Chapter 13 Answers A.69 Chapter 13 Answers 13.1 Vectors in the Plane 31

Chapter 13 Answers A.69 Chapter 13 Answers 13.1 Vectors in the Plane 31. y 1. (4,9) 3. (-15, 3) 3 5. Y = 1 7. No solution 9. No solution 11. No solution 13. a = 4, b = - I 15. a = 0, b = I 17. (XI, YI) + (0,0) = (XI + 0, YI + 0) = (XI' YI) 3 x 19. [(XI' YI) + (X2, Yz)] + (x3' YJ) = (XI + X2 + X3, YI + Yz + Y3) = (XI' YI) + [(X2, Yz) + (x3, Y3)] 21. a(b(x, Y» = a(bx, by) = (abx, aby) = ab(x, y) 33. y 23. y 2 • • • • •• • • • • •• x x • •• • • • 4 • • • • • • 35.

25. (a) k(l,3) + 1(2,0) = m(1, 2) (b) k + 21 = m and 3k + 0 = 2m x (c) k = 4,1 = 1 and m = 6; i.e. 4S03 + S2 = 6S02 27. (a) d (b) e 29. (a) c + d = (6,2) -6

37. (a) 2 .r

Q x 3 6 vAw Pl....-).R u

234 x

(b) -2e+a=(-1,0) (b) v=(l,2); w=(1, -2); u=(-2,0) (c) 0 y 39. (a)

-2e+a x x

(b) (0, I) A.70 Chapter 13 Answers

(d) (0, -2)

11.

w -2

(e) (I, y) (f) v=(O,y) 41. (a) Yes (c) Eliminate rand s. w (b) v = -(s/r)w (d) Solve linear equations.

13. - i + 2j + 3k 15. 7i + 2j + 3k 13.2 Vectors in Space 17. i - k 19. i - j + k 21. i + 4j, B ~ 0.24 radians east of north 1. 23. (a) 12:03 P.M. (b) 4.95 kilometers 25. .\" F = 50 lb.

F, = 50 sin (50') lb ~ 38.3 lb.

(3. 27. The points have the forms (O,y,O), (O,O,z), (x, y, 0), and (x, 0, z). 29. y

5. (11,0, II) 7. (-3, -9, -15) Chapter 13 Answers A.71

31. 9i+ 12j+ 15k 33. 26i + 16j + 38k 33. (I/13)i + (I/13)j + (1/13 )k, (1/ Ii)i + (1/ Ii)k 35. a = !, b = - ! 37. a = 5, b = 2 35.13 39. (4.9,4.9,4.9) and (-4.9, -4.9,4.9) newtons. 37. Ii 41. (a) Letting x, y, and z coordinates be the number 39. (i) ~14t2 - 121 + 4 of atoms of C, H, and 0 respectively, we get (ii) t = 3/7 p(3,4,3) + q(0,0,2) = r(l,0,2) + s(0,2, I). (iii) ~1O/7 (b) P = 2, r = 6, s = 4, q = 5 41. 13 knots (c) 0 43. Solve one equation for I and substitute. The line is vertical when XI = X2' 2(3.4.3)+ 5(0.0.2) = 6(1.0,2) + 4(0,2, I). 45. When the angle between the vectors is O.

6(1,0.2)

4(0,2. I) 13.4 The Dot Product 1.4 3. 0 10 H 5. ~ 0.34 radian 7. 7T /2 radians 9. (l/{S)i + (2/{S)j 10 11. 0.955 radians c

43. (a) P~I=(-I,-I,-I),Po=(I,O,O), PI = (3, I, I), P2 = (5,2,2)

(b) x (i+j+k)

.\'

13. Use Figure 13.4.2. 15. 142 17. '150/11 19. x + Y + z = 0 21. x = 0 (c) The line through (I, 0, 0) parallel to the vector 23. - x + y + z - I = 0 (2, I, I). 25. (2/ff4)i + (3/ff4)j + (l/ff4)k 27. (I/Ii)i + (1/Ii)j 29. x + Y + z - I = 0 13.3 Lines and Distances 31. x = I + t, Y = I + I, z = I + I 1. Use vectors with tails at the vertex containing the 33. (I, - 1/2,3/2), ff4 /2 two sides. 35. x = (22 - 9t)/7, Y = (- 6 - 2t)/7, z = I 3. Use the distributive law for scalar multiplication. 37. 313. 5. (I, - t) 39. Ii /2. 7. x = I - I, Y = I - I, z = I 41. Letting (a,b) and (c,d) be the given points, the 9. x = t, Y = t, z = t equation of the line is (a - c)x + (b - d)y 11. x = I - t,y = I - I, z = I = (l/2)(a 2 - c2 + b2 - d 2). Use this to show that 13. x = - I + 3t, y = - 2 - 2t the two points are equidistant from points on the 15. (-2, -J,O) 17. No line. 43. (a) 3 19. 13 21. Ii (b) -2 23. 21i 25. ::!:: Ii46 (c) 213 27. 112i + j + 2kll = 3, which is less than 13 + Ii· (d) 3 29. One solution is u = i, v = - i, w = i. 45. Letting PI = (p,q) and P2 = (r,s), we have 31. Each side has length Ii. (r - p)x + (s - q)y = (r2 + s2 _ p2 - q2)/2. A.72 Chapter 13 Answers

47. To show that v and w = (v . el)el + (v . e2)e2 are 11. Compute the two determinants. equal, show that v - w is orthogonal to both el 13. Compute the two determinants. and e2' 15. 0 19. -6 49. FI = - (F12)(i + j) and F2 = (F12)(i - j) 17. 4 21. 9 51. (a) F = (3v'2 i + 3v'2 j) 23. abc (b) ~ 0.322 radians j (c) 18v'2 25. - I ~1=-i-3j+3k 53. Use the component formula for the dot product. Ii I 55. (a) [12.5)2 + (16.7)2 - (20.9)21/[(12.5)(16.7)] IS j close to O. 27. I ~I=i-j+k (b) 0.54% Il I 57. (a) Let s = tVa2 + b2 + c2 j 29·1 0 1= -2j (b) Use the fact that Ilull = I. ~ 0 -> 2 (c) Use IIull = I. 31. 6 (d) For LI and L 2 , 33. 12 coso: = 1//3 so 0: = cos-I(l//3), 35. Compute and simplify. cos f3 = 1//3 so f3 = cos-I(I//3), 37. Compute both determinants and compare. cos y = 11/3 so y = cos-I(ll /3). 39. Use Example 8 after renumbering the vectors. For L3 and L 4 , 41. coso: = 11m so 0: = cos-I(l/m), cos f3 = 11m so f3 = cos-I(l/m) and cosy = 91m so y = cos-I(9Im). (e) Only the line t(l, I, I). 13.5 The Cross Product I.j+k 3. 2i - 2j + 4k 5.9i+18j 7. 6i - 2k 9. -i + k 11. 3v'2 13. 2 15. -(l/v'2)j + (1/v'2)k 17. - (I I v'2)i - (I I v'2)j 19. (v'2/6)i - (v'2/6)j + (2v'2/3)k 21. 2x + 3y + 4z = 0 23. x - 3y + 2z = 0 25. 3v'2/2 27. The points are collinear, so the area is zero. 29. Substitute component expressions for VI and V2' 31. The angle between the vectors is () - 1/1. Now use property I in the box on p. 679. 33. Use the result of Exercise 32. 35. Show that M satisfies the defining properties of RxF. 37. Show that nl(N X a) and n2(N X b) have the same magnitude and direction. 39. (a) Draw a figure showing the two lines and the plane in the hint. (b) v'2 41. If F is the gravitational force, the gyroscope ro• tates to the left (viewed from above). 43. Substitute the expressions for x and y in the equa• 13.6 Matrices and Determinants tions. 45. Substitute the given expressions for x, y, and z in 1. 2 3.0 the equations. 5. -2 7. 25 47. x=37/13,y= -3/13 9. ac 49. Compute both determinants and compare. Chapter 13 Answers A.73

51. Subtract four times row 1 from row 2, subtract 49. This is a (double) cone with vertex at the origin. seven times row 1 from row 3, expand by column I and then evaluate the 2 X 2 determinant. 53. 3, -6

Review Exercises for Chapter 13

y 1. (2,8) 3. (- 1,,- 2, 17) 5. Iii + j - k 7. -4i + 7j - 11k 9.6 11. -2k 13. i - 2j 51. (a) Draw a vector diagram. (b) Use c X c = O. 15. 2i + j - 3k (c) Use part (b). 17. (-2//IT)i + (3//IT)j + (3//IT)k 53. Use the dot product to show that the vectors a - b 19. 0 (the three vectors lie in a plane). and - a - b are perpendicular. 21. (a) (6,6) 55. 3 57. 1 59. -2 61. 0 63. y38T 65. 29/2

(b) 1/3 69. Use the fact that IIall 2 = a' a, expand both sides and use the definition of c. 71. x=3/7,y= -29/21,z=23/21 73. -162 75. Each side equals 2 2xy - tyz + 5z 2 - 48x + 54y - 5z - 96. (Or switch the first two columns and then subtract the first row from the second.) 77. v is orthogonal to i (b) (9,7) 79. (a) 4k 23. Put the triangle in the xy-plane; use cross products (b) 20,12 i + 20,12 j with k. 81. (a) Substitute i, j, and k for w. 25. (1825 - 600y''2) I /2 R:: 31.25 km /hr. (b) (u - v) . w = O. 27. (a) 70 cos B + 20 sin B (c) Repeat the reasoning in (a) (b) (21 If + 6) ft.-lbs. (d) Apply (c) to u - v. 29. x = 1 + t, y = 1 + t, Z = 2 + t 83. (a) 31. x = 1 + t, y = 1 - t, Z = 1 - t 33. -x + Y = 0 35. x - Y - Z - I = 0 37. x = - t, Y = t, Z = 3 3~x=2+~y=3-~z=l-t 41. (1/v'38)i - (6/v'38)j + (1/v'38)k 43. (2//5)i - (l//5)j 45. (If /2)i + (1 /2,12)j + (1/2,12)k 47. It is parallel to the z-axis. 123456 x A.74 Chapter 14 Answers

y y

(c)

(d) The set is the entire plane. x

85. (a) (d)

y (I + 21T) (I, 2) 14 (4+1T)(I,2) 12 10 8 6 x 4 2

12345678 x 87. U = sin - 1(.J8 /3)

Chapter 14 Answers 14.1 The Conic Sections

1. Foci at (± 4/f, 0). 3.

y y 2 2

-6 6 -' 2 x -2

-2 Chapter 14 Answers A.7S

5. y 3. y

-2 2 x

x -2

5. A circle of radius 1 centered at (1,0)

.r 9. y=x2/16 11. (0, 1/4),y = -1/4 13. (1/4, 0), x = - 1/4 15. x 2 + y2 = 25 17. Y = x 2/4 x 2 19. y2 - 3"" = 1 x 21. On the axis, 2/15 meters from the mirror. 23. 4/{f5 25. Use the dot product to find an expression for the cosine of the incident and reflected angles.

14.2 Translation and Rotation of Axes 7. An ellipse shifted to (0, 3/4).

1. y

x A.76 Chapter 14 Answers

9. A hyperbola with asymptotes y = ± x, shifted to 21. (-1, -1).

23. x 2 + y2 - 4x - 6y = 12 x 2 25. Y = (x - 1)2 27. (y - 1)2 - 3 = I.

29. X2+ y 2+(l-!3)X+(-I-!3)Y=2. 31. For translations, A = if and C = C. For rotations use equations (9) to compute A + C. 33. The area of the rotated ellipse is 'TT If .{f .

11. x = X /2 - !3 Y /2, Y = !3 X /2 + Y /2, X = x /2 + !3 Y /2, Y = -!3 x /2 + y /2. 14.3 Functions, Graphs and 13. x = 0.97 X - 0.26 Y, Y = 0.26X + 0.97 Y, Level Surfaces X = 0.97x + 0.26y, Y = -0.26x + 0.97y. 15. Hyperbola 1. All (x, y) with x =1= 0; 0, I. 17. Ellipse 3. All (x, y) with x 2 + y2 =1= I; 2, O. 5. All (x, y, z) with x 2 + y2 + z2 =1= I; I, - 2/3 19. 7. All (x, y) with x =1= 'TT + 2n'TT, n an integer; -If /4, 'TT(2 -1f)/2. 9. ~) z

(0,0, I) I x I -2 -I I I }------+---\-- I / (0,1,0) y / (-~ 0.0,0) x Chapter 14 Answers A.77

11. 17. Lines through the origin, excluding points on the line x = y.

c =-2 c=2

c = I y x

13. 19.

