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eple
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1.
Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi].
2.
f(x,y) = sec(pi/14)*sqrt(x^2 + y^2)
p0 = (6,6)
3.
I use the relation D_u = grad(f) * u, where u is the elementary vector <cos(theta),sin(theta)>.
grad(f) at (6,6) is <sec(pi/14)/sqrt(2), sec(pi/14)/sqrt(2)>
Using these we have D_u = 1 = sec(pi/14)/sqrt(2)*(cos(theta) + sin(theta))
Rearranging:
cos(theta) + sin(theta) = sqrt(2) * cos(pi/14)
Isolating theta on LHS by using a relevant angle formula:
sqrt(2)*cos(theta - gamma) = sqrt(2) * cos(pi/14), where gamma = atan(1). Here, atan(1) can be pi/4 or -3*pi/4.
using cos^-1 on L- and RHS.
theta = pi/14 + gamma
Giving the answers:
theta1 = pi/14 + pi/4 = 9*pi/28
theta2 = pi/14 - 3*pi/4 = -19*pi/28
Somehow these angles are incorrect, but I am unable to locate my error in calculating them. Any help in guidance in the right direction will be greatly appreciated.
Given a function f(x,y) at (x0,y0). Find the two angles the directional derivative makes with the x-axis, where the directional derivative is 1. The angles lie in (-pi,pi].
2.
f(x,y) = sec(pi/14)*sqrt(x^2 + y^2)
p0 = (6,6)
3.
I use the relation D_u = grad(f) * u, where u is the elementary vector <cos(theta),sin(theta)>.
grad(f) at (6,6) is <sec(pi/14)/sqrt(2), sec(pi/14)/sqrt(2)>
Using these we have D_u = 1 = sec(pi/14)/sqrt(2)*(cos(theta) + sin(theta))
Rearranging:
cos(theta) + sin(theta) = sqrt(2) * cos(pi/14)
Isolating theta on LHS by using a relevant angle formula:
sqrt(2)*cos(theta - gamma) = sqrt(2) * cos(pi/14), where gamma = atan(1). Here, atan(1) can be pi/4 or -3*pi/4.
using cos^-1 on L- and RHS.
theta = pi/14 + gamma
Giving the answers:
theta1 = pi/14 + pi/4 = 9*pi/28
theta2 = pi/14 - 3*pi/4 = -19*pi/28
Somehow these angles are incorrect, but I am unable to locate my error in calculating them. Any help in guidance in the right direction will be greatly appreciated.