Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 935: 11

$$\eqalign{ & x + y + z - 8 = 0 \cr & 3x + 4y + z - 12 = 0 \cr} $$

$$\eqalign{ & xy + xz + yz - 12 = 0;\,\,\,\left( {2,2,2} \right){\text{ and }}\left( {2,0,6} \right) \cr & {\text{Let }}F\left( {x,y,z} \right) = xy + xz + yz - 12 \cr & {\text{Calculate the partial derivatives }}{F_x}\left( {x,y,z} \right){\text{, }}{F_y}\left( {x,y,z} \right){\text{ }} \cr & {\text{and }}{F_z}\left( {x,y,z} \right){\text{ }} \cr & {F_x}\left( {x,y,z} \right) = y + z \cr & {F_y}\left( {x,y,z} \right) = x + z \cr & {F_z}\left( {x,y,z} \right) = x + y \cr & {\text{Evaluate at the point }}\left( {2,2,2} \right){\text{ and }}\left( {2,0,6} \right) \cr & {\text{at }}{F_x}\left( {x,y,z} \right),\,{F_y}\left( {x,y,z} \right){\text{ and }}{F_z}\left( {x,y,z} \right) \cr & {F_x}\left( {2,2,2} \right) = 4 \cr & {F_y}\left( {2,2,2} \right) = 4 \cr & {F_z}\left( {2,2,2} \right) = 4 \cr & and \cr & {F_x}\left( {2,0,6} \right) = 6 \cr & {F_y}\left( {2,0,6} \right) = 8 \cr & {F_z}\left( {2,0,6} \right) = 2 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & {F_x}\left( {a,b,c} \right)\left( {x - a} \right) + {F_y}\left( {a,b,c} \right)\left( {y - b} \right) + {F_z}\left( {a,b,c} \right)\left( {z - c} \right) = 0 \cr & {\text{For the point }}\left( {2,2,2} \right) \cr & 4\left( {x - 2} \right) + 4\left( {y - 2} \right) + 4\left( {z - 2} \right) = 0 \cr & 4x - 8 + 4y - 8 + 4z - 8 = 0 \cr & x + y + z - 8 = 0 \cr & {\text{For the point }}\left( {2,0,6} \right) \cr & 6\left( {x - 2} \right) + 8\left( {y - 0} \right) + 2\left( {z - 6} \right) = 0 \cr & 6x - 12 + 8y + 2z - 12 = 0 \cr & 6x + 8y + 2z - 24 = 0 \cr & 3x + 4y + z - 12 = 0 \cr} $$