15. Circles with centers on the line y = x and passing through the points (± Ii /2, =+= Ii /2), excluding x points on the circle x 2 + y2 = I.

21.

2 y

x A.78 Chapter 14 Answers

23. 27. y c =.2

y z z ;;: 2

z = 1 x

29.

25. ! \ ;t------~T---) I

c =-2 ~~:::::::=_c=O c=2 x x 31.

c=2

l' x

c::: -2 33.

y Chapter 14 Answers A.79

35. (b) No level curve. (c)

o x x

37. (d) In polar coordinates, the equation is f(r,O) = e -1/,.2. This is independent of O. (e) The graph looks like a plane gradually sloping down to a pit in the center.

39.

1/2

x 43. (a) z = exp[( - l2)(x + y)/25(x - y)] (b) 41. (a) y E

x -1

-3

-5 A.SO Chapter 14 Answers

14.4 Quadric Surfaces 9. z

x 11. z x

Y x

13.

7. z

y x

x Chapter 14 Answers A.81

15. z 19.

21. z x 17. (a) =

x y 23. Substituting z = 1 gives x 2 + y2 = 1, a circle. 25. (a) They are ellipses. (b) In each case the cross section is two straight lines. (c) If (xo, Yo, zo) satisfies the equation, so does (txo, tyo, tz )· x o 27. Substitute x = Xo + ~u, + 1jV" Y = Yo + ~U2 + 1jV2, (b) Rotate the x and y axes by 45°. Z = Zo + ~U3 + 1jV3 into x 2 + y2 = z2, where u = (u" U2, U3) and v = (v" V2, V3)' 14.5 Cylindrical and Spherical Coordinates

1. (If, - '1T /4, 0)

x -I V2-;-c.;;--~----- .:~ 1f Y (1,-1,0) 1-"4

x A.82 Chapter 14 Answers

3. (m, -0.588, J) 11. (0,0,6)

z z

(0.0,6)

/~------~-----. (3,-2,1) / }' v ~ //,/ 1/// 1£'':---

5. (6,0, -2) 13.

6 I I • (6,0,-2) x

7. (0, J, 0)

(0, 1,0) y-plane. y 17, Stretching by a factor of 2 away from the z-axis, and fI reflection through the xr-plane. 19. Right circular cylinder with radius r; vertical plane x making an angle {j with the .);'z-plane horizontal plane containing (0,0, z).

9. (- f3 /2, - J /2, 4) 21. 'iT /4}

,(_.J3 _J.. 4) I 2' 2 ' I I I I I I I I

y y rr/2 rr/6 I I I x x Chapter 14 Answers A.S3

23. ({I4, 2.68, 2.50) 31. (0,0, - 8)

0.64

y I Y I ',- I I I 2.68 '~ I I (-2, I ,-3) I ,I I x I • (0,0, -8) 25. (y'29,3.73,2.41) x

z 33. p2 = 2/sin2cpcosO 35. The vertical half plane with positive y-coordinates and making a 45° angle with the xz-plane. 37. It moves each point twice as far from the origin along the same line through the origin. 39. The unit circle in the xy-plane. Y 41. (3, 'IT /2,4), (5, 'IT /2,0,64)

(-3,-2,-4) x (0,3,4) 27. (0,0, - 3) , I I I I

x 7f I I I 43. (0,0,0), (O,O,CP) for any O,cp, I I I • (0,0.-3) x

29. (0,0,3)

y (0,0,3) (0,0,0)

x A.B4 Chapter 14 Answers

45. (4, 77T /6,3), (5, 77T /6,0.93) 59. (a) z = r 2cos 2e (b) 1 = p tan sin cos 2e z 61. (a) The length of xi + yj + zk is • (-2v3,-c,3) (x2 + y2 + z2)1/2 = P I I (b) cos = z/(x2 + y2 + z2)1/2 I I (c) cose = x/(x2 + y2)1/2 I I I 63. 0 .;;; r .;;; G, 0 .;;; e .;;; 27T means that (r, e, z) is inside -----7 the cylinder with radius G centered on the z-axis, / I / and b means that it is no more than a / / Izl.;;; I / distance b from the xy plane. / // 65. p .;;; 0.;;; and L ____ -/<-/____ .. -(d/6)cos.;;; d/2, e .;;; 27T, 7T - cos-I(I/3).;;; .;;; 7T. y 67. This is a surface whose cross-section with each surface z = c is a four-petaled rose. The petals shrink to zero as lei changes from 0 to I. x 47. ([2/2,[2/2,1)1 ([2, 7T/4, 7T/4) 14.6 Curves in Space 49. (0,0, 1), (1, 7T /4,0) 51. (0,2, I), (15, 7T /2, 1.11) 1. 53. (0,0, - 1), (0, e, -1) for any e y

-I x

i y • (0,0,-1) -4

x 3.

55. (0,0,0), (0, e, 0) for any e

57. (1/2,0, -/3 /2), (1/2,0, -/3 /2) z

y ~(Io_v'i) 2" 2

x Chapter 14 Answers A.8S

5. 17. [t 3cos t( - 2 + csc2t) - 3t2sin t(2 + csc2t)]i + [t 2e- l(3 - t) + 2t2e l(t + 3)]j + [e'csct(l - colt) - e-'(cost + sint)]k 19. e' [2e'i + (sin t + cos t)j + t 2(3 + t)k] 21. ~ Ilu'(t)11 2 = 2u'(t)· u"(t) = O. dt 23. (a) cos ti - 4 sin tj (b) - sin ti - 4 cos tj (c) ";cos2t + 16 sin2t 25. (a) 2i + j + k (b) 0 (c) !6 27. (a) - i + j + 2tk (b) 2k (c) !f+4i2 29. (a) -4sinti+2costj+k (b) -4costi-2sintj 2( t" = -x (c) .js + 12 sin 31. (a) i - (1/(2)j + k (b) (2/(3)j (c) ~ /t 2 ( 3.3.0) 33. (6,6(,3(2); (0.6. 6(); (x, y. z) = t(6. O. 0) 35. (-2sintcost.3 - 3t2.1); (-2cos2f. -6(.0); (x. y.z) = (1,0.0) + {(0.3. I) 37. (/I.e l, -e-'); (O.el.e-'); (x. y.z) = (0.1, I) + (/I. I. -1) 39. (2e, 0, cos I-sin 1) 41. (a) t(l,O,l)

7. (a) (b)

(4.0.0)

(b) (1.2.3)+ t(-I.I, I)

---1-.1' t' =- - .r

11. (a) An ellipse in the plane spanned by v and w and passing through the tip of u. The ellipse has semi-major axis 4 and semi-minor axis 2. (b) I, - 1 + 4 cos t + 8 sin t, and 4 cos t - 8 sin t. 13. u'(t) = - 3 sin ti - 8 cos tj + elk; u"(t) = - 3 cos ti + 8 sin tj + elk. 15. (e l - e-I)i + (cost - csctcOlt)j - 3t2k. A.86 Chapter 14 Answers

y x

(iii) z x

43. (a) The curve is a right circular helix with axis parallel to the z-axis. (b)

(b) Each curve is the line segment joining (0,0,0) to (-1,2,3). It is covered once by (i) and (iii) y and twice by (ii). The velocity is constant in (i), variable in (ii) and (iii). 49. In each case, verify that x 2 + y2 + z2 = I, so the curve lies on the sphere.

(a) ( c) The curve becomes a circle in the xy plane with center (2,0, 0) and radius I. 45. (a) Substitute (b) (Alcost + Blsint,A2cost + B2sint,A3cost + B3sint) where AI' A 2, A 3, B I , B2 and B3 are constants.

47. (a) (i) y

(b)

y x

x Chapter 14 Answers A.B7

51. The set of points above or below Po have coordi• [-1/2/'f + cos(m/12)/2/'f + sin('lTt/12)/4]j + nates (xo, Yo, z) where z = cos - Ixo + 2mr if Xo ;;. 0 [-1/2/'f + cos('lTt/12)//'f - sin('lTt/12)/4jk or z = - cos - Ixo + 2mr if xo"; 0, n an integer. (c) (x, y,z) = (-1/2/'f)(i + 2j + 3k) + The vertical distance is 2'IT. (-'IT/48)(i+j+k)(t-12) 53. Let 0"1 = iIi + gd + hlk, 0"2 = fzi + g2j + h2k. 3. Td would be longer. Form 0"1 0"2 and differentiate using the sum rule + 5. The "exact" formula is - tan I sin a = for scalar functions. cos(2'ITt / T )[tan(2'ITt / Ty)tan(2wt / T ) - cos aj. 55. Using notation in the answer to Exercise 53, form d d 7. A = 9.4° . 0"1 X 0"2 and differentiate using the product rule 9. The equator would receive approximately six for scalar functions. times as much solar energy as Paris. 57. (a) O"(t) describes a curve in the plane through the origin perpendicular to u. (b) Same as (a), except that the plane need not go through the origin. Review Exercises for Chapter 14 (c) O"(t) describes a curve lying in the cone with u as its axis and vertex angle 2 cos - lb. 1. 14.7 The Geometry and Physics y of Space Curves 1. 2'IT 15 3. 4/'f - 2 -3 3 x 5. ~3.326 7. -0.32n:2r, where r is the vector from the center to the particle. 9. (6.05 X 103) seconds. 11. (a) From ma = GmM/ R2, g = GM/ R2 = (6.67 x 10- 11 )(5.98 x 1024)/(6.37 X 106)2 = 9.83 m/sec2• 3. (b) The acceleration is - 9.8k if k points upward. 13. (a) x" = (qb / em)y'; y" = (- qb / em)x'; z" = O. (b) x = -(ame/qb)cos(qbt/me) + (ame/qb) + I, y Y = (ame/qb)sin(qbt/me), z = ct. (c) r = ame/qb, the axis is the line parallel to the 8 z-axis through (ame / qb + 1,0,0). 6 15. The circle parametrized by arc length is O"(s) = (r cos(s / r), r sin(s / r». Calculate T = dO" / ds and dT/ds. 17. k = 1//'f(2y 2 + x 2 /2)3/2. x 19. Assume that the curve is parametrized by arc length and show that v is constant. 21. Force magnitude = (mass) x (speed)2 x (curvature). -6 23. (a) 0 is the normal to the plane. Since a', a", a'" are -8 perpendicular to 0, their triple product is zero. By Exercise 22(e), , = O. (b) By Exercise 22(e), dB/dt = O. By Exercise 22(a), B lies in the direction of v x a. 25. Use the hint for the second equation and Exercise 22 (a) for the third. 5. y 14.S Rotations and the 0.1 Sunshine Formula 1. (a) IIlo = (1/,16 )(i + j + 2k), -0.1 0.1 x --0.1 00 = (I/2y3)(i + j - k) (b) r=[cos('lTt/12)/2/'f +sin('lTt/12)/4ji+ A.SS Chapter 14 Answers

7. y 15. y

20 2 15

x 2

-15 -20 17. The level surfaces are parallel planes.

9. y

(l,5)

y (0,2) (2,2)

(l,-I)

11. y

19. The level surfaces are spheres of radius ~.

c = I

13.

x Chapter 14 Answers A.S9

21. Ellipsoid with intercepts (± 1,0,0), (0, ± 1/2,0) and 27. Elliptic paraboloid with intercepts (± 1,0,0), (0,0, ± I). (0, ± 1/2,0) and (0,0, I).

1/2

23. Elliptic hyperboloid with intercepts (± 1,0,0) and (0, ± 1/2,0). 29. (a) (x/a)2+(y/b?= I + (z/c)2, which are el• lipses. (b) x constant; Ixl < a gives a hyperbola opening along the y-direction, Ixl = a gives two lines, and Ixl > a gives a hyperbola opening along z-direction. (c)

x x

25. Elliptic paraboloid with intercepts (± 1,0,0), (0, ± 1/2,0) and (0,0, -I). y z -2-

x:o x: I

y y

-I

x:2 x:3 x A.90 Chapter 14 Answers

31. (a) 31. (c)

level sets for c = -10,-1,0 ! y 3 lare empty v2

(b) For x = ± I and x = 2, the equations are z = 2y2 + 2 and z = 2y2 + 5 which give parabo• las opening upward in planes parallel to the x yz-plane. For y = ± I and y = 2, the equations are z = x 2 + 3 and z = x 2 + 9 which give parabolas opening upward in planes parallel to the xz-plane.

Rectangular Cylindrical Spherical Coordinates Coordinates Coordinates

33. (I, -I, I) ({2 , - '11' /4, I) ({3, -'11'/4,cos-I(l/{3)) 35. (5cos('11'/12), 5 sin('11'/12),4) (5,'11'/12,4) (/

33. 35. z 37.

