P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-15 JWDD027-Salas-v1 December 5, 2006 16:40 758 SECTION 15.1 CHAPTER 15 SECTION 15.1 1. dom (f ) = the first and third quadrants, including the axes; range (f ) = [0, ∞) 2. dom (f ) = the set of all points (x, y) with xy ≤ 1; the two branches of the hyperbola xy = 1 and all points in between; range (f ) = [0, ∞) 3. dom (f ) = the set of all points (x, y) except those on the line y = −x; range (f )=(−∞, 0) ∪ (0, ∞) 4. dom (f ) = the set of all points (x, y) other than the origin; range (f ) = (0, ∞) 5. dom (f ) = the entire plane; range (f )=(−1, 1) since e x − e y e x + e y = e x + e y − 2e y e x + e y =1 − 2 e x−y +1 and the last quotient takes on all values between 0 and 2. 6. dom (f ) = the set of all points (x, y) other than the origin; range (f ) = [0, 1] 7. dom (f ) = the first and third quadrants, excluding the axes; range (f )=(−∞, ∞) 8. dom (f ) =the set of all points (x, y) between the branches of the hyperbola xy = 1; range (f )=(−∞, ∞) 9. dom (f ) = the set of all points (x, y) with x 2 <y —in other words, the set of all points of the plane above the parabola y = x 2 ; range (f ) = (0, ∞) 10. dom (f )= the set of all points (x, y) with −3 ≤ x ≤ 3, −1 ≤ y ≤ 1 (a rectangle); range (f ) = [0, 3] 11. dom (f ) = the set of all points (x, y) with −3 ≤ x ≤ 3, −2 ≤ y ≤ 2 (a rectangle); range (f )=[−2, 3] 12. dom (f ) = all of space; range (f )=[−3, 3] 13. dom (f ) = the set of all points (x, y, z) not on the plane x + y + z = 0; range (f )= {−1, 1} 14. dom (f ) = the set of all points (x, y, z) with x 2 = y 2 —that is, all points of space except for those which lie on the plane x − y = 0 or on the plane x + y = 0; range (f )=(−∞, ∞) 15. dom (f ) = the set of all points (x, y, z) with |y| < |x|; range (f )=(−∞, 0] 16. dom (f ) =the set of all points (x, y, z) not on the plane x − y = 0; range (f )=(−∞, ∞) 17. dom (f ) = the set of all points (x, y) with x 2 + y 2 < 9 —in other words, the set of all points of the plane inside the circle x 2 + y 2 = 9; range (f )=[2/3, ∞) 18. dom (f ) = all of space; range (f ) = [0, ∞) 19. dom (f ) = the set of all points (x, y, z) with x +2y +3z> 0 — in other words, the set of all points in space that lie on the same side of the plane x +2y +3z = 0 as the point (1, 1, 1); range (f )=(−∞, ∞) 20. dom (f ) = the set of all points (x, y, z) with x 2 + y 2 + z 2 ≤ 4 — in other words, the set of all points inside and on the sphere x 2 + y 2 + z 2 = 4; range (f ) = [1,e 2 ] 21. dom (f ) = all of space; range (f ) = (0, 1]
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Calculus one and several variables 10E Salas solutions manual ch15
Calculus one and several variables 10E Salas solutions manual
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU
JWDD027-15 JWDD027-Salas-v1 December 5, 2006 16:40
758 SECTION 15.1
CHAPTER 15
SECTION 15.1
1. dom (f) = the first and third quadrants, including the axes; range (f) = [0,∞)
2. dom (f) = the set of all points (x, y) with xy ≤ 1; the two branches of the hyperbola xy = 1 and allpoints in between; range (f) = [0,∞)
3. dom (f) = the set of all points (x, y) except those on the line y = −x; range (f) = (−∞, 0) ∪ (0,∞)
4. dom (f) = the set of all points (x, y) other than the origin; range (f) = (0,∞)
5. dom (f) = the entire plane; range (f) = (−1, 1) since
ex − ey
ex + ey=
ex + ey − 2ey
ex + ey= 1 − 2
ex−y + 1and the last quotient takes on all values between 0 and 2.
