I have seen examples on how to solve this but each example had it for the value 1, in which case they made use of the fact that the components of the unit vector equal 1.
For my example, it is find the directions in which the directional derivative of $f = x^2 + xy^3$ at the point (2,1) has the value 2.
I know that this is asking us to solve:
$2 = \nabla f \bullet u$
I also understand that $\nabla f = <2x + y^3, 3xy^2>$
Therefore, $\nabla f$ at the point $(2,1)$ equals, $<5,6>$
We can put these values into the original equation now:
$2 = <5,6> \bullet <cos\theta, sin\theta>$
$2 = 5cos\theta + 6sin\theta$
This is where I get confused. How do I solve for theta? Is there a cleaner way than what I started doing?
Best Answer
No, $u$ is a unit vector i.e let $u = (u_1,u_2)$. Now solve the system $5u_1+6u_2 = 2$ and $u_1^2 + u_2^2 = 1$.