(4.83,1.29,4) p r --;--,--- (1.84, -1.06, I 7 / y 2.12) ..... cjJ I ~P ..... y I / / I ',/ y all I / / o __ i'I 1/

x x

41. Rotate 180 0 around the z-axis and 90 0 away from the positive z-axis. 43. The rod is described by 0 .;; r .;; 5, 0 .;; 0 .;; 2'11', and 0.;; z .;; 15. The winding is described by 5 .;; r.;; 6.2, 0.;; 0 .;; 2'11', and 0 .;; z .;; 15. 45. 1/Y3

y line x + y = 0, z = 0

x Chapter 15 Answers A.91

47. z 61. 1 + In2 63. (9/4, -sin(I/2) - (I/2)cos(I/2), -2e l / 2) 65. (a) x"(t) = -(k/m)x(t),y"(t) = -(k/m)y(t), and z"(t) = -(k/m)z(t). (b) x(t) = O,y(t) = (2m/k)sin(kt/m), and z(t) = (m/ k)sin(kt / m). y 67. (8/8I)cos2t/(20sin2t + 16)3/2, where x = 2cost, y=4/3sint 69. k = 1f"(x)I/[I + U'(x»2]3/2 x 71. (t4 + 3t2 + 8)1/2/(t2 + 2)3/2

49.

51. z 73. (a) The curve is x 2 + z2 = 2 which can be ex• pressed as x = /2 cos t, Y = I, z = /2 sin t. (b) (x, y, z) = (1, 1, 1) + t( -1,0, I)

(2" PI 2 2 2 (c) Jo (-v2sint) +(0) +(/2cost) dt

v0J3 y = 2/2 17. 75. (a) a(O) = u .. a( 17 /2) = U2 (b) a(t) lies on the unit sphere and in the plane determined by UI and U2 x (c) cOS-I(UI·U2) (d) 50"/2 ~r-I---2-u-1.-U-2 s-in-t-c-os-t dt 53. (x, y,z) = (2, I/e,O) + (t - 1)(3, -I/e, -17/2) 55. eli + costj - sintk; eli - sintj - costk (e) Let w be the unit vector in the direction of 57. [e l + 2t/(1 + t2)2]i + (cost + I)j - sintk; (UI X U2) X UI. Let w = ~ cos -I(UI . U2). Then [e l + (2 - 6t2)/(I + t2)3]i - sintj - costk 17 59. x = t + I, Y = (t - 1)/2, z = (t - 2)/3. al(t) = UI cos(wt) + w sin(wt).

Chapter 15 Answers ------15.1 Introduction to 9. fx=yz,fAI,I,I)= I;.£, = xz,.£,(l, 1,1)= I; .fz = xy,.fz(I, 1, I) = 1. Partial Derivatives 11: fx = - y 2sin(xy2) + 3yze3xyz , fA17, 1, 1) = 3e3,,; .£, = -2xysin(xy2) + 3xze3xyz ,.£,(17, I, 1) = 317e3,,; 1. fx = y, fAI, 1) = I;.£, = x, JiI, I) = 1 .fz = 3xye3xyz ,.fz(17, 1, 1) = 317e3". 3. fx = 1/(1 + (x - 3y2)2], fA 1, 0) = 1/2; 13. az/ax = 6x; az/ay = 4y . .£, = - 6y /(1 + (x - 3y2)2], .£,(1, 0) = o. 15. az/ax = 2/3y + 7/3; az/ay = -2x/3y2. 5. fx = yexYsin(x + y) + eXYcos(x + y), fAO,O) = I; 17. au/ax = e- xyZ [ - yz(xy + xz + yz) + (y + z)]; fy = xexYsin(x + y) + eXYcos(x + y), .£,(0, 0) = 1. au/ay = e-xyZ [ -xz(xy + xz + yz) + (x + z)]; 7. fx= -3x2/(x3 +y3)2,fA-I,2)= -3/49; au/az = e-xyZ [ -xy(xy + xz + yz) + (x + y)] . .£,= -3y2/(X3+y3f,Ji-I,2) = -12/49. A.92 Chapter 15 Answers

19. au/ax = e'cos(yz2); au/ay = -z2e X sin(yz2); 75. (a) Look at the function restricted to the X-, yo, au/az = -2yze X sin(yz2) and z-axes. 21. (xyexeY - xexe" + xeY + eX)/(ye X + 1)2 77. (a) Substitute x = °into f,. 23. 16b(mx + b2)7 (b) Substitute y = °into Iv to get Iv(x, 0) = x 25. 12 + (2/9)cos(2/9) - 27e2. (c) f,.x(O, 0) = lim [(j,(0, y) - f,(O, O»/y), etc. 27. -4cos(l) + 3 - 3e . y-->o 29. (a) -x(sinx)e- xy (d) Notice that f, and 1.. are not continuous (b) 0,0, -'!T/2, _('!T/2)e- 172 /4 at (0,0). . 31. I 33. 1/6 - 3sec2(6) 15.2 Linear Approximations and 35. 1/7 + 3sec2(-15) Tangent Planes 37. (tu 2)e stu' (- J.!sinAJ.!)(l + A2 + J.!2) - 2ACOSAJ.! 1. z= -9x+6y-6. 3. z = 1 39. 2 5. z = 2x + 6y - 4 7. z = 1 (I + A2 + J.!2) 9. z = x - y + 2 l1.z=x+y-1 41. fz= lim {[j(x,y,z+~z)-f(x,y,z)l/~z}. Ilz-->O 13. -(\/v'3)(i - j - k) 15. -(\/v'3)(i+j-k) 43. The rate of change is approximately zero. 17. -0.415 19. - 2.85 45. (a) I/(l + R,/ R2 + R,/ R3)2 21. 1.00 (b) 36/121 times as fast. 23. Increasing, decreasing, increasing. 47. a2z/ax2 = 6, a2z/ay2 = 4, 25. 1-~a/6+~v a2z/axay = a2z/ayax = ° 27. (a) 2 49. a2z/ax2 = 0, a2z/ay2 = 4x/3/, (b) A parabola in the yz-plane, opening upward a2z/axay = a2z/ayax = -2/3y2 with vertex at (l, 0, I). 51. fxv = 2x + 2y, fvz = 2z, ix = 0, fxvz = ° (c) (0, 1,2) 53. a2u/ax2 = 24xy(x2 - y2)/(x2 +y2)4, a2u/ayax 29. See Example I; in this case we are dealing with = a2u/axay = -6(X4 - 6xy2 + y4)/(X2 + y2)4, the lower hemisphere. a2u/ay2 = -24xy(x2 _ y2)/(x2 + y2)4. 55. a2u/ax2 = y4e -X,.2 -+; 12 xy 3, a2u/ayax 15.3 The Chain Rule = a2u/axay = e -xv (- 2y + 2xy3) + 12x 3y2, a2u/ay2 = 2xe- xy2(2xy2 - I) + 6xy. 1. (48 + 128t)cos(3 - 2t) - 8(3 - 2t)2cos(3 + 8t) + 57. Take 8 = Eo 2(3 + 8t)2sin(3 - 2t) + (12 - 8t)sin(3 + 8t) 3. (e 2t - e- 2t )(ln(e2t + e- 2t ) + I). 59. ° 61. 52/m =4m 5. -sint + 2costsint + 3t2• 7. et- t2 (\ - 2t) + e t1 - t3 (2t - 3t2) + e tJ - t(3t2 - I). 63. -e 9. Letf(x,y)=x/y. 65. ° 67. g'( to) = - 2 cos tosin to + 2e 2to 11. (x/Jx2 + y2 + 2y2)(dx/du) + 69. Evaluate the derivatives and add. (y / Jx2 + y2 + 4xy)(dy / du). 71. (a) Evaluate the derivatives and compare. 13. ai + bj + ck where - 2a - 4b + c = 0. (b) 15. (a) XX(\ + Inx) (b) XX(\ + lnx) (c) One author prefers (a), the other (b). f'(t)g(t)h(t) + f(t)g'(t)h(t) - f(t)g(t)h'(t) 17. 2 [h (t)] 19. 6.843 21. The half-line lies in its own tangent line. The cone in Example 6 is such a surface, as is any other surface obtained by drawing rays from the origin to the points of a space curve. 15.4 Matrix Multiplication and the Chain Rule x = ( 1. [32) 3. [44) 73. (a) 170 units (b) 276 units. This is the marginal productivity of 5. [s~: ~< :~~~ v ]; (Si~ 1 ~] 9. [l~ In capital per million dollars invested, with a labor force of 5 people and investment level of 7. xz 11. three million dollars. [~z Yl [i i n [~ ~] 75. (a) Look at the function restricted to the X-, yo, and z-axes. Chapter 15 Answers A.93

13. Undefined, the first matrix has two columns and (b) Eliminate B to find a relation between x, y, z, the second matrix has three rows. and ep. (c) Look at the ratio y / x. 15. [~] 17. [~ ~] (d) Find o(x, y,z)/a(u,ep,O) and evaluate its de• terminant. 21. oz/ax = 26x + 6y + 70; oz/oy = 6x + 2y + 14. 47. Express IA II B I as a sum of 36 terms. 23. oz/ax = cos(3x 2 - 2y)(6x)cos(x - 3y) + I sin(3x 2 - 2y)[ - sin(x - 3y)]; 4 az/oy = -2cos(3x2 - 2y)cos(x - 3y) + 2 3 sin(3x 2 - 2y)sin(x - 3y). 4 25. (a) [ I I], [2X 2y ] I - I 2x - 2y 4

(b) u = (t + S)2 + (t - S)2, V = (t + S)2 - (t - S)2, 2 4 o(u,v)/o(t,s) =[1; :n I (c) Multiply the matrices in (a) and express in terms of sand t. Review Exercises for Chapter 15 27. (a) [~ ~ l [6 _0 I ] (b) u = ts, v = - ts, o(u, v)/o(t,s) = [s t] 1. gx = '1Tcos('1Tx)/(1 + y2); -s - t g, = -2ysin('1Tx)/(I + y2f (c) Multiply the matrices in (a). 3. kx = Z2 + z3sin(xz3); kz = 2xz + 3xz2sin(xz3). 29. ou/or = cosBsincp(au/ox) + sinBsincp(ou/oy) + 5. hx = z; hy = 2y + z; hz = x + y coscp(ou/oz), ou/oB= -rsinBsincp(ou/ax)+ 7. Ix = -[cos(xy) + ysin(xy)]/[e X + cos(xy)]; r cos B sin cp(ou/oy), i .. = - x sin(xy)/[e X + cos(xy)]; iz = O. (ou/ocp) = rcosBcoscp(ou/ax) + 9. gx = z + x 2e x + z ; g.. = 0; gz = X + e Z (Xt2e' dt . Jo rsinBcoscp(ou/ay) - rsincp(ou/oz) 11. gx .. = gyX = -2'1Tycos('1Tx)/(I + y2)2 13. kx~ = (x = 2z + 3z2sin(xz 3) + 3xz5cos(xz 3) 15. hxz = hzx = I 17. I 19. - sin(2) 21. (a) 35.25 minutes (b) oT/oxl(27.65) = -0.598 minutes/foot; this means that in diving from 27 to 28 feet, your time decreases about 36 seconds. 0 T /0 VI(27) 0.542 minutes/cubic foot; this means that m I m = 33. AB = 2: ai · (I/m) = - 2: ai' the average of the bringing an extra cubic foot of air will give i=l m i=l you about 33 seconds more diving time. entries of A. 23.4 29. z = I 35. Multiply B (found in Exercise 34) by A. 25.0 31. 9.00733 27. z = 2x + 2y - 2 33. 0.999 37. [5 - 2] - 2 I 35. 5.002 39. Use the relations between areas, volumes, 37. t = v14 (- 3 + 2v7Q9)/70 and determinants in Section 13.6. 39. d[f(o(t)JI dt = 2t/[(I + t 2 + 2cos2t)(2 - 2t2 + t4)] 41. (a) - 16 - 4t(t2 - I )In(l + t 2 + 2 cos 2t) / (2 - 2t2 + t 4)2 (b) 8 - 4 cos t sin t /[(1 + t2 + 2 cos 2t)(2 - 2t 2 + t4)] (c) - 128 = - 16 . 8 41. (a) Use the chain rule with x - ct as intermediate variable. (b) It shifts with velocity c along the x axis, without changing its shape. 43. The radius is increasing by 15 cm/hr. 45. [f'(t)g(t) + i(t)g'(t)]exp[f(t)g(t)] 47. (I + 2y - 2x)exp(x + 2xy) 49. y= -x/6+7/12 51. [4] 53. [~ 6] 45. (a) Substitute and use cos2 + sin2 = I. 57. [ ~ 3 ~ ~ -; I 1 A.94 Chapter 16 Answers

3 I 67. (a) One may solve for any of the variables III 59. [ 5 - I ~ 3] terms of the other two. I I -I (b) aT /3 P = ( v - f3) / R; aP /a v = - RT/( V - f3)2 + 20:/ v 3 ; 61. 3z/ax = 4(e-2x-2y+2xY)(I + y)/(e- 2x - 2y - e2xY i, 3 v /3 T = R/[( v - f3)(RT /( V - f3)2 - 20:/ V 3) 3z/3y = 4(e-2y-2x+2xY)(I + x)/(e-2x-2y _e2xy )2 (c) Multiply and cancel factors. 63. 3z /3u = 3z /3x 3z /3y, + 69. Notice that y = x 2, so if Y is constant, x cannot be 3z/3v = 3z/3x - 3z/ay. a variable. 65. (a) n=PV/RT; P=nRT/V; T=PV/nR; 71. 33u/3x3yaz = 33u/3y3x3z = 33u/ay3z3x V= nRT/ P 73. Differentiate and substitute. (b) 3P/3T represents the ratio between the 75. Use the chain rule. change !::.P in pressure and the change !::. T in 77. Yes. The second partial derivatives are not contin• temperature when the volume and number of uous at the origin; the graph has a 'crinkle' at the moles of gas are held fixed. origin. (c) 3V/3T= nR/P; aT/3P= V/nR; aP/3V = - nRT/ V 2. Multiply, remembering that PV= nRT.