6. dom (f) = the set of all points (x, y) other than the origin; range (f) = [0, 1]
7. dom (f) = the first and third quadrants, excluding the axes; range (f) = (−∞,∞)
8. dom (f) =the set of all points (x, y) between the branches of the hyperbola xy = 1;range (f) = (−∞,∞)
9. dom (f) = the set of all points (x, y) with x2 < y —in other words, the set of all points of the planeabove the parabola y = x2; range (f) = (0,∞)
10. dom (f) = the set of all points (x, y) with −3 ≤ x ≤ 3, −1 ≤ y ≤ 1 (a rectangle);range (f) = [0, 3]
11. dom (f) = the set of all points (x, y) with −3 ≤ x ≤ 3, −2 ≤ y ≤ 2 (a rectangle);
range (f) = [−2, 3]
12. dom (f) = all of space; range (f) = [−3, 3]
13. dom (f) = the set of all points (x, y, z) not on the plane x + y + z = 0; range (f) = {−1, 1}
14. dom (f) = the set of all points (x, y, z) with x2 �= y2 —that is, all points of space except for thosewhich lie on the plane x− y = 0 or on the plane x + y = 0; range (f) = (−∞,∞)
15. dom (f) = the set of all points (x, y, z) with |y| < |x|; range (f) = (−∞, 0 ]
16. dom (f) =the set of all points (x, y, z) not on the plane x− y = 0; range (f) = (−∞,∞)
17. dom (f) = the set of all points (x, y) with x2 + y2 < 9 —in other words, the set of all points of theplane inside the circle x2 + y2 = 9; range (f) = [ 2/3,∞)
18. dom (f) = all of space; range (f) = [0,∞)
19. dom (f) = the set of all points (x, y, z) with x + 2y + 3z > 0 — in other words, the set of all points inspace that lie on the same side of the plane x + 2y + 3z = 0 as the point (1, 1, 1); range (f) = (−∞,∞)
20. dom (f) = the set of all points (x, y, z) with x2 + y2 + z2 ≤ 4 — in other words, the set of all pointsinside and on the sphere x2 + y2 + z2 = 4; range (f) = [1, e2]
21. dom (f) = all of space; range (f) = (0, 1]
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SECTION 15.1 759
22. dom (f) = the set of all points (x, y, z) with −1 ≤ x ≤ 1, −2 ≤ y ≤ 2, −3 ≤ z ≤ 3 (a rectangularsolid); range (f) = [0, 3]
23. dom (f) = {x : x ≥ 0}; range (f) = [0, ∞)
dom (g) = {(x, y) : x ≥ 0, y real}; range (g) = [0, ∞)
dom (h) = {(x, y, z) : x ≥ 0, y, z real}; range (h) = [0, ∞)
24. dom (f) = the entire plane, range (f) = [−1, 1]
dom (g) = all of space, range (g) = [−1, 1]
25. limh→0
f(x + h, y) − f(x, y)h
= limh→0
2(x + h)2 − y − (2x2 − y)h
= limh→0
4xh + 2h2
h= 4x
limh→0
f(x, y + h) − f(x, y)h
= limh→0
2x2 − (y + h) − (2x2 − y)h
= −1
26. limh→0
f(x + h, y) − f(x, y)h
= limh→0
xy + hy + 2y − (xy + 2y)h
= limh→0
y = y.
limh→0
f(x, y + h) − f(x, y)h
= limh→0
xy + xh + 2y + 2h− (xy + 2y)h
= limh→0
(x + 2) = x + 2
27. limh→0
f(x + h, y) − f(x, y)h
= limh→0
3(x + h) − (x + h)y + 2y2 − (3x− xy + 2y2)h
= limh→0
3h− hy
h= 3 − y
limh→0
f(x, y + h) − f(x, y)h
= limh→0
3x− x(y + h) + 2(y + h)2 − (3x− xy + 2y2)h
= limh→0
−xh + 4yh + 2h2
h= −x + 4y
28. limh→0
f(x + h, y) − f(x, y)h
= limh→0
x sin y + h sin y − x sin y
h= lim
h→0sin y = sin y.
