Section 14.1 Functions of Several Variables

(1)

Section 14.1 Functions of Several Variables

382 ¤ CHAPTER 14 PARTIAL DERIVATIVES

27.  = sin , a cylinder. 28. = 2 − ²− ², a circular paraboloid opening downward with vertex at (0 0 2).

29.  = ²+ 4²+ 1, an elliptic paraboloid opening upward with vertex at (0 0 1).

30. =

4²+ ² so 4²+ ²= ² and  ≥ 0, the top half of an elliptic cone.

31.  =

4 − 4²− ² so 4²+ ²+ ²= 4 or ²+² 4 +²

4 = 1 and  ≥ 0, the top half of an ellipsoid.

32. (a) ( ) = 1

1 + ²+ ². The trace in  = 0 is  = 1

1 + ², and the trace in  = 0 is  = 1

1 + ². The only possibility is graph III. Notice also that the level curves of  are 1

1 + ²+ ² =  ⇔ ²+ ²= 1

 − 1, a family of circles for

  1.

(b) ( ) = 1

1 + ²². The trace in  = 0 is the horizontal line  = 1, and the trace in  = 0 is also  = 1. Both graphs I and II have these traces; however, notice that here   0, so the graph is I.

(c) ( ) = ln(²+ ²). The trace in  = 0 is  = ln ², and the trace in  = 0 is  = ln ². The level curves of  are ln(²+ ²) =  ⇔ ²+ ²= ^, a family of circles. In addition,  is large negative when ²+ ²is small, so this is graph IV.

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES ¤ 383 (d) ( ) = cos

²+ ². The trace in  = 0 is  = cos

² = cos || = cos , and the trace in  = 0 is

 = cos√

²= cos || = cos . Notice also that the level curve ( ) = 0 is cos

²+ ² = 0 ⇔

²+ ²= 2 + 2

, a family of circles, so this is graph V.

(e) ( ) = ||. The trace in  = 0 is  = 0, and the trace in  = 0 is  = 0, so it must be graph VI.

(f) ( ) = cos(). The trace in  = 0 is  = cos 0 = 1, and the trace in  = 0 is  = 1. As mentioned in part (b), these traces match both graphs I and II. Here  can be negative, so the graph is II. (Also notice that the trace in  = 1 is

 = cos , and the trace in  = 1 is  = cos .)

33.The point (−3 3) lies between the level curves with -values 50 and 60. Since the point is a little closer to the level curve with

 = 60, we estimate that (−3 3) ≈ 56. The point (3 −2) appears to be just about halfway between the level curves with

-values 30 and 40, so we estimate (3 −2) ≈ 35. The graph rises as we approach the origin, gradually from above, steeply from below.

34. (a)  (Chicago) lies between level curves with pressures 1012 and 1016 mb, and since  appears to be located about one-fourth the distance from the 1012 mb isobar to the 1016 mb isobar, we estimate the pressure at Chicago to be about 1013 mb.  lies very close to a level curve with pressure 1012 mb so we estimate the pressure at Nashville to be

approximately 1012 mb.  appears to be just about halfway between level curves with pressures 1008 and 1012 mb, so we estimate the pressure at San Francisco to be about 1010 mb.  lies close to a level curve with pressure 1016 mb but we can’t see a level curve to its left so it is more difﬁcult to make an accurate estimate. There are lower pressures to the right of  and  is a short distance to the left of the level curve with pressure 1016 mb, so we might estimate that the pressure at Vancouver is about 1017 mb.

(b) Winds are stronger where the isobars are closer together (see Figure 13), and the level curves are closer near  than at the other locations, so the winds were strongest at San Francisco.

35.The point (160 10), corresponding to day 160 and a depth of 10 m, lies between the isothermals with temperature values of 8 and 12^◦C. Since the point appears to be located about three-fourths the distance from the 8^◦Cisothermal to the 12^◦C isothermal, we estimate the temperature at that point to be approximately 11^◦C. The point (180 5) lies between the 16 and 20^◦Cisothermals, very close to the 20^◦Clevel curve, so we estimate the temperature there to be about 195^◦C.

36.If we start at the origin and move along the -axis, for example, the -values of a cone centered at the origin increase at a constant rate, so we would expect its level curves to be equally spaced. A paraboloid with vertex the origin, on the other hand, has -values which change slowly near the origin and more quickly as we move farther away. Thus, we would expect its level curves near the origin to be spaced more widely apart than those farther from the origin. Therefore contour map I must correspond to the paraboloid, and contour map II the cone.

