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13. J(L, Y) y", 1(-1, 1) = 8 – 2x2 – y². For the following level curves f(x, y) = C and 50-51. Level curves Consider the paraboloid f(x, points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 50. f(x, y) = 5; (a, b) = (1, 1) 51. f(x, y) = 0; (a, b) = (2, 0)

13. J(L, Y) y", 1(-1, 1) = 8 – 2x2 – y². For the following level curves f(x, y) = C and 50-51. Level curves Consider the paraboloid f(x, points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 50. f(x, y) = 5; (a, b) = (1, 1) 51. f(x, y) = 0; (a, b) = (2, 0)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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42. J(2,9) 2, 1(1, 2); u V2 12 43. g(x, y) = x²y°; P(-1, 1); u = 13' 13 V3 1. 44. f(x, y) = ; Р(), 3); и %3D y2 2 3 45. h(x, y) = V2 + x2 + 2y² ; P(2, 1); u = 4 5' 5 4 46. f(x, y, z) = xe'+y´+z; P(0, 1, –2); u = 1 8 9' 9 9 2 3 47. f(x, y, z) = sin xy+ cOs z; P(1, TT, 0); u = 7 7 7 48-49. Direction of steepest ascent and descent a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a unit vector that points in a direction of no change. 48. f(x, y) = ln (1+ xy); P(2, 3) 49. f(x, y) = V4 – x² – y?; P(-1, 1) 50-51. Level curves Consider the paraboloid f(x, y) = 8 – 2x – y'. For the following level curves f(x, y) = C and points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 50. f(x, y) = 5; (a, b) = (1, 1) 51. f(x, y) = 0; (a, b) = (2, 0) 52. Directions of zero change Find the directions in which the function f(x, y) = 4x² – y has zero change at the point (1, 1, 3). Express the directions in terms of unit vectors. 53. Electric potential due to a charged cylinder An infinitely long charged cylinder of radius R with its axis along the z-axis has an electric potential V = k ln R where r is the distance between a variable point P(x, y) and the axis of the cylinder (r² = x² + y²) and k is a physical constant. The electric field at a point (x, y) in the xy-plane is given by E = -VV, where VV is the two-dimensional gradient. Compute the electriç field at a point (x, y) with
Transcribed Image Text:42. J(2,9) 2, 1(1, 2); u V2 12 43. g(x, y) = x²y°; P(-1, 1); u = 13' 13 V3 1. 44. f(x, y) = ; Р(), 3); и %3D y2 2 3 45. h(x, y) = V2 + x2 + 2y² ; P(2, 1); u = 4 5' 5 4 46. f(x, y, z) = xe'+y´+z; P(0, 1, –2); u = 1 8 9' 9 9 2 3 47. f(x, y, z) = sin xy+ cOs z; P(1, TT, 0); u = 7 7 7 48-49. Direction of steepest ascent and descent a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a unit vector that points in a direction of no change. 48. f(x, y) = ln (1+ xy); P(2, 3) 49. f(x, y) = V4 – x² – y?; P(-1, 1) 50-51. Level curves Consider the paraboloid f(x, y) = 8 – 2x – y'. For the following level curves f(x, y) = C and points (a, b), compute the slope of the line tangent to the level curve at (a, b) and verify that the tangent line is orthogonal to the gradient at that point. 50. f(x, y) = 5; (a, b) = (1, 1) 51. f(x, y) = 0; (a, b) = (2, 0) 52. Directions of zero change Find the directions in which the function f(x, y) = 4x² – y has zero change at the point (1, 1, 3). Express the directions in terms of unit vectors. 53. Electric potential due to a charged cylinder An infinitely long charged cylinder of radius R with its axis along the z-axis has an electric potential V = k ln R where r is the distance between a variable point P(x, y) and the axis of the cylinder (r² = x² + y²) and k is a physical constant. The electric field at a point (x, y) in the xy-plane is given by E = -VV, where VV is the two-dimensional gradient. Compute the electriç field at a point (x, y) with
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