Question 3 Consider the function h(x,y, z) = e10-xy + sin(2nz). (а) (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(r v z) = 1

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
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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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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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