Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 936: 20

$$z = y + 2{\text{ and }}z = 5x + 1$$

$$\eqalign{ & z = \sin xy + 2,{\text{ }}\left( {1,0,2} \right)\,\,{\text{and }}\left( {0,5,2} \right) \cr & f\left( {x,y} \right) = \sin xy + 2 \cr & {\text{Calculate the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & {f_x}\left( {x,y} \right) = y\cos xy \cr & {f_y}\left( {x,y} \right) = x\cos xy \cr & {\text{Evaluate at the point }}\left( {1,0,2} \right){\text{ and }}\left( {0,5,2} \right) \cr & {\text{at }}{f_x}\left( {x,y} \right),\,{f_y}\left( {x,y} \right) \cr & {f_x}\left( {1,0} \right) = 0 \cr & {f_y}\left( {1,0} \right) = 1 \cr & and \cr & {f_x}\left( {0,5} \right) = 5 \cr & {f_y}\left( {0,5} \right) = 0 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & z = f\left( {x,y} \right){\text{ at the point }}\left( {a,b,f\left( {a,b} \right)} \right){\text{ is}} \cr & {\text{For the point }}\left( {1,0,2} \right) \cr & z = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_x}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr & z = 0\left( {x - 1} \right) + 1\left( {y - 0} \right) + 2 \cr & z = y + 2 \cr & {\text{For the point }}\left( {0,5,2} \right) \cr & {f_x}\left( {0,5} \right) = 5 \cr & {f_y}\left( {0,5} \right) = 0 \cr & z = 5\left( {x - 0} \right) + 0\left( {y - 5} \right) + 1 \cr & z = 5x + 1 \cr} $$