Chapter 16 Answers 16.1 Gradients and 11. (a) Directional Derivatives

1. (X/Vx2 + y2 + z2 )i + (Y/Vx2 + y2 + Z2)j /' "- 2 + (z/vx + y2 + z2)k /' "'- 3. i + 2yj + 3z2k / \ 2 2 I , 5. [x/(x + y2)]i + [y/(x + y2)]j 7. (I + 2x2)exp(x2 + y2)i + 2xy exp(x2 + y2)j. x \ 9. \ " ,.,1 I ...... /

,,' \ 4 11// "'-.\3 / /// (b) a( - y)/3y 7"= 3(x)/3x ...... , , I /' ,.,..,-- 13 ~ ( I ) = - 2x , etc. -- . ax x 2 + y2 + z2 ,4 --~'l "".--- 15. 2e'cost + cos 2t - sin 2( --- 17. t/1l+f2 -4 -3 -~ -I I 2 3 4 --// /-1 , ...... 19. The angle between the gradient and the velocit~ vector is between 0 and 'TT /2. ----/' /_~ " ...... 21. - II - 16-/3 /' /" / /-3 23. 17//2 /" /" / /-4 \"'" 25. -14/-/3 \\.." 27. 'TT/8 - 1/2 29. (i + 2j)/15 31. e[(sin l)i + (cos 1)j] 33. (a) (1,2,3) (b) -2jl4 e2 degrees per second (c) She should fly outside the cone with vertex, (1, 1, I), axis along (1,2, 3) and sides at a angle of 'TT/3 from the axis. Chapter 16 Answers A.95

35. d l = [-(0.03 + 2bYI)/2aji + yd, d2 = [-(0.03 + 16.2 Gradients, Level Surfaces, 2bYz)/2aji + Yzj where YI and Yz are the solutions 2 and Implicit Differentiation of (a2 + b2)y2 + 0.03by + ( -4003 - a2) = O. 1. Vf(0,0,1)=2k

z z 3

x y

37. 2i x

y 3. V f(O, 0, I) = - i + j + k

x i • H.O.O) I I )----~------~~----y / (0,1,0) /

39. (\/yJ)(i+j+k) 41. Write out each expression in terms of partial deriv• 5. (1//i29)(8i + 8j + k) atives and use the properties of differentiation. 7. k 9. V = Qq I r; the level surfaces are spheres, which 43. (a) (1//2,1//2) are orthogonal to radial vectors. (b) The directional derivative is 0 in the direction 11. x + 2yJ y + 3z = 10. (xoi + yoj)1 ~x5 + Y5 . 13. 3x + 8y + 3z = 20 (c) The level curve through (xo, Yo) must be tan• 15. x + y + z = 3 gent to the line through (0,0) and (xo, Yo). The 17. x + 2y - 3 = 0 level curves are lines or half lines emanating 19. x + y - 'TT 12 = 0 from the origin. 21. (I, I, I) + t(l, I, 1) 45. Vf{l,3) = (2, -2); -21m 23. (I, I, I) + t(l, -I, -I) 25. -xl2y 47. (a) _A._ +2 -2 i + 2y _ {(x Xo _ X xo) (~ ~)j} 27. ylx = 1/10 2:n:eo r l rz r l rz 2 (b) Compute the indicated partial derivatives. 29. 3x /(cos y - 4y 3) 49. The function f must satisfy Laplace's equation: 3l. 1/2 fxx + !v," = O. 33. - I - 2yJ 13 35. At (0,0), the slope of y = IX is infinite. 37. At (0,0), the slope of y = x 1/5 is infinite. 39. dxldt = (-l/y)(dyldt) A.96 Chapter 16 Answers

41. x\dx/dt) + y3(dy/dt) = 0 35. (a) Calculate 3z/3x and 3z/3y and set them 43. (a) dx/dy = -(3z/3y)/(3z/ax) equal to zero. (b) (cos y - 4y 3)/3x2 ; (b) The maximum is at (0, 0) and local maxima - y(2ex +y2 + 3y)/ e x +y2 [resp. minima] occur on circles of radius '2,. 45. (1/ x - I)(dx/ dt) + (- tan y)(dy / dt) = 0 '4, . " [resp. 'I, '3, ... ] where 0 < 'I < '2 47. (a) z = 2x - 4y - 5 < '3 < . .. are the solutions of w, = tan( w,). (b) The slope is the tangent of the angle between (c) Symmetric in every vertical plane through the the upward pointing unit normal vector and origin and under any rotation about the z-axis. the z-axis. The slope in this case is 2{5. 37. (a) Set 3w/3pi=0. This occurs when Ti-I/Ti = (P?!(Pi_IPi+I»I-(I/nl. 49. Crosses at (2,2,0), {5 / 10 seconds later. 51. (a) They are perpendicular. 1',3 n/(n-I) ]1/4 [ (b) If it were not equipotential, there would be (b) PI = (TI T~T3 ) P5P4 places where the force of gravity is not perpen• dicular to the surface and the water would _[(ToTI)n/(n-l) ]1/2 flow to correct this. The rotation of the earth P2 - TzT3 POP4 and tides (among other things) spoil the ap• _[(ToT IT2 )n/(n-l) 3]1/4 proximation. P3- --- POP4 Tj 39. A = 2, B = 1, C = 2 so A > 0 and AC - B Z = 3 16.3 Maxima and Minima > O. Thus the point is a local minimum. 1. Local minimum at (1,0), local maximum at 41. (a) (0,0) is a saddle point. (- 1,0) (b) The behavior changes qualitatively at 3. (0,0) is a local minimum. C = ±2. For -2 < C < 2, (0,0) is a strict minimum; for C 2 or C 2, (0,0) is a 5. (0,0) is a local maximum. < - > saddle point. For C = ± 2, (0,0) is a mini• 7. b/2 mum. 9. The height is 4b2/3/ S2/3. 43. (a) b = 1, m = 4/7 11. Minimum 13. Saddle point y 15. Minimum (although the test per se is inconclusive) 17. (-3,2), minimum 19. (0,0), neither 21. (3,7), minimum 23. (1, I), minimum 25. (4/5, -9/10), minimum 27. (0,0), neither x 29. (0,0), neither 2 3 4 31. (0,0) is a saddle point. 33. The second derivative test fails, but from the ac• companying graph, we can see that (0,0) is neither (b) b = 1/6, m = 3/2 a local maximum nor minimum. y

y (0,0)

45. ~~ = -22:(Yi - mXi - b) and

33~ = - 2 2: Xi(Yi - mXi - b); set these equal t zero and use properties of summation (Sectio 4.1). Chapter 16 Answers A.97

47. Compute a2s /am 2, a2s /amab, and a2s /ab2 di• 9. 1/4 is the maximum value. (1/2,1/2) is the criti• rectly. cal point. 49. (a) e=JB2- CA /A 11. /W is the maximum value, - /W is the minimum (b) Y = Ax/(Ae - B), Y = - Ax/(B + Ae) value. (± /W / 10, =+= 3/W / 10) are the critical (c) g(x, y) is positive when (i) y > Ax/(Ae - B) points. and y > - Ax/(B + Ae) or 13. x = Y = 25,000; z = 50,000. (ii) y < Ax/(Ae - B) and y < - Ax/(B + Ae) 15. Horizontal length is hA / p, vertical length IS (d) If A = 0 and B = 0, then AC - B2 = O. Thus B cannot be zero if A = 0 and AC - B2 < O. JpA/q. Rewrite g as g(x, y) = y(2Bx + Cy). Note 17. (Q2/ QI)I/3 that y = 0 and 2Bx + Cy = 0 are two lines 19. (a) (/2 /2,/2 /2,3/2) and (-/2 /2, -/2 /2,3/2) intersecting at the origin. Thus g(x) > 0 in the are maxima, while (-/2 /2,/2 /2,1/2), and region above 2Bx + Cy = 0 and the negative (/2 /2, -/2 /2, 1/2) are minima. x-axis, and the region below 2Bx + Cy = 0 (b) h is increasing at (± 1,0) and decreasing at and the positive x-axis. (0, ± I). 51. (a) fx(O, 0) = 0, h(O, 0) = O. (b) Use one variable calculus on h (c) (/2 /2,/2 /2,3/2), (-/2 /2, -/2 /2, 3/2) are (c) f> 0 if y > 3x2 or y < x 2 and maxima, (0,0,0) is the minimum. f < 0 if Xl < Y < 3Xl. p'lTnrir ( DI Dz ) 21. (a) C = -- - + - (d) f(x, y) = 0 if Y = X Z or y = 3xz. ah x y D - D (b) x = Z I , 2(1 + ..jDz/ DI )

D2 - DI y=----- 2(VDI/ D z + I) 23. (a) V f is parallel to V g. (b) The maximum value of If /9 occurs at (If /3,1f /3,1f /3), (If /3, - If /3, - If /3), (-If /3, - If /3,1f /3) and ., (-If /3,1f /3, - If /3). The minimum value of - If /9 occurs at (-If /3,1f /3,1f /3), (If /3, -If /3,1f /3), (If /3,1f /3, -If /3) and ( - If /3, - If /3, - If /3). (f) The segment on which h is positive shrinks to 0 (c) x = y = 8, h = 4. as (J~O. 53. (0, 0, 0) is closest for a ::;; 1/8; Review Exercises for Chapter 16 (±..j(8a-I)/32, 0, (8a-I)/8)areclosest 1. [y exp(xy) - y sin(xy)]i + [x exp(xy) - x sin(xy)]j fora> 1/8. 3. [2x exp(x 2) + y 2sin(xy2)]i + 2xy sin(xyz)j 55. Let d Z = (x - a)z + (y - b)z + (k(x, y) - e)z and 5. (a) (9//2)cos(3) set a(d2)/ax and a(dz)/ay equal to zero. (b) (i - 2j)/VS 7. (a) /2 / e (b) (- i - 2j) / VS 16.4 Constrained Extrema and 9. 3x + 4y - z = 4 Lagrlhge Multipliers 11. x + Y + z = If 13. (2x + y)(dx/dt) + (x + 2y)(dy/dt) = 0 2 1. The minimum value is 0 (occurs at (0,0)), maxi• 15. (x + yZ)(dx/ dt) + 2xy(dy / dt) = 0 mum value is 3 (occurs at (0, I) and (0, -I)). 17. I 3. The minimum value is 8 (occurs at (0, 1) and (0, -I) 19. I maximum value is 15 (occurs at (1,0) and (-1,0»). 21. (0,0) is a saddle point. 23. (0,0) is a saddle point. 5. J3572 is the maximum value, -J35/2 is the mini• 25. (- 1,0) is a saddle point, (0,0) a local maximum, and mum value. There are no interior critical points. (2,0) a local minimum. 7. J2 is the maximum value, - J2 is the minimum 27. (n,O), n an integer, are saddle points. value. ( ± fij2, ± fij2) are the critical points. 29. (a) II Gil = «a? jax)2 + (ap jay)2)1/2 A.98 Chapter 16 Answers

