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Chapter 12: Q. 16 (page 953)

What does it mean for a function of three variables, f(x,y,z), to be differentiable at a point (a,b,c) ?

Short Answer

Expert verified

The function of three variables to be differentiable at the point is(螖x,螖y,螖z)鈫�(0,0,0)

Step by step solution

01

Introduction

Let f(x,y,z) be a three-variable function defined on an open set including the point (a,b,c), and let 鈭�f(x,y,z)=f(a+鈭�x,b+鈭�y,c+鈭�z)-f(a,b,c) be a function of three variables defined on an If the partial derivatives fx(a,b,c),fy(a,b,c), and fz(a,b,c) exist, the function f is said to be differentiable at (a,b,c).

02

Explanation

鈭�f(x,y,z)=fx(a,b,c)螖x+fy(a,b,c)螖y+fz(a,b,c)螖z+系1螖x+系2螖y+系3螖z Where 系1,系2 and 系3 are a function of 螖x,螖y and 螖z, and all goes to zero as (螖x,螖y,螖z)鈫�(0,0,0).

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Most popular questions from this chapter

Letf(x)beadifferentiablefunctionofx,g(y)beadifferentiablefunctionofy,andh(x,y)=f(x)g(y).Provethat鈭�2h鈭�x鈭�y=鈭�2h鈭�y鈭�x.

Letz=f(x,y)g(x,y).Showthat鈭�z鈭�x=fx(x,y)g(x,y)鈭�f(x,y)gx(x,y)(g(x,y))2and鈭�z鈭�y=fy(x,y)g(x,y)鈭�f(x,y)gy(x,y)(g(x,y))2

Construct examples of the thing(s) described in

the following.

Try to find examples that are different than

any in the reading.

(a) A function z = f(x, y) for which 鈭噁(0, 0) = 0 but f is

not differentiable at (0, 0).

(b) A function z = f(x, y) for which 鈭噁(0, 0) = 0 for every

point in R2.

(c) A function z = f(x, y) and a unit vector u such that

Du f(0, 0) = 鈭噁(0, 0) 路 u.

In Exercises 37-42, use the partial derivatives of role="math" localid="1650186853142" gx,y=tan-1xy2and the point role="math" localid="1650186870407" 1,0specified to

afind the equation of the line tangent to the surface defined by the function in the xdirection,

bfind the equation of the line tangent to the surface defined by the function in the ydirection, and

cfind the equation of the plane containing the lines you found in parts aandb.

Evaluate the following limits, or explain why the limit does not exist.

lim(x,y)鈫�(4,蟺)tan-1yx
See all solutions

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