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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

(a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A(x) of a wafer changes when the side length x changes. Find A′(15) and explain its meaning in this situation.

(b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount x . How can you approximate the resulting change in area A if x is small?

To determine

Part (a):

To find:The rate of change of the area A(x) that means how the area A(x) of a wafer changes when the side length x changes then find A’ (15) and explain its meaning in given situation.

Explanation

1) Concept

Use concept of derivative to find the rate of change of the area A.Here ,  A'x is the rate of change of area of square wafers of silicon with respect to its side length x.

2) Formula:

i) The area of a square with side length x is, Ax=x2

ii) Power Rule:ddxxn=nxn-1

3) Given:

A company makes computer chips from square wafers of silicon and the side length of a wafer very close to 15 mm that means x = 15 mm.

4) Calculation:

To find A'x use power rule of derivative

A'x=ddxx2=2x

Thus, the rate of change of area A’(x) = 2x

To find A'15, substitute x=15 in above step

A

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