Page 1 14.5 Directional Derivatives and Gradient Vectors 1013 EXERCISES 14.5 Calculating Gradients at Points In Exercises 1–4, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. 1. 2. ƒsx, yd = ln sx 2 + y 2 d, s1, 1d ƒsx, yd = y - x, s2, 1d 3. 4. In Exercises 5–8, find at the given point. 5. 6. ƒsx, y, zd = 2z 3 - 3sx 2 + y 2 dz + tan -1 xz, s1, 1, 1d ƒsx, y, zd = x 2 + y 2 - 2z 2 + z ln x, s1, 1, 1d §f gsx, yd = x 2 2 - y 2 2 , A22, 1B gsx, yd = y - x 2 , s -1, 0d 4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 1013 Page 2 14.5 Directional Derivatives and Gradient Vectors 1013 EXERCISES 14.5 Calculating Gradients at Points In Exercises 1–4, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. 1. 2. ƒsx, yd = ln sx 2 + y 2 d, s1, 1d ƒsx, yd = y - x, s2, 1d 3. 4. In Exercises 5–8, find at the given point. 5. 6. ƒsx, y, zd = 2z 3 - 3sx 2 + y 2 dz + tan -1 xz, s1, 1, 1d ƒsx, y, zd = x 2 + y 2 - 2z 2 + z ln x, s1, 1, 1d §f gsx, yd = x 2 2 - y 2 2 , A22, 1B gsx, yd = y - x 2 , s -1, 0d 4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 1013 7. 8. Finding Directional Derivatives In Exercises 9–16, find the derivative of the function at in the direction of A. 9. 10. 11. 12. 13. 14. 15. 16. Directions of Most Rapid Increase and Decrease In Exercises 17–22, find the directions in which the functions increase and decrease most rapidly at Then find the derivatives of the func- tions in these directions. 17. 18. 19. 20. 21. 22. Tangent Lines to Curves In Exercises 23–26, sketch the curve together with and the tangent line at the given point. Then write an equation for the tangent line. 23. 24. 25. 26. Theory and Examples 27. Zero directional derivative In what direction is the derivative of at P(3, 2) equal to zero? 28. Zero directional derivative In what directions is the derivative of at P(1, 1) equal to zero? 29. Is there a direction u in which the rate of change of at P(1, 2) equals 14? Give reasons for your answer. x 2 - 3xy + 4y 2 ƒsx, yd = ƒsx, yd = sx 2 - y 2 d>sx 2 + y 2 d ƒsx, yd = xy + y 2 x 2 - xy + y 2 = 7, s -1, 2d xy =-4, s2, -2d x 2 - y = 1, A22, 1B x 2 + y 2 = 4, A22, 22B §f ƒsx, yd = c hsx, y, zd = ln sx 2 + y 2 - 1d + y + 6z, P 0 s1, 1, 0d ƒsx, y, zd = ln xy + ln yz + ln xz, P 0 s1, 1, 1d gsx, y, zd = xe y + z 2 , P 0 s1, ln 2, 1>2d ƒsx, y, zd = sx>yd - yz, P 0 s4, 1, 1d ƒsx, yd = x 2 y + e xy sin y, P 0 s1, 0d ƒsx, yd = x 2 + xy + y 2 , P 0 s -1, 1d P 0 . A = i + 2j + 2k hsx, y, zd = cos xy + e yz + ln zx, P 0 s1, 0, 1>2d, gsx, y, zd = 3e x cos yz, P 0 s0, 0, 0d, A = 2i + j - 2k ƒsx, y, zd = x 2 + 2y 2 - 3z 2 , P 0 s1, 1, 1d, A = i + j + k ƒsx, y, zd = xy + yz + zx, P 0 s1, -1, 2d, A = 3i + 6j - 2k A = 3i - 2j hsx, yd = tan -1 sy>xd +23 sin -1 sxy>2d, P 0 s1, 1d, A = 12i + 5j gsx, yd = x - s y 2 >xd +23 sec -1 s2xyd, P 0 s1, 1d, ƒsx, yd = 2x 2 + y 2 , P 0 s -1, 1d, A = 3i - 4j ƒsx, yd = 2xy - 3y 2 , P 0 s5, 5d, A = 4i + 3j P 0 ƒsx, y, zd = e x + y cos z + s y + 1d sin -1 x, s0, 0, p>6d ƒsx, y, zd = sx 2 + y 2 + z 2 d -1>2 + ln sxyzd, s -1, 2, -2d 30. Changing temperature along a circle Is there a direction u in which the rate of change of the temperature function (temperature in degrees Celsius, distance in feet) at is Give reasons for your answer. 31. The derivative of ƒ(x, y) at in the direction of is and in the direction of is What is the derivative of ƒ in the direction of Give reasons for your answer. 32. The derivative of ƒ(x, y, z) at a point P is greatest in the direction of In this direction, the value of the derivative is a. What is at P ? Give reasons for your answer. b. What is the derivative of ƒ at P in the direction of 33. Directional derivatives and scalar components How is the derivative of a differentiable function ƒ(x, y, z) at a point in the direction of a unit vector u related to the scalar component of in the direction of u? Give reasons for your answer. 34. Directional derivatives and partial derivatives Assuming that the necessary derivatives of ƒ(x, y, z) are defined, how are and related to and Give reasons for your answer. 35. Lines in the xy-plane Show that is an equation for the line in the xy-plane through the point normal to the vector 36. The algebra rules for gradients Given a constant k and the gradients and use the scalar equations and so on, to establish the following rules. a. b. c. d. e. §a ƒ g b = g§ƒ - ƒ§g g 2 §sƒgd = ƒ§g + g§ƒ §sƒ - gd =§ƒ -§g §sƒ + gd =§ƒ +§g §skƒd = k§ƒ 0 0x sƒgd = ƒ 0g 0x + g 0ƒ 0x , 0 0x a ƒ g b = g 0ƒ 0x - ƒ 0g 0x g 2 , 0 0x skƒd = k 0ƒ 0x , 0 0x sƒ ; gd = 0ƒ 0x ; 0g 0x , §g = 0g 0x i + 0g 0y j + 0g 0z k, §ƒ = 0ƒ 0x i + 0ƒ 0y j + 0ƒ 0z k N = Ai + Bj. sx 0 , y 0 d Asx - x 0 d + Bsy - y 0 d = 0 ƒ z ? ƒ x , ƒ y , D k ƒ D j ƒ, D i ƒ, s§ƒd P 0 P 0 i + j? §ƒ 213. v = i + j - k. -i - 2j? -3. -2j 212 i + j P 0 s1, 2d -3°C>ft ? Ps1, -1, 1d 2xy - yz Tsx, y, zd = 1014 Chapter 14: Partial Derivatives 4100 AWL/Thomas_ch14p965-1066 8/25/04 2:53 PM Page 1014Read More

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