http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Thu, 17 Jul 2014 15:43:34 -0500http://calc3.askbot.com/question/130/directional-derivative-equal-to-10/So with # 4 on the review sheet. $$\nabla f=\langle y^{3},3xy^{2} \rangle$$ So at $(1,2)$ $\nabla f=\langle1,6 \rangle.$ So to determine if $D_{u}f$ is equal to $10$ we solve for $a$ and $b$ in the equation $$\langle1,6 \rangle \cdot \langle a,b \rangle=10$$ Here's where I feel confused. You end up with $a+6b=10.$ Now $a=4, b=1$ is an obvious solution to this equation. However, does $\textbf{u}$ have to be a unit vector to correctly answer this question? This leads to two other questions. If it does need to be unit vector do we find it by normalizing the vector $\langle4,1 \rangle$? If we do normalize this vector the components of the unit vector we construct will not solve the equation $a+6b=10.$ But I feel that since it will still have the same direction this may not matter. As an alternative method we could solve the system of equations below? $$ a+6b=10$$ $$a^{2}+b^{2}=1$$ I haven't bothered trying to solve that system yet because I'm not sure it's necessary. Thoughts?SpaceManSpiffThu, 17 Jul 2014 15:43:34 -0500http://calc3.askbot.com/question/130/