limh→0
f(x, y + h) − f(x, y)h
= limh→0
x sin(y + h) − x sin y
h= x lim
h→0
sin(y + h) − sin y
h= x cos y
29. limh→0
f(x + h, y) − f(x, y)h
= limh→0
cos[(x + h)y] − cos[xy]h
= limh→0
cos[xy] cos[hy] − sin[xy] sin[hy] − cos[xy]h
= cos[xy](
limh→0
cos[hy] − 1h
)− sin[xy] lim
h→0
sinhy
h
= y cos[xy](
limh→0
cos[hy] − 1hy
)− y sin[xy] lim
h→0
sinhy
hy
= −y sin[xy]
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760 SECTION 15.2
and
limh→0
f(x, y + h) − f(x, y)h
= limh→0
cos[x(y + h)] − cos[xy]h
= limh→0
cos[xy] cos[hx] − sin[xy] sin[hx] − cos[xy]h
= cos[xy](
limh→0
cos[hx] − 1h
)− sin[xy] lim
h→0
sinhx
h
= x cos[xy](
limh→0
cos[hx] − 1hx
)− x sin[xy] lim
h→0
sinhx
hx
= −x sin[xy]
30. limh→0
f(x + h, y) − f(x, y)h
= limh→0
(x2 + 2xh + h2)ey − x2ey
h= lim
h→0(2x + h)ey = 2xey.
limh→0
f(x, y + h) − f(x, y)h
= limh→0
x2ey+h − x2ey
h= x2 lim
h→0
ey+h − ey
h= x2ey.
31. (a) f(x, y) =Ay (b) f(x, y) = πx2y (b) f(x, y) = |2 i × (x i + y j)| = 2|y|
32. (a) f(x, y, z) = xy + 2xz + 2yz
(b) f(x, y, z) = cos−1 (i + j) · (x i + y j + z k)‖i + j‖‖x i + y j + z k‖ = cos−1 x + y√
2√x2 + y2 + z2
(c) f(x, y, z) = [i × (i + j)] · (x i + y j + z k) = z
33. Surface area: S = 2lw + 2lh + 2hw = 20 =⇒ w =20 − 2lh2l + 2h
=10 − lh
l + h
Volume: V = lwh =lh(10 − lh)
l + h
34. wlh = 12 =⇒ h =12wl
; C = 4wl + 2(2wh + 2lh) = 4wl +48l
+48w
35. V = πr2h +43πr3
36. A =12[2(12 − 2x) + 2x cos θ] · x sin θ = (12 − 2x + x cos θ) · x sin θ
SECTION 15.2
1. an elliptic cone 2. an ellipsoid
3. a parabolic cylinder 4. a hyperbolic paraboloid
5. a hyperboloid of one sheet 6. an elliptic cylinder
7. a sphere 8. a hyperboloid of two sheets
9. an elliptic paraboloid 10. a hyperbolic cylinder
11. a hyperbolic paraboloid 12. an elliptic paraboloid
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SECTION 15.2 761
13. 14. 15.
16. 17. 18.
19. 20. 21.
22. 23. 24.
