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47-50. Level curves Consider the paraboloid f(x, y) = 16 %3D - - 4 16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 47. f(x, y) = 0; P(0, 16) %3D

47-50. Level curves Consider the paraboloid f(x, y) = 16 %3D - - 4 16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 47. f(x, y) = 0; P(0, 16) %3D

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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37. J(, y) 38. f(z, y)- 8++3y; P(-3,-1) 39. f(r, y)-V2+2+y²; P(V3, 1) 40. f(r, y)-V12--y; P(-1,-1/ V3) 41. f(z, y)-e-; P(-1, 0) 42. f(z, y) = In (1+222+3y); P(3/4,-v3) 43-46. Directions of change Consider the following functions f and points P. Sketch the xy-plane showing P and the level curve through P. Indicate (as in Figure 15.52D the directions of maximum increase, maximum decrease, and no change for f. 43. f(a, y) 8+4x2+2y%; P(2,-4) 44. f(x, y) = -4+ 6x2 + 3y; P(-1,-2) 45. Tf(x, y) = 46. T f(x, y) %3D a²+ ay+y +7; P(-3, 3) = tan (2x + 2y); P(T/16, 7/16) 47-50. Level curves Consider the paraboloid f(x, y) : = 16 – 4 16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 47. f(x, y) = 0; P(0, 16) 48. f(x, y) = 0; P(8, 0) 49. f(x, y) = 12; P(4, 0) 50. f(x, y) = 12; P(2/3, 4) 51-54. Level curves Consider the upper half of the ellipsoid f(z, y) = V1- and the point P on the given level curve of f. 16 y? 4 Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 51. f(x, y) 1 52. f(x, y) V2 53. f(x, y): ; P(0, v8) %3D ; P(7 0)
Transcribed Image Text:37. J(, y) 38. f(z, y)- 8++3y; P(-3,-1) 39. f(r, y)-V2+2+y²; P(V3, 1) 40. f(r, y)-V12--y; P(-1,-1/ V3) 41. f(z, y)-e-; P(-1, 0) 42. f(z, y) = In (1+222+3y); P(3/4,-v3) 43-46. Directions of change Consider the following functions f and points P. Sketch the xy-plane showing P and the level curve through P. Indicate (as in Figure 15.52D the directions of maximum increase, maximum decrease, and no change for f. 43. f(a, y) 8+4x2+2y%; P(2,-4) 44. f(x, y) = -4+ 6x2 + 3y; P(-1,-2) 45. Tf(x, y) = 46. T f(x, y) %3D a²+ ay+y +7; P(-3, 3) = tan (2x + 2y); P(T/16, 7/16) 47-50. Level curves Consider the paraboloid f(x, y) : = 16 – 4 16 and the point P on the given level curve of f. Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 47. f(x, y) = 0; P(0, 16) 48. f(x, y) = 0; P(8, 0) 49. f(x, y) = 12; P(4, 0) 50. f(x, y) = 12; P(2/3, 4) 51-54. Level curves Consider the upper half of the ellipsoid f(z, y) = V1- and the point P on the given level curve of f. 16 y? 4 Compute the slope of the line tangent to the level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 51. f(x, y) 1 52. f(x, y) V2 53. f(x, y): ; P(0, v8) %3D ; P(7 0)
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