Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 935: 9

$$\eqalign{ & 2x + y + z - 4 = 0 \cr & 4x + y + z - 7 = 0 \cr} $$

$$\eqalign{ & {x^2} + y + z = 3;\,\,\,\left( {1,1,1} \right){\text{ and }}\left( {2,0, - 1} \right) \cr & {\text{Let }}F\left( {x,y,z} \right) = {x^2} + y + z - 3 \cr & {\text{Calculate the partial derivatives }}{F_x}\left( {x,y,z} \right){\text{, }}{F_y}\left( {x,y,z} \right){\text{ }} \cr & {\text{and }}{F_z}\left( {x,y,z} \right){\text{ }} \cr & {F_x}\left( {x,y,z} \right) = 2x \cr & {F_y}\left( {x,y,z} \right) = 1 \cr & {F_z}\left( {x,y,z} \right) = 1 \cr & {\text{Evaluate at the point }}\left( {1,1,1} \right){\text{ and }}\left( {2,0, - 1} \right) \cr & {\text{at }}{F_x}\left( {x,y,z} \right),\,{F_y}\left( {x,y,z} \right){\text{ and }}{F_z}\left( {x,y,z} \right) \cr & {F_x}\left( {1,1,1} \right) = 2 \cr & {F_y}\left( {1,1,1} \right) = 1 \cr & {F_z}\left( {1,1,1} \right) = 1 \cr & and \cr & {F_x}\left( {2,0, - 1} \right) = 4 \cr & {F_y}\left( {2,0, - 1} \right) = 1 \cr & {F_z}\left( {2,0, - 1} \right) = 1 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & {F_x}\left( {a,b,c} \right)\left( {x - a} \right) + {F_y}\left( {a,b,c} \right)\left( {y - b} \right) + {F_z}\left( {a,b,c} \right)\left( {z - c} \right) = 0 \cr & {\text{For the point }}\left( {1,1,1} \right) \cr & 2\left( {x - 1} \right) + \left( {y - 1} \right) + \left( {z - 1} \right) = 0 \cr & 2x - 2 + y - 1 + z - 1 = 0 \cr & 2x + y + z - 4 = 0 \cr & {\text{For the point }}\left( {2,0, - 1} \right) \cr & 4\left( {x - 2} \right) + \left( {y - 0} \right) + \left( {z + 1} \right) = 0 \cr & 4x - 8 + y + z + 1 = 0 \cr & 4x + y + z - 7 = 0 \cr} $$