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762 SECTION 15.2
25. elliptic paraboloidxy-trace: the originxz-trace: the parabola x2 = 4zyz-trace: the parabola y2 = 9zsurface has the form of Figure 15.2.5
26. ellipsoidxy-trace: the ellipse 9x2 + 4y2 = 36xz-trace: the ellipse x2 + 4z2 = 4yz-trace: the ellipse y2 + 9z2 = 9surface has the form of Figure 15.2.1
27. elliptic conexy-trace: the originxz-trace: the lines x = ±2zyz-trace: the lines y = ±3zsurface has the form of Figure 15.2.4
28. hyperboloid of one sheetxy-trace: the ellipse 9x2 + 4y2 = 36xz-trace: the hyperbola x2 − 4z2 = 4yz-trace: the hyperbola y2 − 9z2 = 9surface has the form of Figure 15.2.2
29. hyperboloid of two sheetsxy-trace: nonexz-trace: the hyperbola 4z2 − x2 = 4yz-trace: the hyperbola 9z2 − y2 = 9surface has the form of Figure 15.2.3
30. hyperbolic paraboloidxy-trace: the lines y = ± 3
2x
xz-trace: the parabola x2 = 4zyz-trace: the parabola y2 = −9zsurface has the form Figure 15.2.6
31. hyperboloid of two sheetsxy-trace: the hyperbola 9x2 − 4y2 = 36xz-trace: the hyperbola x2 − 4z2 = 4yz-trace: nonesee Figure 15.2.3
32. hyperboloid of one sheetxy-trace: the hyperbola x2 − 9y2 = 9xz-trace: the circle x2 + y2 = 9yz-trace: the hyperbola z2 − 9y2 = 9surface has the form of Figure 15.2.2,rotated 90◦ about the x-axis.
33. elliptic paraboloidxy-trace: the parabola x2 = 9yxz-trace: the originyz-trace: the parabola z2 = 4ysurface has the form of Figure 15.2.5
34. elliptic conexy-trace: the lines x = ±2yxz-trace: the originyz-trace: the lines z = ±3ysurface has the form of Figure 15.2.4,rotated 90◦ about the x-axis.
35. hyperboloid of two sheetsxy-trace: the hyperbola 9y2 − 4x2 = 36xz-trace: noneyz-trace: the hyperbola y2 − 4z2 = 4see Figure 15.2.3
36. elliptic paraboloidxy-trace: the parabola y2 = 4xxz-trace: the parabola z2 = 9xyz-trace: the originsurface has the form of Figure 15.2.5,but opening along the positive x-axis.
37. paraboloid of revolutionxy-trace: the originxz-trace: the parabola x2 = 4zyz-trace: the parabola y2 = 4zsurface has the form of Figure 15.2.5
38. ellipsoidxy-trace: the ellipse 4x2 + y2 = 4xz-trace: the ellipse 9x2 + z2 = 9yz-trace: the ellipse 9y2 + 4z2 = 36the surface has the form of Figure 15.2.1,rotated 90◦ about the x-axis.
39. (a) an elliptic paraboloid (vertex down if A and B are both positive, vertex up if A and B are bothnegative)
(b) a hyperbolic paraboloid
(c) the xy-plane if A and B are both zero; otherwise a parabolic cylinder
40. The xz-plane and all planes parallel to the xy-plane.
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SECTION 15.2 763
41. x2 + y2 − 4z = 0 (paraboloid of revolution)
42. c2x2 + c2y2 − b2z2 = b2c2 (hyperboloid of revolution, one sheet)
43. (a) a circle
(b) (i)√x2 + y2 = −3z (ii)
√x2 + z2 = 1
3y
44. (a) the ellipse b2x2 + y2 = b2
(b) ellipse approaches parallel lines x = ±1 in the plane z = 1
(c) paraboloid approaches parabolic cylinder z = x2
45. x + 2y + 3(x + y − 6
2
)= 6 or 5x + 7y = 30, a line
46. 3x + y − 2(4 − x + 2y) = 1, or 5x− 3y = 9, a line
47.x2 + y2 + (z − 1)2 = 3
2
x2 + y2 − z2 = 1
}(z2 + 1) + (z − 1)2 =
32; (2z − 1)2 = 0, z =
12
so that x2 + y2 =54
48. z2 + (z − 2)2 = 2 =⇒ 2(z − 1)2 = 0 =⇒ z = 1 =⇒ x2 + y2 = 1, a circle.
out the boundary lines), boundary = {(x, y) : x = −2,
x = −1, x = 1, or x = 2} (four vertical lines); set is closed.
