Chapter 13 - Functions of Several Variables - 13.7 Exercises - Page 933: 16

$$4\sqrt 2 y + 8z = \sqrt 2 \pi + 4\sqrt 2 $$

$$\eqalign{ & h\left( {x,y} \right) = \cos y,{\text{ }}\left( {5,\frac{\pi }{4},\frac{{\sqrt 2 }}{2}} \right) \cr & {\text{Let }}z = h\left( {x,y} \right) \cr & z = \cos y \cr & z - \cos y = 0 \cr & {\text{Considering }} \cr & F\left( {x,y,z} \right) = z - \cos y \cr & {\text{Calculate the partial derivatives }} \cr & {F_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {z - \cos y} \right] = 0 \cr & {F_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {z - \cos y} \right] = \sin y \cr & {F_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {z - \cos y} \right] = 1 \cr & {\text{At the point }}\left( {5,\frac{\pi }{4},\frac{{\sqrt 2 }}{2}} \right){\text{ the partial derivatives are:}} \cr & {F_x}\left( {5,\frac{\pi }{4},\frac{{\sqrt 2 }}{2}} \right) = 0 \cr & {F_y}\left( {5,\frac{\pi }{4},\frac{{\sqrt 2 }}{2}} \right) = \sin \left( {\frac{\pi }{4}} \right) = \frac{{\sqrt 2 }}{2} \cr & {F_z}\left( {5,\frac{\pi }{4},\frac{{\sqrt 2 }}{2}} \right) = 1 \cr & {\text{An equation of the tangent plane at }}\left( {{x_0},{y_0},{z_0}} \right){\text{ is}} \cr & {F_x}\left( {{x_0},{y_0},{z_0}} \right)\left( {x - {x_0}} \right) + {F_y}\left( {{x_0},{y_0},{z_0}} \right)\left( {y - {y_0}} \right) \cr & + {F_z}\left( {{x_0},{y_0},{z_0}} \right)\left( {z - {z_0}} \right) = 0 \cr & {\text{Substituting}} \cr & 0\left( {x - 5} \right) + \frac{{\sqrt 2 }}{2}\left( {y - \frac{\pi }{4}} \right) + \left( {z - \frac{{\sqrt 2 }}{2}} \right) = 0 \cr & {\text{Simplifying}} \cr & 0\left( {x - 5} \right) + \frac{{\sqrt 2 }}{2}\left( {y - \frac{\pi }{4}} \right) + \left( {z - \frac{{\sqrt 2 }}{2}} \right) = 0 \cr & \frac{{\sqrt 2 }}{2}y - \frac{{\sqrt 2 \pi }}{8} + z - \frac{{\sqrt 2 }}{2} = 0 \cr & {\text{Multiplying both sides by }}8 \cr & 8\left( {\frac{{\sqrt 2 }}{2}y - \frac{{\sqrt 2 \pi }}{8} + z - \frac{{\sqrt 2 }}{2}} \right) = 0 \cr & 4\sqrt 2 y - \sqrt 2 \pi + 8z - 4\sqrt 2 = 0 \cr & 4\sqrt 2 y + 8z = \sqrt 2 \pi + 4\sqrt 2 \cr} $$