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Consider the function h(x,y,z) = e10-xy + sin(2nz). (a) (i) Calculate the gradient of h. Hence, determine the directional derivative of h at the point (2,5,1) in the direction 3i – j+k. (ii) (b) Now, consider the level surface h(x,y,z) = 1. (i) Verify that (2,5, 1) is a point on the level surface. Comment about the relation between the gradient of h at the point (2,5, 1) and the level surface. (ii) (iii) Hence, or otherwise, determine the maximum rate of change of h at the point (2,5, 1).

Consider the function h(x,y,z) = e10-xy + sin(2nz). (a) (i) Calculate the gradient of h. Hence, determine the directional derivative of h at the point (2,5,1) in the direction 3i – j+k. (ii) (b) Now, consider the level surface h(x,y,z) = 1. (i) Verify that (2,5, 1) is a point on the level surface. Comment about the relation between the gradient of h at the point (2,5, 1) and the level surface. (ii) (iii) Hence, or otherwise, determine the maximum rate of change of h at the point (2,5, 1).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 3 Consider the function h(x,y,z) = e10-xy + sin(2nz). (a) (i) Calculate the gradient of h. Hence, determine the directional derivative of h at the point (2,5, 1) in the direction 3i – j+k. (ii) (b) Now, consider the level surface h(x, y,z) = 1. (i) Verify that (2,5, 1) is a point on the level surface. (ii) Comment about the relation between the gradient of h at the point (2,5, 1) and the level surface. (iii) Hence, or otherwise, determine the maximum rate of change of h at the point (2,5, 1).
Transcribed Image Text:Question 3 Consider the function h(x,y,z) = e10-xy + sin(2nz). (a) (i) Calculate the gradient of h. Hence, determine the directional derivative of h at the point (2,5, 1) in the direction 3i – j+k. (ii) (b) Now, consider the level surface h(x, y,z) = 1. (i) Verify that (2,5, 1) is a point on the level surface. (ii) Comment about the relation between the gradient of h at the point (2,5, 1) and the level surface. (iii) Hence, or otherwise, determine the maximum rate of change of h at the point (2,5, 1).
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