(b) According to Newton's second law of motion, is 5; the plane contains the line through the G creates a force on the air mass which pro• point (1,0,2) parallel to the y-axis. duces a proportionate acceleration in the di• 45. Equate the four partial derivatives equal to zero. rection of G. Eliminate AI by subtracting two equations and A1 (c) by dividing two equations. 47. 600 North 49. Approximately 570 l low 51. (4,2) ~pressure 53. (a) dy/dx= -(2x)/(3y 1+eY ) (b) (aFI/aYz)(aF2/aX) - (aF1/aYz)(aFI/ax dy , / dx = -c::--=--:-:---:-=-:=--:-::--,----:-c---:-,--,,--,---,--- ~ ~ ""pressure (aFdaYz)(aF,/ay,) - (aF1/ay,)(aF,/ay

(aF,/ay,)(aFdax) - (aF,/ax)(aF2/ay, dYz / dx = -:-::-=--=--:-~:--:-c--,---,-c-----c,------:--- QBP (aF,/aYz)(aF2/ay,) - (aF2/aYz)(aF,/ay L-..--"I<'---+-----.~----jJ------East (c) dy,/dx = -(2x + sinx)/2y" dYz/dx = (cos x - 2x)/Yz. wind 55. (a) Use equality of mixed partials of I(x, y) (b) Use equality of mixed partials of I(x, y, z) (d) If, in the Southern Hemisphere, you stand (c) No with your back to the wind, the high pressure 57. (a) x 2exp(xy1) + C is on your left and the low pressure is on your (b) No function exists. right. (c) ln(l + x 2 + /) + C. North (d) No function exists. y 59. (a) (xo, -Yo) high pressure (b) (Yo, xo) QBP (c) g(x, y) = xy low (d) Since Vg is normal to the level curves of g, the ~ pressure tangent to these curves is normal to level curves of f. (e) © y '---~e..----J'--+---,t------East x

wind

(± 31. (a) I, 0, 0) are closest, distance is I. f= constant (b) (-v'4T /6,v'4T /6,1/6). (v'4T /6, - v'4T /6, 1/6) are closest,

the distance is 183 /6. x (c) (- I, - I, I), ( - I, I, - I), (I, - I, - I) and (I, I, I) are closest, the distance is 13. 33. 0.382 is the minimum value, 2.618 is the maximum value. 35. 0.540 is the minimum value, I is the maximum value. 37. The partials of k(x, y, A) = x + 2y sec 0 + A(Xy + y1tanO - A) must all vanish 39. (a) Use the second derivative test. 61. (a) Ix = -2xy3/(x1 + y1)1, (b) Since the maximum and minimum must occur Iv = y1(3x2 + y2)/(x1 + y2f. Ix (0, 0) = 0 and on the boundary (by (a», both are zero. Hence t(O,O) = I (compute the latter two using lim• I is everywhere zero. its). 41. (a) (1,1,2), x + Y + 2z = 6; (b) (a/ar)I(rcosO,rsinO) = (2,3,2), 2x + 3Y + 2z = 9. cosZO( - 2 sin 30 + 3 sin20 + I) is defined for all (b) 0 = 0.47 O. (c) (1,1,2) + t( -4,2, I) (c) Vj(0, 0) . (cos 0, sin 0) = sin 0 disagrees with 43. (a) A normal vector to the tangent plane is the formula in (b). There is no contradiction I) + !;,j - k. because the partial derivatives are not continu• (b) The slope of the plane relative to the xy plane ous at (0,0) (see Exercise 20, p. 783). Chapter 17 Answers A.99 Chapter 17 Answers 17.1 The Double Integral and 21. 25/6 grams. 23. Use partition points obtained from subrectangles Iterated Integral for both the functions being added. 25. (a) 0.88(1 - sin2a cos2(hT /365)]1/2cos(2'IT1 /24)] 1. 12 + 0.67 sin a cos(2'ITT /365), where a = 23.5° 3. (a) Divide D into DI = [ - 1,0] X [2,3], (b) This is the total solar energy received in the D2 = [-1,0] X (3,4], and D3 = (0,1] X [2,4]. state between times tl and 12 on day T. Let g(x, y) be -4 on D I , -5 on D 2 , and 0 on D 3• (b) The part of the integral for x .;; 0 is the negative of the part for x ;;, o. 17.2 The Double Integral Over z General Regions 1. Both a type 1 and a type 2 region.

5. 18 7. 16/9 9. e/2 - I/2e 11. 50 13. 4/3 15. 45/4 + (15 /2)In(3 /2) 3. Both a type I and a type 2 region. 17. 0; this agrees with the answer in Exercise 3(b). 19. 76/3

5. 7//1.2 35 9. I: 1" + A.100 Chapter 17 Answers

II. Type?; 104/45 25. The result of the first integration is the length of a section of the region. 27. Type I states include: Wyoming, Colorado. Nevada, New Mexico, Kansas, and Ohio. Type 2 states include: Wyoming, Colorado, Kansas, North Dakota, and Vermont. 17.3 Applications of the Double Integral 13. rype I; 3J 140 1. 127T 3. (4/5)(2 + 93,(4) 5. 10 + 8/n

IS. !: 71

(1,2,0)

7. (16e 2 - 16 + 7T 4)/32 9. 243/80 11. (7T 2 - sine 7T 2)l/ 7T 3 13. 2 15. 2/3 17. 2a2/3 19. (11/18.65/126) 21. (7/5.0) 23. (4/15)(9y3 - 8/2 + I) 25. /2 7T /4 27. Compare the formulas for average value and cen• ter of mass.

29. (a) Write z = ± ..jr2 - x 2 - y2 over the region 0.;; x.;; r, -p=-;r .;; y.;;p=-;r. (b) The result is independent of r. 31. $503.64 b 33. 21 f f (X) ~[j(x)f - y2 dydx = 7Tlb[f(X)fdx. a - f(x) a (This is the disk method; see p. 423.) 17.4 Triple Integrals 1. 7 3. -8 5. Type I 7. Type I 11. 1/2 9. 25";2/3 7T 13. 0 15. as /20 17. 0 19. 3/10 21. The double integral of F over the base of the box times the height of the box. 23. Interchange x and z in Example 4. 25. The region under the graph is of type I. 27. (a) Use iterated integrals and the constant multi• 21. ple rule for definite integrals. 2.l 19 (b) e X+ v + z = eXeYe Z. Chapter 17 Answers A.101

17.5 Integrals in Polar, Rearrangement of the formula for z gives the first line of the equation. The next step is Cylindrical, and justified by the additivity property of integrals Spherical Coordinates (see rule 2 on p. 841). By symmetry, we can replace z by - z and integrate in the region 1. 64'11"/5 above the xy-plane. Finally, we can factor the 3. V7T /10 minus sign outside the second integral and since p(x,y,z)=p(u,v, -w), we are subtract• 5. 1287T ing the second integral from itself. Thus, the 4 7. 57T(e - 1)/2 answer is O. 9. 27T[,12 -In(l +/2)] (c) In part (b), we showed that z times the mass of 11. 47Tln(a/b) W is O. Since the mass must be positive, z 13. (7T /6)(8 - 3V3) must be O. 15. 47T/6 (d) By part (c), the center of mass must lie in both 17. 1/';;;; planes. 17. Follow the pattern on p. 880 for the case of 19. (a) Use the linear approximations constant density. Be sure to use different symbols ax ax ay ay for the density and the spherical coordinate ~x ~ au ~u + at: ~t: and ~y ~ au ~u + at: ~t:. Jx 2 + y2 + Z2. (b) Use the fact that I a(x, Y) 1= r. a(r,O ) Review Exercises for Chapter 17 21. The conditions 0 « y « I - x and 0 « x « I are equivalent to 0 « x + y « I and 0 « -y- « I. x+y 1. 960 - sin (I I ) + sine 10) + sin(7) - sin(6) ~ 961.4 3. 15/2 5. 64/3 v 7. 7T In(sec I + tan I) 9. 1/48 11. 10/3 13. 47Tabc /3 15. 27T(l - 1/ra+T)/3 17. (25 + 1Ov's)7T /3 19. (47T /3)(1 - /2 /2) II 21. 407T /3

17.6 Applications of Triple (-3.lT. 10) Integration

1. (a) p, where p is the (constant) mass density. (b) 41/3 '3. (1/2,1/2,1/2) 5. ~ (as in Example 2) (-3.lT,0) 7. (a) kc 2 (b) Along the sphere x 2 y2 Z2 = c 2 + + ( I. O. 0) 1l:::.:,Y<------":':.l' 9. 1/4 ( l.lT.O) 11. Letting d be density, the moment of inertia IS d10k 10 277 10a sec p4sin\pdpdO d1>

13. 1.00 X 108(m/s)2 15. (al The only plane of symmetry for the body of .r an automobile is the one dividing the left and 23. 10/3 righ t sides of the car. 25. Cut with the planes x + y + z =3,jk/n, (b) z.J J JwP(x,y,z)dxdydz is the z-coordinate I « k « n - I, k an integer. of the center of mass times the mass of W. 27. Use the formula for surface area. A.102 Chapter 18 Answers

29. (0,0, - 0.203) (b) Same as in (a). 31. 2.,,(2/2 - 1)/3 47. Using d for density, the potential is given by 2."dG [(P2f - R 2/3 - 2(PI)3 /3R]. 33. .,,(5/5 - 1)/6 35 . .,,2/8 49. Show that the double integral of fxy - /yx over every rectangle is zero. 37. (." / 4)ln(2) 51. + = 1 39. V(O,O,R) = (4.71 x lO22)G/ R u; u; 41. (a) 32/3 (b) 64." 43. J.,,/5 45. (a) It is equal to the average of the values of the function at the vertices.

Chapter 18 Answers 18.1 Line Integrals 11. The two line integrals are equal since P dx + Q dy is exact. 1. 5 3.0 5.3/2 13. xeY + yeX = 2 7. -h 9 • ." 11. h 15. x 3 + xy + y3/3 = 17/3. 13. h 2 15. -I 17. 0 17. Not exact 19. 2x3y + xy2 = C 19. 2 + e 21. I 21. (a) -I 23. 2/3 25. -1/2 (b) Y = x 27. (cos 3)/3 + 5/12 29.0 23. Y = ±x 31. (5/5 - 1)/12 33. 52/I4 25. p. = x 35. Use the chain rule and make a change of variables 27. x 2/2 + x/y = C in the integral. 29. In p. = f[(Nx - My)/ M]dy 18.2 Path Independence 31. If f is an antiderivative of P dx + Q dy, then 1. 3 3. 8 5. 1/2 fx = aax f P dx; now integrate. 7. ." 9. 0 11. 256 13. JMm(r2- 3 - rl- 3)/3 17. Yes; xy + sin y + C 15. No 19. Yes; x 2 + x 2cos Y + C 18.4 Green's Theorem 21. Calculate fx = - Y /(x2 + y2) and 1. Each side gives 1/12. /y = x/(x2 + y2). 3. Each side gives -.". 2 23. h 5. 25.0 27. (a) The integrals along the four sides are 0, I, -I, y and O. (b) f(x, y,z) = z3x + xy + C. 29. (1/2)3 - (/2 /2i + sin(3/2 .,,/4) 31. The field is not conservative. 33. x exp(yz) + C 35. If Vf=Vg then V(f - g) = O. Show thatf - g is constant, using the second box on p. 896. x 37. Pick a point P and draw an arc from P to each region. If the arc crosses the circles an even num• 7. ber of times, color the region red, otherwise color y the region blue. 18.3 Exact Differentials 1. No 3. No 5. Not exact 7. Exact 9. J is the integral of P dx + Q dy along the line segment from (0,0) to (0, y) followed by the seg• x ment from (0, y) to (x, y). Chapter 18 Answers A.103

