Consider the function Question 1: Answer the following questions: 3rz 2² + y² + 2² Compute the gradient of f at any point (r, y, z). Use the gradient to find the unit vector pointing in the direction of steepest ascent at the point P (1.0,-2). • Use the gradient to find the unit vectors pointing along a level curve at the point P(1,0,-2).

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 60E
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4 The gradient
Consider the function
f(x, y, z) =
Question 1: Answer the following questions:
3rz
2² + y² + 2²
Compute the gradient of f at any point (r, y, z).
. Use the gradient to find the unit vector pointing in the direction of steepest ascent at the point P
(1,0,-2).
Use the gradient to find the unit vectors pointing along a level curve at the point P(1,0,-2).
• What is the directional derivative of f in the direction of steepest descent, at the point P (1,0,-2)?
Question 2: Consider the surface defined by the equation f(x, y, z)=-6/5. Show that P(1,0,-2) lies
on this surface, then find the tangent plane to the surface at the point P.
Question 3: Transform the function f into spherical polar coordinates. What is the gradient when
expressed in this new coordinate system?
Transcribed Image Text:4 The gradient Consider the function f(x, y, z) = Question 1: Answer the following questions: 3rz 2² + y² + 2² Compute the gradient of f at any point (r, y, z). . Use the gradient to find the unit vector pointing in the direction of steepest ascent at the point P (1,0,-2). Use the gradient to find the unit vectors pointing along a level curve at the point P(1,0,-2). • What is the directional derivative of f in the direction of steepest descent, at the point P (1,0,-2)? Question 2: Consider the surface defined by the equation f(x, y, z)=-6/5. Show that P(1,0,-2) lies on this surface, then find the tangent plane to the surface at the point P. Question 3: Transform the function f into spherical polar coordinates. What is the gradient when expressed in this new coordinate system?
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