Chapter 13 - Functions of Several Variables - Review Exercises - Page 961: 43

$$\eqalign{ & \nabla f\left( {x,y} \right) = 2xy{\bf{i}} + {x^2}{\bf{j}} \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = - 50 \cr} $$

$$\eqalign{ & f\left( {x,y} \right) = {x^2}y,{\text{ }}P\left( { - 5,5} \right),{\text{ }}{\bf{v}} = 3{\bf{i}} - 4{\bf{j}} \cr & {\text{Calculate }}\left| {\bf{v}} \right| \cr & \left| {\bf{v}} \right| = \left| {3{\bf{i}} - 4{\bf{j}}} \right| = \sqrt {9 + 16} = 5 \cr & {\bf{v}}{\text{ is not a unit vector, the unit vector in the direction of }}{\bf{v}}{\text{ is:}} \cr & {\bf{u}} = \frac{{\bf{v}}}{{\left| {\bf{v}} \right|}} = \frac{{3{\bf{i}} - 4{\bf{j}}}}{5} = \frac{3}{5}{\bf{i}} - \frac{4}{5}{\bf{j}} \cr & {\text{Calculate }}\nabla f\left( {x,y} \right) \cr & \nabla f\left( {x,y} \right) = {f_x}\left( {x,y} \right){\bf{i}} + {f_y}\left( {x,y} \right){\bf{j}} \cr & {f_x}\left( {x,y} \right) = 2xy \cr & {f_y}\left( {x,y} \right) = {x^2} \cr & \nabla f\left( {x,y} \right) = 2xy{\bf{i}} + {x^2}{\bf{j}} \cr & {\text{Evaluate }}\nabla f\left( { - 5,5} \right) \cr & \nabla f\left( { - 5,5} \right) = 2\left( { - 5} \right)\left( 5 \right){\bf{i}} + {\left( { - 5} \right)^2}{\bf{j}} \cr & \nabla f\left( { - 5,5} \right) = - 50{\bf{i}} + 25{\bf{j}} \cr & {\text{The directional derivative at }}\left( { - 5,5} \right){\text{ in the direction of }}{\bf{u}}{\text{ is}} \cr & {D_{\bf{u}}}f\left( {x,y} \right) = \nabla f\left( {x,y} \right) \cdot {\bf{u}} \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = \nabla f\left( { - 5,5} \right) \cdot \left( {\frac{3}{5}{\bf{i}} - \frac{4}{5}{\bf{j}}} \right) \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = \left( { - 50{\bf{i}} + 25{\bf{j}}} \right) \cdot \left( {\frac{3}{5}{\bf{i}} - \frac{4}{5}{\bf{j}}} \right) \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = - 50\left( {\frac{3}{5}} \right) + 25\left( { - \frac{4}{5}} \right) \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = - 30 - 20 \cr & {D_{\bf{u}}}f\left( { - 5,5} \right) = - 50 \cr} $$