37.Near , the level curves are very close together, indicating that the terrain is quite steep. At , the level curves are much farther apart, so we would expect the terrain to be much less steep than near , perhaps almost ﬂat.

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES ¤ 383 (d) ( ) = cos

²+ ². The trace in  = 0 is  = cos

²= cos || = cos , and the trace in  = 0 is

 = cos√

²= cos || = cos . Notice also that the level curve ( ) = 0 is cos

²+ ²= 0 ⇔

²+ ²= 2 + 2

, a family of circles, so this is graph V.

(e) ( ) = ||. The trace in  = 0 is  = 0, and the trace in  = 0 is  = 0, so it must be graph VI.

(f) ( ) = cos(). The trace in  = 0 is  = cos 0 = 1, and the trace in  = 0 is  = 1. As mentioned in part (b), these traces match both graphs I and II. Here  can be negative, so the graph is II. (Also notice that the trace in  = 1 is

 = cos , and the trace in  = 1 is  = cos .)

33. The point (−3 3) lies between the level curves with -values 50 and 60. Since the point is a little closer to the level curve with

 = 60, we estimate that (−3 3) ≈ 56. The point (3 −2) appears to be just about halfway between the level curves with

-values 30 and 40, so we estimate (3 −2) ≈ 35. The graph rises as we approach the origin, gradually from above, steeply from below.

34. (a)  (Chicago) lies between level curves with pressures 1012 and 1016 mb, and since  appears to be located about one-fourth the distance from the 1012 mb isobar to the 1016 mb isobar, we estimate the pressure at Chicago to be about 1013 mb.  lies very close to a level curve with pressure 1012 mb so we estimate the pressure at Nashville to be

approximately 1012 mb.  appears to be just about halfway between level curves with pressures 1008 and 1012 mb, so we estimate the pressure at San Francisco to be about 1010 mb.  lies close to a level curve with pressure 1016 mb but we can’t see a level curve to its left so it is more difﬁcult to make an accurate estimate. There are lower pressures to the right of  and  is a short distance to the left of the level curve with pressure 1016 mb, so we might estimate that the pressure at Vancouver is about 1017 mb.

(b) Winds are stronger where the isobars are closer together (see Figure 13), and the level curves are closer near  than at the other locations, so the winds were strongest at San Francisco.

35. The point (160 10), corresponding to day 160 and a depth of 10 m, lies between the isothermals with temperature values of 8 and 12^◦C. Since the point appears to be located about three-fourths the distance from the 8^◦Cisothermal to the 12^◦C isothermal, we estimate the temperature at that point to be approximately 11^◦C. The point (180 5) lies between the 16 and 20^◦Cisothermals, very close to the 20^◦Clevel curve, so we estimate the temperature there to be about 195^◦C.

36. If we start at the origin and move along the -axis, for example, the -values of a cone centered at the origin increase at a constant rate, so we would expect its level curves to be equally spaced. A paraboloid with vertex the origin, on the other hand, has -values which change slowly near the origin and more quickly as we move farther away. Thus, we would expect its level curves near the origin to be spaced more widely apart than those farther from the origin. Therefore contour map I must correspond to the paraboloid, and contour map II the cone.

37. Near , the level curves are very close together, indicating that the terrain is quite steep. At , the level curves are much farther apart, so we would expect the terrain to be much less steep than near , perhaps almost ﬂat.

49. The level curves are ^= or  = ^−, a family of exponential curves.

50.The level curves are  − arctan  =  or

 = (arctan ) + , a family of vertical translations of the graph of the inverse tangent function  = arctan .

51. The level curves are³

²+ ²=  or ²+ ²= ³ ( ≥ 0), a family of circles centered at the origin with radius ^32.

52.For  6= 0 and ( ) 6= (0 0),  = 

²+ ² ⇔

²+ ²−

 = 0 ⇔ ²+

 −2¹

2

= 1

4², a family of circles with center

0_2¹ and radius_2¹ (without the origin). If  = 0, the level curve is the -axis.

53. The contour map consists of the level curves  = ²+ 9², a family of ellipses with major axis the -axis. (Or, if  = 0, the origin.)

The graph of ( ) is the surface  = ²+ 9², an elliptic paraboloid.

If we visualize lifting each ellipse  = ²+ 9²of the contour map to the plane

 = , we have horizontal traces that indicate the shape of the graph of .