9. -4fb (a/b) b'-y' xydxdy = 0 18.6 Flux and the -b)(a/b) b'-y' Divergence Theorem 11. 67/6 13. -20 1. 3x2 - x 2cos(xy) 15. 0 3. ycos(xy) - x 2 sin(x2 y) 17. (a) Recall that a zero dot product implies orthogo• 5. -4 nality. 7.0 (b) Use Green's Theorem. 9. (a) A,C 19. In applying the hint, note that the terms involving (b) B,D Vu' Vv cancel. 11. Y exp(xy) - x exp(xy) + yexp(yz) 21. 9 13. 3 23. Simplify A = 1J> dy - Y dx to 15. I2w /5 17. I 3a2 (27Tsin220dO; now use the double angle for- 19. Use Gauss' theorem and V • n = O. 8 )0 21. Apply the divergence theorem to f~ using mula to integrate. V·(j~)=Vf·~+ fV·~. 25. 5/12 23. (i) Use the vector identity V ·(V X H) = O. 27. A horizontal line segment divides the region into (ii) Since charge is conserved, the rate at which three regions to which Green's theorem applies; charge is entering equals the rate at which see Example 2. charge is leaving; the total flux is therefore 29.0 zero. By Gauss' theorem, the integral of V·J 31. au/ay = Q by the· iamental theorem of calcu• over any region is zero, so V·J = O. lus; similarly for ~ , JX = P. 25. (a) Calculate div Vcp on a region excluding a small 33. The device is run around the perimeter of the ball near q. region and the mechanism evaluates the integral (b) Use (a). 1faD (x dy - Y dx). Review Exercises for Chapter 18 18.5 Circulation and Stokes' Theorem 1. 10/3 - 2cos2 + 2sin2 - 2sin 1 3. 2e 1. -2 5. -8 2 3. 4xy /(x + y2)2 7. 1- e 5. 8 9. -cos 5 + cos 3 + (In 4)2/4 + (44 In 2)/3 + 83/18 7.0 11. (a) sin(ln(5/4» - sin(ln(3/4» - (In(5/4))2 + 9. z3j + eZj + (y sin xy)k. (In(3/4))2 2 3 3 11. (2/(x + y2 + z2)2)(x j + (x) - yz2)j - z k) (b) 0 13. Let ~ = pj + Qj + R k, write out curl (j~), and (c) 0 use the product rule for derivatives. 13. 43/54 15. V X V f = 0 is a vector identity; the integral of an 15. -1 exact differential about a closed loop is zero. 17. (e - 1)/3 17. Compute that V x~ = 0 and use Stokes' theorem. 19. Not conservative 19. -1/2 21. Not conservative 21. Use Stokes' theorem and ~(u(s» • u'(s) = O. 23. Yes; 3zy + x) + C 23. -17w!f 25. Exact; xeYsin x + C 25·1.H (r)'dr= Jfs(VXH)'ndA = Jfs J • ndA 27. Not exact 29. ysinx + x 2e" + 2y = w2/4 27. The component of the curl of the velocity of the 31. - x + x) + Y = I fluid along a vector n is the circulation around n 33. Not exact per unit area (p. 921); this is maximized when the 35. Not exact curl and n are aligned. 37. (a) JD(l)dxdy, the area of D 29. Partition the surface into the upper and lower J hemispherical pieces; apply Stokes' theorem to (b) JL (- I) dy dx, the negative of the area of D each piece and add. (c) J L(O)dydx = 0

39. curl ~ = [y /(x + zf]i - [y /(x + z)2]k; div~ = I/(x + z) 41. curl ~ = -4xi - 2yj + 2zk; div~ = O. A.104 Chapter 18 Answers

43. (a) V X F 2x}>zi - 3xy2zj + 2k, V· F x»2 = = 59. f !aj(Vj).ndA = f f fw(fVj" + Vf' Vf)dxdydz (b) 2'lT (c) 1/12 gives 0 = f f fw(Vf' Vf)dxdydz 45. Is(V XF)· ndA, where C is the boundary of S. f = f f fwllVfll2dxdydz, so Vf= 0 and thus 47.0 f is constant. 49. -2'lT/3 61. (V X CIJ) • ndA = 0 if S is the union of the two 51. Use Gauss' theorem. f Is 53. Use Gauss' theorem over a small region; divide by surfaces. the volume of the region and use the mean value theorem for integrals. 55. -8'17 57. (a) Letu'= foxa(t,O,O)dt+ {c(x,O,t)dt+

foYb(x, t, z)dt, so au'lay = Q. Permute x, y, z to give u" with au" laz = R, and u'" with au'" lax = P. Use Stokes' theorem to show that u' = u" = u"'. (b) xyz - cosx + C Index Includes Volumes I, II and III

Note: Pages 1-336 are in Volume I; pages 337--644 are in Volume II; pages 645-934 are in Volume III.

Abbot, Edwin A. 883 of power 130, 338 Abel, Nils Hendrik 172 rules 337, 338 absolute value 22 of sum 130, 338 function 42, 72 of trigonometric function 340 properties of 23 of trigonometric functions 269 absolutely convergent 574 Apollonius 696 accelerating 160 Apostol, Tom M. 582 acceleration 102, 131, 741 approaches 58 gravitational 446 approximation vector 741 first-order (see linear approximation) Achilles and tortoise 568 linear (see linear approximation) addition formulas 259 arc length 477 addition of ordered pairs 646 of curve in space 745 air resistance 136 parametrized by 749 Airy's equation 640 in polar coordinates 500 algebraic Archimedes 3,5,6 operations on power series 591 area 4, 251 rules 16 between curves 853 alternating series test 573 between graphs 211, 241 amplitude 372 between intersecting graphs 242 analytic 600 of graph 857 angular of n-sided polygon 934 frequency 373 of parallelogram 683 momentum 506, 748 in polar coordinates 502 annual percentage rate 382 of region bounded by a curve antiderivative 104, 128, 897 914 of if 323, 342 of sector 252 of constant mutiple 130, 338 of surface 482 of exponential 342 of revolution 483 of hyperbolic functions 389 signed 215 of inverse trigonometric function 341 of triangle 934 of l/x 323, 342 under graph 208, 212, 229 of polynomial 130 of step function 210 1.2 Index

argument 40 Buys-Ballot's law 834 arithmetic mean 188 arithmetic-geometric mean inequality 436 associative 679 Calculator discussion 49, 112, 166, 255, astigmatism 821 257,265,277,309,327,330,541 astroid 198 calculator symbol 29 astronomy 9 Calculus Unlimited iii, 7(fn) asymptote 165 calculus horizontal 165, 513, 535 differential I of hyperbola 698 fundamental theorem of 4, 225, 237 vertical 164, 518, 531 integral I ,3 asymptotic 164 Calder, Nigel 756 average 3 capacitor equation 406 power 464,465 Captain Astro 802, 804, 816 rate of change 100 carbon-14 383 value 434, 854, 878 Cardano, Girolamo 172 velocity 50 cardiac vector 658 weighted 437 cardioid 298 axes 29 cartesian coordinates 255 rotation of 705, 707 catastrophe translation of 703 cusp 176 axial symmetry 423 theory 176 axis catenary 402 major 696 Cauchy, Augustin-Louis 6,521,908 minor 696 mean value theorem 526 of symmetry 440 Cauchy-Riemann equations 835 Cauchy-Schwarz-B uniakowski inequality 669 B-8 definition of limit 516 Cavalieri, Bonaventura 8, 425 ball 421 principle 843 Bascom, Willard 306(fn) center of mass 437, 693, 857, 876 base of logarithm 313 in the plane 439 basis vectors, standard 656 of region under graph 441 bearing 659 of triangular region 445 beats 628 centripetal force 747 Beckman, P. 251, (fn) chain rule 112, 779 Berkeley, Bishop George 6(fn) for partial derivatives 800 Bernoulli, J. 252(fn), 521 physical model 116 equation 414 change numbers 643 average rate of 100 Bessel, F.W. 639 instantaneous rate of 10 equation 639 linear or proportional 100 functions 643 proportional 95 Binder, S. M. 836 rate of 2, 100, 101, 247 binomial series 600 of sign 146 binormal vector 753 total 244 bird 692 of variables 877 bisection, method of 142, 145 chaos and Newton's method 547 blows up 399 characteristic equation 617 Boltzmann's constant 823 charge 930 bouncing ball 549 chemical equation 648, 651, 660 boundary 848,908 chemical reaction rates 407 bounded above 575 circle 34, 44, 120, 251, 421 Boyce, W. 401 as section of cone 695 Boyer, C. 7(fn) , 252(fn) equations of 37 Braun, Martin 380,401,414,626 parametric equations of 490 Burton, Robert 8 circuit, electric 413 bus, motion of 49, 202, 207, 225 circular functions 385 Index 1.3 circulation 914 of rational functions 140 circumference 251 continuous function 63, 139, 770 city integrability of 219 Fat 116 continuously compounded interest 331, Thin 115 382,416 Clairaut 767 convergence climate 180 absolute 574 closed curve 889 conditional 574 closed interval 21 radius of 587 closed interval test 181 of series 562 closed rectangle 839 of Taylor series 597 Cobb-Douglas production function 831 convergent integral 529 College, George 383 convex function 199 common sense 61, 193 cooling, Newton's law of 378 commutative 788 Cooper, Henry S. Jr. 682 comparison test 570 coordinates 29, 648, 653 for improper integrals 530 cartesian 255 for limits 518 polar 253, 255, 791, 869 for sequences 543 spherical 731 completing the square 16, 17, 463 Coriolis force 499 complex number 607, 609 cosecant 256 argument of 609 inverse 285 conjugate of 609 cosine 254 imaginary part of 609 derivative of 266 length or absolute value of 609 direction 676 polar representation of 612 hyperbolic 385 properties of 610 inverse 283 real part of 609 law of 258, 676 component 648 series for 600 functions 738 cost, marginal 106 composition of functions 112, 113, 779 cotangent 256 compressing fluid 926 inverse 285 computer-generated graph 716,717, 720, Coulomb's law 805 721,813,819,821,822,833,834, Cramer, Gabriel 690 837 rule 690 concave Creese, T.M. 401 downward 158 critical points 151, 814 upward 158 critically damped 621 concavity, second derivative test for 159 cross product 674, 679, 754 conditionally convergent 574 cross-derivative test 898, 904 cone, elliptic 728, 793 Crowe, M. J. 657 conic sections 695 cubic function 168 connected 897 general, roots and graphing 172 conoid 486 curl conservation of energy 372 of a vector field 917 conservative vector field 895 scalar 915 consolidation principle 438 curvature 749, 750, 821 constant function 41, 192 curve 31 derivative of 54 closed 889 rule for antiderivatives 130 geometric 889 rule for derivatives 77 level 712 rule for limits 62, 511 parametric 124, 298, 489 rule for series 566 in space 735 constrained extrema 825 regular 749 first derivative test for 826 in space, arc length of 745 consumer's surplus 248 cusp 170 continuity 63, 72, 770 catastrophe 176 equation 953 cycloid 497 1.4 Index

cylinder 715, 722 of polynomial 75, 79 parabolic 714, 723 of power 75, 119 cylindrical coordinates 728 of a function 110, 119 triple integrals in 872 of product 82 of quadratic function 54 of quotient 85 dam 454 of rational power 119 damping 377 of a function 119 in forced oscillations 626 of reciprocal 85 in simple harmonic motion 415 of sum 78 Davis, Phillip 550 of vector function 739 day of vX71 length of 30, 302 as a limit 69 shortening of 303 directional 801 sidereal 757 formal definition of 70 solar 757 Leibniz notation for 73 decay 378 logarithmic 117, 322, 329 decelerating 160 matrix 784, 786 decimal approximations 538 partial 765 decrease, rate of 101 second 99, 104, 157 decreasing function 146 second partial 768 definite integral 232 summary of rules 88 constant multiple rule for 339 determinant 683, 685 endpoint additivity rule for 339 jacobian 792 inequality rule for 339 Dido 182 power rule 339 Dieterici's equation 795 properties of 234, 339 difference quotient 53, 766 by substitution 355 differentiable 70 sum rule for 339 differential degree algebra 356 as angular measure 252 calculus 1 of polynomial and rational functions 97 equation 369 delicatessen, Cavalieri's 425 Airy's 369 delta 50(fn) Bessel's 639 demand curve 248 Euler's 796 Demoivre, Abraham 614 first order 369 formula 614 of growth and decay 379 density 440 harmonic oscillator 370 uniform 440 Hermite's 636 dependent, linearly 89 Legendre's 635 depreciation 109 linear first order 369 derivative 3, 53, 70 of motion 369 of 17" 318 numerical methods for 405 of composition 113 partial 898 of constant multiple 77 second-order 399 of cosine 266 second-order linear 617 of hyperbolic functions 388 separable 398, 399 of implicitly defined function 122 series solution of 632 of integer power 87 solution of 369 of integral with respect to endpoint 236 spring 370 of integral, endpoint a given Tchebycheff's, 640 function 236 form 893, 902 of inverse function 278 geometry 749 of inverse hyperbolic function 396 notation 351, 359, 374, 398 of inverse trigonometric functions 285 differentiation 3, 53, 122, 201 of linear function 54 implicit 120, 398, 810 of logarithmic function 321 logarithmic 117, 322, 329 of lIx 71 partial 767 Index 1.5

