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\(f(x,y)=-x^2y+xy^2+xy\)
\(\nabla f=<-2xy+y^2+y, -x^2+2xy+x> @\ P=(2,1)\)
\(\nabla f(2,1)=<-2,2>\) a). \(D_\vec{u}F=\nabla f \cdot \vec{u}\) The unit vector of \(\vec{v}=<\frac{3}{5}, \frac{4}{5}>\) \(D_\vec{u}F=<-2,2>\cdot<\frac{3}{5}, \frac{4}{5}>=\frac{2}{5}\)

2. \(\vec{PQ}=<1-2,-1-1>=<-1,-2>\)
   The unit vector of \(\vec{PQ}=<\frac{-1}{\sqrt 5}, \frac{-2}{\sqrt5}>\) \(D_\vec{u}F=<-2,2>\cdot<\frac{-1}{\sqrt 5}, \frac{-2}{\sqrt5}>=-\frac{2 \sqrt5}{5}\)

Although I have a master degree in mathematics and teach college math courses since then, I am still benefit from this class. I have learned how to use LaTex which is extremely useful for my own classes and it will definitely help me with my future Data Science classes. It also helped me review the topics of Linear Algebra which I am rusty for.