Chapter 14.3, Problem 73E

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

### Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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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