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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A metal plate is situated in the xy-plane and occupies the rectangle 0 ≤ x 10, 0 ≤ y ≤ 8, where x and y are measured in meters. The temperature at the point (x, y) in the plate is T(x, y), where T is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table.

(a) Estimate the values of the partial derivatives Tx(6, 4) and Ty(6,4). What arc the units?

(b) Estimate the value of DuT(6, 4), where u = ( i + j ) / 2 .

(c) Estimate the value of Txy(6,4).

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(a)

To determine

To estimate: The value of the partial derivatives Tx(x,y)andTy(x,y) at (6,4) and its unit.

Explanation

Given:

The function is, T(x,y) .

Where T is temperature and measured in degree Celsius.

xandy are measured in meters.

Calculation:

Find the value of Tx(6,4) as follows,

Tx(6,4)=limh0T(6+h,4)T(6,4)h

Let h=2 and find the value Tx(6,4) ,

Tx(6,4)=limh0T(6+2,4)T(6,4)2=limh0T(8,4)T(6,4)2=86802=3

Let h=2 and find the value Tx(6,4) ,

Tx(6,4)=limh0T(62,4)T(6,4)2=limh0T(4,4)T(6,4)2=72802=4

Compute the average value of Tx(6,4)=3 and Tx(6,4)=4 .

Tx(6,4)=3+42=72=3.5

Thus, the value of Tx(6,4)=3

(b)

To determine

To estimate: The directional derivative of T(x,y) at the point (6,4) in the direction unit vector u=(i+j)2 and interpret it.

(c)

To determine

To estimate: The value of Txy(x,y) at the point (6,4) .

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Chapter 14 Solutions

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