Chapter 12 - Functions of Several Veriables - 12.7 Tangent Planes and Linear Approximation - 12.7 Exercises - Page 935: 10

$$\eqalign{ & 2x + 4z - 6 = 0 \cr & - 2x + 4z - 6 = 0 \cr} $$

$$\eqalign{ & {x^2} + {y^3} + {z^4} = 2;\,\,\,\left( {1,0,1} \right){\text{ and }}\left( { - 1,0,1} \right) \cr & {\text{Let }}F\left( {x,y,z} \right) = {x^2} + {y^3} + {z^4} - 2 \cr & {\text{Calculate the partial derivatives }}{F_x}\left( {x,y,z} \right){\text{, }}{F_y}\left( {x,y,z} \right){\text{ }} \cr & {\text{and }}{F_z}\left( {x,y,z} \right){\text{ }} \cr & {F_x}\left( {x,y,z} \right) = 2x \cr & {F_y}\left( {x,y,z} \right) = 3{y^2} \cr & {F_z}\left( {x,y,z} \right) = 4{z^3} \cr & {\text{Evaluate at the point }}\left( {1,0,1} \right){\text{ and }}\left( { - 1,0,1} \right) \cr & {\text{at }}{F_x}\left( {x,y,z} \right),\,{F_y}\left( {x,y,z} \right){\text{ and }}{F_z}\left( {x,y,z} \right) \cr & {F_x}\left( {1,1,1} \right) = 2 \cr & {F_y}\left( {1,1,1} \right) = 0 \cr & {F_z}\left( {1,1,1} \right) = 4 \cr & and \cr & {F_x}\left( { - 1,0,1} \right) = - 2 \cr & {F_y}\left( { - 1,0,1} \right) = 0 \cr & {F_z}\left( { - 1,0,1} \right) = 4 \cr & \cr & {\text{An equation of the plane tangent to the surface is}} \cr & {F_x}\left( {a,b,c} \right)\left( {x - a} \right) + {F_y}\left( {a,b,c} \right)\left( {y - b} \right) + {F_z}\left( {a,b,c} \right)\left( {z - c} \right) = 0 \cr & {\text{For the point }}\left( {1,0,1} \right) \cr & 2\left( {x - 1} \right) + 0\left( {y - 0} \right) + 4\left( {z - 1} \right) = 0 \cr & 2x - 2 + 4z - 4 = 0 \cr & 2x + 4z - 6 = 0 \cr & {\text{For the point }}\left( { - 1,0,1} \right) \cr & - 2\left( {x + 1} \right) + 0\left( {y - 0} \right) + 4\left( {z - 1} \right) = 0 \cr & - 2x - 2 + 4z - 4 = 0 \cr & - 2x + 4z - 6 = 0 \cr} $$