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Find an equation of the plane tangent to the following surface at the given point:

                            7xy +  yz  +  4xz   –  48  =  0;  ( 2,  2,  2 )

The aim of this question is to understand the partial derivatives of a surface and their significance in terms of finding the tangent planes.

Once we have partial derivative equations, we simply put the values in following equation to obtain the equation of the tangent plane:

\[ ( \ x \ – \ x_1 \ ) \dfrac{ \partial }{ \partial x } f(x_1,y_1,z_1) \ + \ ( \ y \ – \ y_1 \ ) \dfrac{ \partial }{ \partial y } f(x_1,y_1,z_1) \ + \ ( \ z \ – \ z_1 \ ) \dfrac{ \partial }{ \partial z } f(x_1,y_1,z_1) \ = 0\]

Where, $( \ x_1, \ y_1, \ z_1 \ )$ is the point where tangent equation is to be calculated.

Expert Answer

Step (1) – Calculating the partial derivative equations:

\[ \dfrac{ \partial }{ \partial x } f(x,y,z) = \dfrac{ \partial }{ \partial x } ( 7xy \ + \ yz \ + \ 4xz  ) = 7y \ + \ 4z \]

\[ \dfrac{ \partial }{ \partial y } f(x,y,z) = \dfrac{ \partial }{ \partial y } ( 7xy \ + \ yz \ + \ 4xz  ) = 7y \ + \ y \]

\[ \dfrac{ \partial }{ \partial z } f(x,y,z) = \dfrac{ \partial }{ \partial z } ( 7xy \ + \ yz \ + \ 4xz  ) = y \ + \ 4x \]

Step (2) – Evaluating the partial derivatives at at $( \ 2, \ 2, \ 2 \ )$:

\[ \dfrac{ \partial }{ \partial x }  f(2,2,2) \ = \ 7(2) \ + \ 4(2) \ = \ 22 \]

\[ \dfrac{ \partial }{ \partial y }  f(2,2,2) \ = \ 7(2) \ + \ (2) \  = \ 16 \]

\[ \dfrac{ \partial }{ \partial z }  f(2,2,2) \ = \ (2) \ + \ 4(2) \ = \ 10 \]

Step (3) – Deriving the equation of tangent plane:

\[ ( \ x \ – \ x_1 \ ) \dfrac{ \partial }{ \partial x } f(x_1,y_1,z_1) \ + \ ( \ y \ – \ y_1 \ ) \dfrac{ \partial }{ \partial y } f(x_1,y_1,z_1) \ + \ ( \ z \ – \ z_1 \ ) \dfrac{ \partial }{ \partial z } f(x_1,y_1,z_1)  = 0\]

\[  \Rightarrow ( \ x \ – \ 2 \ ) \dfrac{ \partial }{ \partial x } f(2,2,2) \ + \ ( \ y \ – \ 2 \ ) \dfrac{ \partial }{ \partial y } f(2,2,2) \ + \ ( \ z \ – \ 2 \ ) \dfrac{ \partial }{ \partial z } f(2,2,2) = 0\]

\[  \Rightarrow ( \ x \ – \ 2 \ ) ( 22 ) \ + \ ( \ y \ – \ 2 \ ) ( 16 ) \ + \ ( \ z \ – \ 2 \ ) ( 10 ) = 0\]

\[  \Rightarrow \ 22x \ – \ 44 \ + \ 16y \ – \ 32 \ + \ 10z \ – \ 20 \ = 0 \]

\[  \Rightarrow \ 22x \ + \ 16y \ + \ 10z \ – \ 96 \ = 0 \]

Which is the equation of the tangent.

Numerical Result

\[  \ 22x \ + \ 16y \ + \ 10z \ – \ 96 \ = 0 \]

Example

Find an equation of the plane tangent to the following surface at the given point:

\[ \boldsymbol{ x \ + \ y \ = \ 0; \ ( \ 1, \ 1, \ 1 \ ) } \]

Calculating the partial derivatives:

\[ \dfrac{ \partial }{ \partial x } (x+y) = y = 1 @ ( \ 1, \ 1, \ 1 \ ) \]

\[ \dfrac{ \partial }{ \partial y } (x+y) = x = 1 @ ( \ 1, \ 1, \ 1 \ ) \]

Equation of tangent is:

\[ 1(x-1) + 1(y-1) = 0 \]

\[ \Rightarrow x-1+y-1 = 0 \]

\[ \Rightarrow x+y-2 = 0 \]

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