of power series 590 hyperboloid of one sheet 760 rules for vector functions 740 integral 417, 506, 507 under integral sign 883 paraboloid 728 diminishing returns, law of 106 endpoints 181 dipole 693 of integration 217 Diprima, Richard 390, 401 energy 201, 445 direction conservation of 372 angles 676 equation 753 cosines 676 potential 446 field 403 equation directional derivative 801 chemical 648,651,652,660 directrix 700 differential 369 (see also differential discriminant 17 equation) disk 421 indicial 638 method 423 of circle and parabola 37 displacement 230 of ellipse 696 vector 657 of hyperbola 698 distance formula of line 662 in the plane 30 of parabola 701 on the line 23 of plane 671 divergence 925 of plane in space 672 free 926 of straight line 32 theorem 694, 924 of tangent line 90 divergent integral 529 parametric 124, 298 dog saddle 722 simultaneous, 37 domain 41 spring 376 dot product 668 equipotential surfaces 816 double integral 839, 850 error function 558 applications of 853 Eudoxus 4 over general regions 847 Euler, Leonhard 251(fn), 252(fn), 369 in polar coordinates 870 differential equation 796 properties of 841 equation 636 double-angle formulas 259 formula 608 drag 136, 414 method 404 dummy index 203 evaluating 40 even function 164, 175 exact differential 901, 903 e 319, 325 exhaustion, method of 5, 7 as a limit 330 existence theorem 180, 219 e-A definition of limit 513 expansion by minors 687 £-8 definition of limit 511, 769 exponent zero 23 ear popping 116 exponential and logarithmic functions, earth, rotation of 756 graphing problems 236 earth's axis, inclination of 301 exponential functions 307 eccentricity 702 derivative of 320 economics 105, 830 limiting behavior of 328 electric circuits 399, 413 exponential growth 332 element 21 exponential series 600 elementary regions 848, 864 exponential spiral 310, 333, 751 ellipse 696 exponentiation 23 equation of 696 exponents focus of 696 integer 23 reflection of property of 702 laws of 25 as section of cone 695 negative 26 shifted 703 rational 27, 118 ellipsoid 724, 793 real, 308 elliptic extended product rule for limits 62 cone 728, 273 extended sum rule for limits 62, 69 1.6 Index

extensive quantity 445 exponential 307 extreme value theorem 180 graph of 41, 44 extremum, local 813 greatest integer 224 harmonic 774 homogeneous 796 factoring 16 hyperbolic 384, 385 falling object 412, 414 identity 40, 277, 384, 385 Faraday's law 922 integration of 217 Feigenbaum, M. J. 548 inverse 272, 274 Feinberg, M. 836 inverse hyperbolic 392 Ferguson, Helaman 602 inverse trigonometric 281, 285 Fermat, Pierre de 8 linear 192 Fine, H.B. 468 odd 164, 175 first derivative test 153, 814 piecewise linear 480 for constrained extrema 826 power 307 first-order approximation (see linear rational 63 approximation) squaring 41 Fisher, Chris 884 step 140, 209, 210, 839, 861 fluid 914, 926 of three variables 712 flux 924 trigonometric, antiderivative of 269 law 805 trigonometric, graphs of 260 of a vector field 925 of two variables 711 flying saucer 430 vector 737 focus of ellipse 696 zero 41 focusing property of parabolas 36, 95, 97, fundamental integration method 226 701 fundamental set 630 football 453 fundamental theorem of calculus 4, 225, force 448, 659, 675, 885, 886 237 centripetal 747 alternative version of 236 on a dam 454 gravitational 834 resultant 659 Galileo 8 forced oscillations 415, 624 gamma function 643 four-petaled rose 730 gas Fourier coefficients 506 ideal 795 fractals 499 Van der Waals 795 fractional exponents (see rational Gauss, Carl Friedrich 205, 613, 908 exponents) divergence theorem in the plane 925 fractional powers (see rational powers) in space 927 Frenet formulas 753 gaussian integral 870, 871 frequency 259 Gear, C. W. 405 friction 377 Gelbaum, Bernard R. 576, 600 Friedrichs, Kurt 694 general solution 618, 623 Frobenius, George 636 geometric curve 889 frustrum 485 geometric mean 188,436 function 1, 39 geometric series 564, 600 absolute value 42, 72, 73 geometry, differential 749 average value of 434 Gibbs, J. Willard 657 circular 385 global 141, 177 component 738 maximum 813 composition of 112, 113, 779 Goldman, M. J. 658 constant 41, 192 Goldstein, Larry 172 continuous 63 Gould, S.H. 6(fn) convex 199 gradient 197, 798 cubic 168 and Laplacian in polar coordinates 836 definition of 41 and level curves 808 differentiation of 268 pressure 833 even 164, 175 and tangent planes 806 Index 1.7

vector fields 896 hyperbolic cosine 385 graph 41, 163 hyperbolic functions 384, 385 area between 241 antiderivatives of 389 area under 212, 229 derivatives of 388 computer-generated 716,717,720,721, inverse 392 813, 819, 821, 822, 833, 834, 837 hyperbolic paraboloid 719, 720, 728 of function 41, 44 hyperbolic sine 385 of two variables 711 inverse of 393 surface area of 857 hyperboloid graphing in polar coordinates 296 of one sheet 725 graphing problems elliptic 760 exponential and logarithmic of revolution 725 functions 236 as ruled surface 763 trigonometric functions 292 of two sheets 724 graphing procedure 163 gravitational acceleration 446 gravitational force 834 I method 361 inside a hollow planet 880 ice ages 756 gravitational potential 878, 882, 883 ideal gas 795 greatest integer function 224 identity function 40, 277 Green, George 908 rule for limits 60 identities 933 identity, trigonometric 257 theorem 908, 911 illumination 183 growth 378 imaginary axis 609 and decay equation, solution of 379 imaginary numbers 18 exponential 332 implicit differentiation 120, 122, 398, 810 gyroscope 682 implicit function theorem 810 improper integrals 528, 529 comparison test 530 half-life 381, 383 inclination, of the earth's axis 301 hanging cable 401 incompressible 926 Haralick, R.M. 401 increase, rate of 10 1 Hardin, Garrett 416 increasing function 146 harmonic series 567 test 148 harmonic function 774 theorem 195 heat increasing on an interval 149 conduction 772 increasing sequence property 575 equation 775, 933 indefinite integral (see antiderivative) flow 933 test 233 helix, right circular 736 independent variable 40 Henderson, James 831 indeterminate form 521 Hermite polynomial 636 index Hermite's equation 636 dummy 203 herring 156 substitution of 205 Hipparchus 256(fn) indices of refraction 682 Hofstadter, Douglas 548 indicial equation 638 Holder condition 559 induction, principle of 69 homogenous equation 623 inequalities 18 homogenous function 796 properties of 19 Hooke's Law 99, 295 inequality horizontal asymptote 165, 513, 535 arithmetic-geometric mean 188, 436 horizontal tangent 193 Cauchy-Schwarz-Buniakowski 669 horsepower 446 Minkowski's 365 horserace theorem 193 Schwartz 669 hyperbola 698 triangle 665 asymptotes of 698 infinite limit 66 equation of 698 infinite series 561 shifted 703 infinite sum 561 1.8 Index

infinitesimal 73 numerical 550 parallelograms 856 of power series 590 infinitesimals, method of 6, 8,419, 428, intensity of sunshine 451 441,477,482,495,856,872 interest, compound 244, 331 infinity 21 interior 839 inflection point 159 intermediate value theorem 141, 142 test for 160 intersecting graphs, area between 242 initial conditions 371, 398 intersection of points 39 inner product 668 intertia, moment of 877 instantaneous quantity 445 interval 21 instantaneous velocity 50, 51 closed 21 integer power rule for derivatives 87 open 19 integers 15 inverse sum of the first n 204 cosecant 285 integrability of continuous function 219 cosine 283 integrable 217,848,861 cotangent 285 integral 129, 217, 861 function 272, 274 calculated "by hand" 212 integral of 362 calculus 1 rule 278 convergent 529 test 276 definite 232 hyperbolic functions 392 definition of 217 derivatives 396 divergent 529 integrals 396 double &39, 850 hyperbolic sine 393 elliptic 417 secant 285 gaussian 870, 871 sine 281 of hyperbolic function 389 tangent 283 improper 528, 529 trigonometric functions 281, 285 indefinite 129 (see also antiderivative) invertibility, test for 275, 276 of inverse function 362 irrational numbers 16 iterated 842 isobars 833 reversing order of 851 isoquants 830 Leibniz notation for 132 isotherms 710 line 888, 893 iterated integrals 842 mean value theorem for 239, 435 reduction to 843, 862 of rational expression in sin x and ith term test 567 cos x 475 of rational function 469 Riemann 220 Jacobi identity 682 sign 129, 132, 217 Jacobian determinant 792 differentiation under 883 Jacobian matrix 792 surface 916 joule 445 tables 356 trigonometric 457, 458 triple 860, 861, 865 of unbounded function, 531 Kadanoff, Leo 548 wrong way 235 Katz, V.J. 908 integrand 129 Kazdan, Jerry 716 integrating factor 905 Keisler, H. J. 7(fn), 73(fn) integration 33, 129, 201, 851 Kelvin, Lord 594, 908 applications of 420 Kendrew, W.G. 180 by parts 358, 359 Kepler, Johannes 8 by substitution 347, 348, 352 first law 753 endpoint of 217 second law 506 limit of 217 kilowatt-hour 446 method, fundamental 226 kinetic energy 446, 859, 886 methods of 337 Kline, Morris 182 multiple 839 Korteweg-de Vries equation 783 Index 1.9

I'Hopital, Guillaume 521 real number 18 l'Hopital's rule 522, 523, 525 secant 51, 191 labor 106 slope of 52 ladder 190 slope-intercept form 32 Lagrange straight 31(fn), 125 interpolation polynomial 556 tangent 2, 191, 741 multiplier 826, 827 linear approximation 90, 91, 92, 158, 159, Laguerre functions 640 601,775,776 Lambert, Johann Heinrich 251(fn) linear combination 675 Laplace equation 796 linear function 192 Laplacian 933 derivative of 54 latitude 300, 732 linear or proportional change 100 least squares 823 linearized oscillations 375 Legendre, Adrien Marie 251(fn) linearly dependent 652, 689 equation 635 linearly independent 652 polynomials 635 Lipschitz condition 559 Leibniz, Gottfried 3, 73, 193, 594 Lissajous figure 507 notation 73, 104, 132, 217 local 141, 151, 177 for derivative 73 extremum 813 for integral 132 maximum point 151, 157 lemniscate 136 minimum point 151, 157,813 length logarithm 313 of curves 477 base of 313 of days 300, 302 defined as integral 326 of parametric curve 495 and exponential functions, word properties of 665 problems 326 of vector function, derivative of 321 level curve 712, 808 laws of 314 level surface 713 limiting behavior of 328 librations 506 natural 319 lima<;on 298 properties of 314 limit 6, 57, 59 series for 600 comparison test 518 logarithmic differentiation 117, 322, 329 of (cos x - 1)/x 265 logarithmic spiral 534, 535 derivative as a 69 logistic equation 506 derived properties of 62 logistic law 407 e-3 definition of 509, 769 logistic model for population 335 of functions 509 longitude 732 infinite 66 Lotka-Voltera model 400 at infinity 65, 512 love bugs 535 of integration 217 lower sum 210, 840, 861 method 6 Lucan 8(fn) one-sided 65, 517 of powers 542 product rule 511 properties of 60, 511 Maclaurin, Colin 594, 690 reciprocal rule 511 polynomials for sin x 602 of sequences 537, 540 series 594, 596 properties of 563 MACSYMA 465 of (sin x)/x 265 magnetic field 752 line 31(fn) major axis 696 equation of 32 majorize 199 integral 886, 893 Mandelbrot, Benoit 499 of a scalar function 895 marginal parametric equation of 664, 665 cost 106 perpendicular 33 productivity 106 point-point form of 32 profit 106 point-slope form of 32 revenue 106 1.10 Index