5. interior = {(x, y) : 1 < x2 < 4} =
{(x, y) : −2 < x < −1} ∪ {(x, y) : 1 < x < 2}(two vertical strips without the boundary lines),
boundary = {(x, y) : x = −2, x = −1, x = 1,
or x = 2} (four vertical lines); set is neither open
nor closed.
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776 SECTION 15.5
6. interior = the entire set (region below the parabola
y = x2), boundary = the parabola y = x2; the set
is open.
7. interior = region below the parabola y = x2,
boundary = the parabola y = x2; the set is closed.
8. interior = the inside of the cube; boundary = thefaces of the cube; set is neither open nor closed (upperface of cube is omitted)
9. interior = { (x, y, z) : x2 + y2 < 1, 0 < z ≤ 4}(the inside of the cylinder), boundary = the totalsurface of the cylinder (the curved part, the top, andthe bottom); the set is closed.
10. interior = the entire set (the inside of the ball of radius 12 , centered at (1,1,1)),
boundary = the spherical surface; set is open.
11. (a) φ (b) S (c) closed
12. interior = the entire set, boundary = {1, 3}; set is open.
13. interior = {x : 1 < x < 3}, boundary = {1, 3}; set is closed.
14. interior = {x : 1 < x < 3}, boundary = {1, 3}; set is neither open nor closed.
15. interior = the entire set, boundary = {1}; set is open.
16. interior ={x : x < −1}, boundary = {−1}; set is closed.
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SECTION 15.6 777
17. interior = {x : |x| > 1}, boundary = {1,−1}; set is neither open nor closed.
18. interior = φ, boundary = the entire set; set is closed.
19. interior = φ, boundary = {the entire set} ∪ {0}; the set is neither open nor closed.
20. (a) φ is open because it contains no boundary points,
φ is closed because it contains its boundary (the boundary is empty).
(b) X is open because it contains a neighborhood of each of its points,
X is closed because it contains its boundary (the boundary is empty).
(c) Suppose that U is open. Let x be a boundary point of X − U . Then every neighborhood of
x contains points from X − U . The point x can not be in U because U contains a
neighborhood of each of its points. Thus x ∈ X − U . This shows that X − U contains its
boundary and is therefore closed.
Suppose now that X − U is closed. Let x be a point of U . If no neighborhood of x lies entirely
in U , then every neighborhood of x contains points from X − U . This makes x a boundary
point of X − U and, since X − U is closed, places x in X − U . This contradiction shows that
some neighborhood of x lies entirely in U . Thus U contains a neighborhood of each of its points
and is therefore open.
(d) Set U = X − F and note that F = X − U . By (c)
F = X − U is closed iff X − F = U is open.
SECTION 15.6
1.∂2f
∂x2= 2A,
∂2f
∂y2= 2C,
∂2f
∂y∂x=
∂2f
∂x∂y= 2B
2.∂2f
∂x2= 6Ax + 2By,
∂2f
∂y2= 2Cx,
∂2f
∂y∂x=
∂2f
∂x∂y= 2Bx + 2Cy
3.∂2f
∂x2= Cy2exy,
∂2f
∂y2= Cx2exy,
∂2f
∂y∂x=
∂2f
∂x∂y= Cexy(xy + 1)
4.∂2f
∂x2= 2 cos y − y2 sinx,
∂2f
∂y2= 2 sinx− x2 cos y,
∂2f
∂y∂x=
∂2f
∂x∂y= 2(y cosx− x sin y)
5.∂2f
∂x2= 2,
∂2f
∂y2= 4(x + 3y2 + z3),
∂2f
∂z2= 6z(2x + 2y2 + 5z3)
∂2f
∂x∂y=
∂2f
∂y∂x= 4y,
∂2f
∂z∂x=
∂2f
∂x∂z= 6z2,
∂2f
∂z∂y=
∂2f
∂y∂z= 12yz2
6.∂2f
∂x2= − 1
4(x + y2)3/2,
∂2f
∂y2=
x
(x + y2)3/2,
∂2f
∂x∂y=
∂2f
∂y∂x= − y
2(x + y2)3/2
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