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Chapter 12: Q. 12. (page 963)

If a function f(x,y)is differentiable at (a,b), explain how to

use the gradient 鈭�f(a,b)to find the equation of the plane

tangent to the graph of fat (a,b).

Short Answer

Expert verified

The equation of tangent plane to the surface f(x,y)at (a,b)is:

z-f(a,b)=鈭�f(a,b)路鉄�(x,y)-(a,b)鉄�

Step by step solution

01

Given information

Let, f(x,y)is a function of two variables defined on an open set containing the point (a,b)

Let 螖z=f(a+螖x,b+螖y)-f(a,b)

If both fx(a,b)and fy(a,b)exist, the function fis said to be differentiable and

螖z=fx(a,b)螖x+fy(a,b)螖y+蔚1螖x+蔚2螖y鈥�鈥�(1).where 蔚1and蔚2 are functions of 螖xand 螖yand

both are zero when(螖x,螖y)鈫�(0,0)

02

The objective is to find the equation of the plane tangent to the graph of f(x, y) at (a, b) 

Substitute 螖x=x-a,螖y=y-band 螖z=z-f(a,b)in (1)and eliminate 蔚1螖xand 蔚2螖yterms

z-f(a,b)=fx(a,b)(x-a)+fy(a,b)(y-b)鈬�鈭�z=(fx(a,b),fy(a,b))路((x-a),(y-b))=鈭�f(a,b)路((x,y)-(a,b))

Hence, the equation of tangent plane to the surface f(x,y)at (a,b)is:

z-f(a,b)=鈭�f(a,b)路鉄�(x,y)-(a,b)鉄�

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