   Chapter 14.3, Problem 73E

Chapter
Section
Textbook Problem

Use the table of values of f(x, y) to estimate the values of fx(3, 2), fx(3, 2.2),and fxy(3, 2). To determine

To estimate: The values of fx(3,2) , fx(3,2.2) and fxy(3,2) by using the table of values of f(x,y) .

Explanation

Definition used:

If f is a function of two variables, its partial derivatives fx and fy are defined as a limit by,

fx(x,y)=limh0f(x+h,y)f(x,y)hfy(x,y)=limh0f(x,y+h)f(x,y)h

Calculation:

By definition, fx(3,2)=limh0f(3+h,2)f(3,2)h .

Assume h=0.5 and use the given Table to obtain the value of fx(3,2) .

fx(3,2)f(3+0.5,2)f(3,2)0.5f(3.5,2)f(3,2)0.5=22.417.50.5=4.90.5=9.8

Then, assume h=0.5 and use the given Table to obtain the values of fx(3,2) .

fx(3,2)f(30.5,2)f(3,2)0.5f(2.5,2)f(3,2)0.5=10.217.50.5=7.30.5=14.6

Obtain the average of above two values of fx(3,2) ,

fx(3,2)=9.8+14.62=24.42=12.2

Thus, the value of fx(3,2)=12.2 .

The partial derivative fxy(3,2) is defined as,

fxy(3,2)=y[fx(3,2)]=limh0fx(x,y+h)fx(x,y)h=limh0fx(3,2+h)fx(3,2)h

If h=0.2 , then fxy(3,2)=limh0fx(3,2.2)fx(3,2)h . (1)

If h=0.2 , then fxy(3,2)=limh0fx(3,1.8)fx(3,2)h . (2)

By definition, fx(3,2.2)=limh0f(3+h,2.2)f(3,2.2)h .

Assume h=0.5 and use the given Table to obtain the value of fx(3,2.2) .

fx(3,2.2)f(3+0.5,2.2)f(3,2

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