Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 936: 17

$$z = - 2x - 4y + 4{\text{ and }}z = 4x + 2y + 7$$

$$\eqalign{ & z = 4 - 2{x^2} - {y^2};\,\,\,\left( {2,2, - 8} \right){\text{ and }}\left( { - 1, - 1,1} \right) \cr & f\left( {x,y} \right) = 4 - 2{x^2} - {y^2} \cr & {\text{Calculate the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & {f_x}\left( {x,y} \right) = - 4x \cr & {f_y}\left( {x,y} \right) = - 2y \cr & {\text{Evaluate at the point }}\left( {2,2, - 8} \right){\text{ and }}\left( { - 1, - 1,1} \right) \cr & {\text{at }}{f_x}\left( {x,y} \right),\,{f_y}\left( {x,y} \right) \cr & {f_x}\left( {2,2} \right) = - 8 \cr & {f_y}\left( {2,2} \right) = - 4 \cr & and \cr & {f_x}\left( { - 1, - 1} \right) = 4 \cr & {f_y}\left( { - 1, - 1} \right) = 2 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & z = f\left( {x,y} \right){\text{ at the point }}\left( {a,b,f\left( {a,b} \right)} \right){\text{ is}} \cr & {\text{For the point }}\left( {2,2, - 8} \right) \cr & z = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_x}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr & z = - 2\left( {x - 2} \right) - 4\left( {y - 2} \right) - 8 \cr & z = - 2x + 4 - 4y + 8 - 8 \cr & z = - 2x - 4y + 4 \cr & {\text{For the point }}\left( { - 1, - 1,1} \right) \cr & z = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_x}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr & z = 4\left( {x + 1} \right) + 2\left( {y + 1} \right) + 1 \cr & z = 4x + 4 + 2y + 2 + 1 \cr & z = 4x + 2y + 7 \cr} $$