as, (x1)2+(y1)2=c+2.
This gives us circles centered at the point, (1,1), each with a radius of 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)=x2y+z,c=4

49) w(x,y,z)=x2+y2+z2,c=9

Answer
x2+y2+z2=9, a sphere of radius 3

50) w(x,y,z)=x2+y2z2,c=4

51) w(x,y,z)=x2+y2z2,c=4

Answer
x2+y2z2=4, a hyperboloid of one sheet

52) w(x,y,z)=9x24y2+36z2,c=0

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

53) f(x,y)=14x2y2,P(0,1)

Answer
4x2+y2=1,

54) g(x,y)=y2arctanx,P(1,2)

55) g(x,y)=exy(x2+y2),P(1,0)

Answer
1=exy(x2+y2)

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/x2+y2, 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)=kx2+y2

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=\dfrac{k}{40}, \quad x^2+y^2=\dfrac{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) added the contour plots to answers for problems 17, 21 and 29.

 


Functions of Multiple Variables (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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