1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find two angles where the directional derivative is 1 at p0

  1. Feb 18, 2015 #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].


    f(x,y) = sec(pi/14)*sqrt(x^2 + y^2)
    p0 = (6,6)


    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))


    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.
  2. jcsd
  3. Feb 18, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper

    9*pi/28 is corrrect. Things start getting sloppy in the trig part. sqrt(2)(cos(t)+sin(t))=cos(t+pi/4). The other angle for atan(t)=1 doesn't work. It gives negative signs on the sin and cos. And watch out when you use inverse cos. There's more than one angle on (-pi,pi) that has the same cos.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Find two angles where the directional derivative is 1 at p0