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) = In (1+2x² + 4y²), P.-√3) a. Find the gradient and evaluate it at P. The gradient at P is. b. Find the angles 0 (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associa with the direction of maximum increase? (Type your answer in radians. Type an exact answer in terms of x. Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? (Type your answer in radians. Type an exact answer in terms of x. Use a comma to separate answers as needed.) What angles are associated with the direction of zero change? (Type your answer in radians. Type an exact answer in terms of x. 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) = (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of 0 between 0 and 2x that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? 0= 8 V
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) = In (1+2x² + 4y²), P.-√3) a. Find the gradient and evaluate it at P. The gradient at P is. b. Find the angles 0 (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associa with the direction of maximum increase? (Type your answer in radians. Type an exact answer in terms of x. Use a comma to separate answers as needed.) What angles are associated with the direction of maximum decrease? (Type your answer in radians. Type an exact answer in terms of x. Use a comma to separate answers as needed.) What angles are associated with the direction of zero change? (Type your answer in radians. Type an exact answer in terms of x. 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) = (Simplify your answer. Type an exact answer, using radicals as needed.) d. Find the value of 0 between 0 and 2x that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)? 0= 8 V
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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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) = In (1 + 2x² + 4y²), P (1.-√3)
a. Find the gradient and evaluate it at P.
The gradient at P is
0.
b. Find the angles 0 (with respect to the positive x-axis) between 0 and 2x associated with the directions of maximum increase, maximum decrease, and zero change. What angles are associate
with the direction of maximum increase?
(Type your answer in radians. Type an exact answer in terms of . Use à comma to separate answers as needed.)
What angles are associated with the direction of maximum decrease?
(Type your answer in radians. Type an exact answer in terms of л. Use a comma to separate answers as needed.)
What angles are associated with the direction of zero change?
(Type your answer in radians. Type an exact answer in terms of x. Use a comma to separate answers as needed.)
c. Write the directional derivative at P as a function of 8; call this function g(0).
g(0) =
(Simplify your answer. Type an exact answer, using radicals as needed.)
d. Find the value of 0 between 0 and 2x that maximizes g(0) and find the maximum value. What value of 0 maximizes g(0)?
0=
8
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