1

(2)

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES ¤ 387

54. The contour map consists of the level curves  =

36 − 9²− 4² ⇒ 9²+ 4²= 36 − ²,  ≥ 0, a family of ellipses with major axis the

-axis. (Or, if  = 6, the origin.)

The graph of ( ) is the surface  =

36 − 9²− 4², or equivalently the upper half of the ellipsoid 9²+ 4²+ ²= 36. If we visualize lifting each ellipse  =

36 − 9²− 4²of the contour map to the plane  = , we have horizontal traces that indicate the shape of the graph of .

55. The isothermals are given by  = 100(1 + ²+ 2²)or

²+ 2²= (100 − ) [0   ≤ 100], a family of ellipses.

56. The equipotential curves are  = 

²− ²− ² or

²+ ²= ²− 



2

, a family of circles ( ≥ ).

Note: As  → ∞, the radius of the circle approaches .

57.  ( ) = ²− ³

The traces parallel to the -plane (such as the left-front trace in the graph above) are parabolas; those parallel to the -plane (such as the right-front trace) are cubic curves. The surface is called a monkey saddle because a monkey sitting on the surface near the origin has places for both legs and tail to rest.

58.  ( ) = ³− ³ The traces parallel to either the

-plane or the -plane are cubic curves.

59.  ( ) = ^−(²^+²^)3

sin(²) + cos(²)

60.  ( ) = cos  cos 

The traces parallel to either the

- or -plane are cosine curves with amplitudes that vary from 0 to 1.

61.  = sin() (a) C (b) II

Reasons: This function is periodic in both  and , and the function is the same when  is interchanged with , so its graph is symmetric about the plane  = . In addition, the function is 0 along the - and -axes. These conditions are satisﬁed only by C and II.

62.  = ^cos  (a) A (b) IV

Reasons: This function is periodic in  but not , a condition satisﬁed only by A and IV. Also, note that traces in  =  are cosine curves with amplitude that increases as  increases.

63.  = sin( − ) (a) F (b) I

Reasons: This function is periodic in both  and  but is constant along the lines  =  + , a condition satisﬁed only by F and I.

64.  = sin  − sin  (a) E (b) III

Reasons: This function is periodic in both  and , but unlike the function in Exercise 63, it is not constant along lines such as

 =  + , so the contour map is III. Also notice that traces in  =  are vertically shifted copies of the sine wave  = sin , so the graph must be E.

SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES ¤ 389

65.  = (1 − ²)(1 − ²) (a) B (b) VI

Reasons: This function is 0 along the lines  = ±1 and  = ±1. The only contour map in which this could occur is VI. Also note that the trace in the -plane is the parabola  = 1 − ²and the trace in the -plane is the parabola  = 1 − ², so the graph is B.

66.  =  − 

1 + ²+ ² (a) D (b) V

Reasons: This function is not periodic, ruling out the graphs in A, C, E, and F. Also, the values of  approach 0 as we use points farther from the origin. The only graph that shows this behavior is D, which corresponds to V.

67.  =  + 3 + 5is a family of parallel planes with normal vector h1 3 5i.

68.  = ²+ 3²+ 5²is a family of ellipsoids for   0 and the origin for  = 0.

69. Equations for the level surfaces are  = ²+ ². For   0, we have a family of circular cylinders with axis the -axis and radius√

. When  = 0 the level surface is the -axis. (There are no level surfaces for   0.)

70. Equations for the level surfaces are ²− ²− ²= . For  = 0, the equation becomes ²+ ² = ²and the surface is a right circular cone with vertex the origin and axis the -axis. For   0, we have a family of hyperboloids of two sheets with axis the -axis, and for   0, we have a family of hyperboloids of one sheet with axis the -axis.

71. (a) The graph of  is the graph of  shifted upward 2 units.

(b) The graph of  is the graph of  stretched vertically by a factor of 2.

(c) The graph of  is the graph of  reﬂected about the -plane.

(d) The graph of ( ) = −( ) + 2 is the graph of  reﬂected about the -plane and then shifted upward 2 units.

72. (a) The graph of  is the graph of  shifted 2 units in the positive -direction.

(b) The graph of  is the graph of  shifted 2 units in the negative -direction.

(c) The graph of  is the graph of  shifted 3 units in the negative -direction and 4 units in the positive -direction.

73.  ( ) = 3 − ⁴− 4²− 10

Three-dimensional view Front view

[continued]