Marsden, Jerrold 582, 615, 710, 810, 826, accuracy of 559 849 and chaos 547 mass action, law of 476 second law of motion 369, 746, 886 matrix 685, 784 nonhomogenous equation 623 derivative 784, 786 noon 301(fn) multiplication 787 normal 669 Matsuoka, Y. 582 vector 671 Mauna Loa 804 principal 750 maxima and minima, tests for 153, 157, normalization 666 181,816 northern hemisphere 30 I maximum notation global 177 differential 351, 359, 374, 398 point 813 Leibniz 73, 104, 132, 217 value 177 summation 203, 204 maximum-minimum nowhere differentiable continuous problems 177 function 578 test for quadratic functions 816 number Maxwell equations 922, 923, 931 complex 607, 609 mean value theorem 191,922 imaginary 18 Cauchy's 526 irrational 16 consequences of 192 natural 15 for integrals 239, 435, 455 rational 15 Meech, L.W. 9 real 15, 16 midnight sun 301(fn) numerical integration 550 minimum points 177 local 813 odd function 164, 175 value 177 Olmsted, J. M. H. 578,600 Minkowski ' s inequality 365 one-sided limit 65,517 minor axis 696 open interval 21 minors, expansion by 687 open rectangle 839 mixed partial derivatives 769 optical focusing property of parabolas 36, mixing problem 413, 414 95,97,701 modulate 628 orbit 702 moment order 18 of a force 682 ordered pairs of inertia 878 addition of 646 momentum 692 multiplication of 646 monkey saddle 719, 721 ordered triples, algebra of 654 motion, simple harmonic 373 orientation 683 with damping 415 orientation quizzes 13 mUltiple integration 839 origin 29 mUltiplication orthogonal 669 matrix 787 decomposition 675 of ordered pairs 646 projection 670 multiplier, Lagrange 826 trajectories 402 oscillations 294, 369 damped forced 628 natural forced 415, 626 growth or decay 380 harmonic 373 logarithms 319 linearized 375 numbers 15 overdamped 621 Newton, Isaac 3(fn), 8(fn), 193(fn), underdamped 621 253(fn), 594 oscillator (see oscillations) iteration 559 oscillatory part 629 law of cooling 378 Osgood, W. 521 law of gravitation 746 Ostrogradsky, Michel 908 method 537, 546 overdamped oscillation 621 Index 1.11 pH 317 point Pappus' theorem for volumes 454 critical 151 parabola 34, 700, 752 inflection 159 equations of 37, 701 intersection 39 focusing property of 36, 95, 97, 701 local maximum or minimum 151, 157 as section of cone 695 point-point form 32 shifted 703 Poisson's equation 931 vertex of 55 polar coordinates 253, 255, 791, 869 parabolic cylinder 714, 723 arc length in 500 paraboloid area in 502 elliptic 728 double integrals in 870 hyperbolic 719, 720, 728 graphing in 296 of revolution 714 gradient and Laplacian in 836 parallel projection rule 653 tangents in 299 parallelepiped, volume of 685 " polar representation of complex parallelogram numbers 614 area of 683 Polya, George 182 infinitesimal 856 polynomial law 650 antiderivative of 130 parameter 489 derivative of 75, 79 parametric curve 124, 287, 489 pond, 74 in space 735 population 117,175,189,195,335,344, length of 495 382,400,407,416 tangent line 491, 492 position 131 parametric equations positive velocity 149 of line 490, 662 Poston, Tim 176 of circle 490 potential 834, 931 parametrized by arc length 749 energy 446 partial derivatives 765 power 445 equality of mixed 769 function 307 second 768 of function rule for derivatives 110 partial differentiation 765, 767 integer 23 partial differential equations 898 negative 26 partial fractions 465, 469, 591 rational 18, 27, 169 partial integration (see parts, integration by) real 308 particular solution 371, 623 rule partition 209 for antiderivatives 130 parts, integration by 358, 359 for derivatives 76, 119 path independence 895 for limits 62 pendulum 376,391,417 series 586 perihelion 702 algebraic operations on 591 period 259 differentiation and integration of 590 of satellite 748 root test for 589 periodic 259 precession 756 perpendicular lines 33 predator-prey equations 400 Perverse, Arthur 367 pressure gradient 833 Perverse, Joe 811 price vector 785 pharaohs 416 principal normal vector 750 phase shift 372, 629 producer's surplus 248 Picard's method 559 product Pierce, J.M. cross 677 Planck's constant 823 dot 668 Planck's law 823 inner 668 plane rule in space, equation of 671, 672 for derivatives 82 tangent 776, 782, 835 for limits 60 planimeter 914 £-8 proof of 520 plotting 29, 43, 163 triple 688 1.12 Index

vector 677 real powers 308 productivity reciprocal rule of labor 106 for derivatives 86 marginal 106 for limits 60 profit 329 reciprocal test for limit 517 marginal 106 rectangle program 40 closed 839 projectile 295, 752 open 839 projection, orthogonal 670 reduction proportional change 95 formula 365 Ptolemy 256(fn) to iterated integrals 843, 862 pursuit curve 499 of order 619 Pythagoras 694 reflecting property theorem of 30 of ellipse 702 of parabola 36, 95, 97, 701 reflection, law of 290 quadratic refraction, indices of 682 formula 16, 17 region function, derivative of 54 between graphs 240 quadratic surfaces 719,723 bounded by a curve, area of 912 Quandt, Richard 831 elementary 848 quantity vector 7115 regular curve 749 quartic function, general, graphing 176 regular tetrahedron 694 quizzes, orientation 13 related rates 124, 815 quotient word problems for 125 derivative of 85 relative rate of change 329 difference 53, 766 relativity 80(fn) rule, for limits 62 repeated roots 620 replacement rule for limits 60 resisting medium 412 radian 252 resonance 415, 626, 629 radius 34 resultant force 659 of convergence 587 revenue, marginal 106 Rado, T. 856 revolution rate hyperboloid of 725 of change 2, 101,247 surface of 482 of decrease 101 rhombus 692 of increase 10 1 Riccati equation 414 relative 329 Richter scale 317 rates, related 124, 811 Riemann, Bernhard 220(fn) ratio comparison test for series 571 integral 220 ratio test sums 220,221,551 for power series 587 right-hand rule 653, 677 for series 582 Rivlin's equation 199 rational Robinson, Abraham 7, 73(fn) exponents 118 rocket propulsion 412 expressions 475 Rodrigues' formula 640 function, continuity of 63, 140 Rolle, Michel 193(fn) numbers 15 theorem 193 power rule for derivatives of a root function 119 splitting 619 powers 118, 119 test 589 rationalizing 228 series 589 sUbsti'tution 474 for series 584 real axis 609 rose 297 real exponents 308 rotation 754, 7,):; real numbers 15, 16 of axes 705, 707 real number line 18 qf the earth 756 Index 1.13

Ruelle, David 548 shell method 429 Ruffini, Paolo 172 shifted ellipse 703 ruled surface 725, 763 shifted hyperbola 703 shifted parabola 703 shifting rule Saari, Donald G. 548 for derivatives 115 saddle for integrals 350 dog 726 sidereal day 757 monkey 719, 721 sigma 203 point 719, 817 sign, change of 146 satellite 747 signed area 215 period of 748 similar triangles 254 scalar 646 Simmons, G.F. 401 curl 915, 916 simple harmonic motion 373 multiplication 646, 649 damped 415 product 668 Simpson's rule 554 scaling rule for integral 350 simultaneous equations 37 Schelin, C.W. 257(fn) sine 254 school year 303 derivative of 266 Schwarz inequality 669 hyperbolic 385 Scott Russell, J. 783 inverse 281 seagull 658 law of 263 secant, 256 series for 600 inverse 285 Sky lab astronauts 682 line 52, 191 slice method 420 second derivative 99, 104, 157 slope 2, 31 test for maxima and minima 157,817 of tangent line 52 test for concavity 159 slope-intercept form 32 second-order approximation 601 Smith, D.E. 193(fn) second-order linear differential Snell's law 305, 682 equations 617 solar day 757 second partial derivatives 768 solar energy 8, 107, 179, 180,221,449, sections, method of 713 846 sector, area of 252 solid of revolution 423, 429 Seeley, Robert T. 883 solution of growth and decay equation 379 separable differential equations 398, 399 solution of harmonic oscillator sequence 537 equation 373 comparison test for 543 space limit of 537, 540, 563 Gauss' divergence theorem in 927 series 581 parametric curve in 735 alternating 572 vector in comparison test for 570 Spearman-Brown formula 520 constant multiple rule for 566 speed 103,497,666,741 convergence of 562 speedometer 95 divergent 562 sphere 421 geometric 564 bands on 483 harmonic 567 spherical coordinates 731 infinite 561 spiral integral test for 580 exponential 310, 333, 751 p 581 logarithmic 534, 535 power (see power series) Spivak, Mike 251(fn) ratio comparison test for 571 spring ratio test for 582 constant 370 root test for 584 equation 370, 376 solutions 632 square, completing the 16, 17,463,704 sum of 562 square root function, continuity of 64 sum rule 566 squaring function 41 set 21 stable eqUilibrium 376 1.14 Index

standard basis vectors 656 tangent standard deviation 453 function 256 steady-state current 520 inverse 284 steepest descent 808 horizontal 193 step function 5, 140, 209, 210, 839, 861 hyperbolic 386 area under graph of 210 line 2, 191,491,741 Stokes, Sir George Gabriel 908 to parametric curves 492 theorem 914, 918 slope of 52 straight line 31(fn), 125 plane 776, 782, 835 stretching rule for derivatives 117 and gradients 806 strict local minimum 151 to surface 806 Stuart, Ian 176 vector, unit 749 substitution vertical 169 of index 205 Tartaglia, Niccolo 172 integration by 347,348,352,355 Taylor, Brook 594 rationalizing 474 series 594 trigonometric 461 convergence of 597 subtraction, vector 650 test 599 sum 203 Tchebycheff's equation 640 collapsing 206 telescoping sum 206 of the first n integers 204 terminal speed 412 infinite 561 tetrahedron 694, 882 lower 210, 840 regular 694 Riemann 220,221,551 volume of 693 rule Thales' theorem 692 for antiderivatives 130 thermodynamics 836 for derivatives 78 third derivative test 160 physical model for 80 Thompson, D'Arcy 423 for limits 60 time 1::-0 proof of 520 of day 301 telescoping 206 of year 301 upper 210, 840 torque 748 summation torus 431,744 notation 201, 203, 204 total change 244 properties of 204, 208 tractrix 499 sun 300 train 55, 80, 291 sunshine transcontinental railroad 569 formula 754 transient 411, 628 intensity 451 transitional spiral 643 superposition 371 translation 793 supply curve 248 of axes 703 surface transpose 691 area of graph 857 trapezoidal rule 552 integral 916 triangle inequality 665 level 713 triangles, similar 254 quadratic 722 trigonometric functions 254, 256 of revolution 482 antiderivatives of 269 area of 483 differentiation of 264, 268 ruled 725, 763 graphs of 260 tangent plane to 806 graphing problem 282 suspension bridge 407 inverse 281, 285 symmetry 163, 296 word problems 289 axis of 440 trigonometric identity 257 principle 440 trigonometric integrals 457, 458 trigonometric substitution 461 triple integral 860, 861, 865 tables of integrals 356 applications of 876 Tacoma bridge disaster 626 in cylindrical coordinates 872 Index 1.15

in spherical coordinates 873 velocity 102,131,230,741 triple product 688 average 50 trisecting angles 172 field 404 Tromba, Anthony 710, 810, 826 instantaneous 50, 51 two-color problem 90 1 of light 823 positive 149 vector 741 unbounded region 528 vertex 55 underdamped oscillations 621 vertical asymptote 164, 518, 531 undetermined coefficients 623 vertical tangent 169 unicellular organisms 423 Viete, Franc;ois 251(fn) uniform density 440 Volterra, Vito 401 uniform growth or decay 381 volume 876 unit tangent vector 749 of bologna 426 unit vector 666 by disk method 423 unstable atmosphere 795 of parallelepiped 685 unstable eqUilibrium 376, 390, 406 by shell method 429 upper sum 210, 861 by slice method 419 uranium 383 of a solid region 419 Urenko, J.B. 548 of tetrahedron 693 by washer method 424

value absolute (see absolute value) washer method 424 maximum 177 water 178,247,772 minimum 177 flowing 131, 144,343,915 van del Waals gas 795 in tank 126 variable watt 446 change of 354, 875 wave 306 independent 40 wave motion 772 variance 453 wavelength 263 variation of parameter or constants 378, Weber-Fechner law 33 624 Weierstrass, Karl 6, 578 vector 645, 648 weighted average 437 acceleration 741 Wien's displacement 823 addition 649 Wilson, E. B. 657 cardiac 658 window seat 291 displacement 657 wobble 756 field 798, 888 word problems curl of 917 integration 247 flux of 925 logarithmic and exponential gradient 896 functions 326 function 737 maximum-minimum 177 derivative of 739 related rates 125 differentiation rules for 740 trigonometric functions 289 length of 664 work 675, 886, 888 moment of 682 wrong-way integrals 235 normal 671 Wronskians 630 principal 750 price 785 product 677 yogurt 279 quantity 785 Yosemite Valley 762 standard basis 568 subtraction 650 unit 666 zero unit tangent 749 exponent 23 velocity 741 function 41