Examples with Partial Derivatives                                           

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

 

Example 3:  Find the equation of the tangent plane,  TP,  to the surface   z  =   f(x, y)   where

 

 f(x, y)  =  x2   +  y2  (paraboloid)  and  point  P has the coordinates (0, 1, 1).

 

 

                                  

 

 

Method 1

 

Strategy:  Use the definition of a plane in space.

 

The equation of the tangent plane at the point   ( a,  b, c )   is    ( here  c  =  f(a,b)  )

 

                        fx(a,b) [ x – a ]  +    fy(a,b) [ y – b ]  -  [z  -  f(a, b)]  =   0

 

Here   (a, b, c)  =  (0, 1, 1)    So   a  =  0 ,  b  =  1  and  f(a, b)   =   02   +  12   =   1

 

  fx(x,y)   =  2 x  and   fy(x,y)   =  2 y     and   fx(0, 1)   =  0,     fy(0, 1)   =   2

 

So            0 [x – 0 ]  +   2 [ y – 1 ]  - [ z – 1 ]  =  0

 

Or                 2 y  - z  -  1  =  0

 

                      z   =  2 y  -  1     (result for tangent plane to surface at P)

 

Click here for a second method of solution.

 


Copyright © 2010 Richard C. Coddington

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