Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 936: 19

$$z = x + y{\text{ and }}z = x + 1$$

$$\eqalign{ & z = {e^{xy}}{\text{ }}\left( {1,0,1} \right)\,\,{\text{and }}\left( {0,1,1} \right) \cr & f\left( {x,y} \right) = {e^{xy}} \cr & {\text{Calculate the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & {f_x}\left( {x,y} \right) = y{e^{xy}} \cr & {f_y}\left( {x,y} \right) = x{e^{xy}} \cr & {\text{Evaluate at the point }}\left( {1,0,1} \right){\text{ and }}\left( {0,1,1} \right) \cr & {\text{at }}{f_x}\left( {x,y} \right),\,{f_y}\left( {x,y} \right) \cr & {f_x}\left( {1,0} \right) = 1 \cr & {f_y}\left( {1,0} \right) = 1 \cr & and \cr & {f_x}\left( {0,1} \right) = 1 \cr & {f_y}\left( {0,1} \right) = 0 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & z = f\left( {x,y} \right){\text{ at the point }}\left( {a,b,f\left( {a,b} \right)} \right){\text{ is}} \cr & {\text{For the point }}\left( {1,0,1} \right) \cr & z = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_x}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr & z = \left( {x - 1} \right) + \left( {y - 0} \right) + 1 \cr & z = x + y \cr & {\text{For the point }}\left( {0,1,1} \right) \cr & {f_x}\left( {0,1} \right) = 1 \cr & {f_y}\left( {0,1} \right) = 0 \cr & z = 1\left( {x - 0} \right) + 0\left( {y - 1} \right) + 1 \cr & z = x + 1 \cr} $$