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A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts (a) through (e). f(x,y) = 16-2x²-3y², P(-3,2) a. Find the gradient and evaluate it at P. The gradient at P is (12-12) b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? (Type any angles in radians between 0 and 2л. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(0) =

A function f and a point P are given. Let correspond to the direction of the directional derivative. Complete parts (a) through (e). f(x,y) = 16-2x²-3y², P(-3,2) a. Find the gradient and evaluate it at P. The gradient at P is (12-12) b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? (Type any angles in radians between 0 and 2л. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(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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A function f and a point P are given. Let 0 correspond to the direction of the directional derivative. Complete parts (a) through (e). f(x,y) = 16-2x²-3y², P(-3,2) a. Find the gradient and evaluate it at P. The gradient at P is (12-12). b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? (Type any angles in radians between 0 and 2. Type an exact answer, using it as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(0) = d. Find the value of 0 that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? (Type any angles in radians between 0 and 2. Type an exact answer, using as needed.) What is the maximum value? g(0) = (Type an exact answer, using radicals as needed.) e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
Transcribed Image Text:A function f and a point P are given. Let 0 correspond to the direction of the directional derivative. Complete parts (a) through (e). f(x,y) = 16-2x²-3y², P(-3,2) a. Find the gradient and evaluate it at P. The gradient at P is (12-12). b. Find the angles 0 (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? (Type any angles in radians between 0 and 2. Type an exact answer, using it as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and 21. Type an exact answer, using as needed. Use a comma to separate answers as needed.) c. Write the directional derivative at P as a function of 0; call this function g(0). g(0) = d. Find the value of 0 that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? (Type any angles in radians between 0 and 2. Type an exact answer, using as needed.) What is the maximum value? g(0) = (Type an exact answer, using radicals as needed.) e. Verify that the value of 0 that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. Are the values from part d consistent with the values from parts a and b?
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