4.10. If y = f(x) = x – 6x, find (a) Ay, (b) dy, and (c) Ay – dy. (a) Ay = f(x + Ax) – f(x) = {(x + Ax)³ –6(x + Ax)} – {x³ – 6x} =x + 3x Ax + 3x(Ax)² + (Ax)³ – 6r – 6Ax – x+ 6x = (3x – 6) Ax + 3x(Ax)² + (Ax)³ (b) dy = principal part of Ay = (3x – 6)Ax = (3x - 6)dx, since by definition Ax = dx. Note that f'(x) = 3x² - 6 and dy = (3x – 6)dx, i.e.; dy/dx = 3x? - 6. It must be emphasized that dy and dx are not necessarily small. (c) From (a) and (b), Ay – dy = 3x(Ax)² + (Ax)³ = €Ax, where e = 3xAx + (Ax)².
4.10. If y = f(x) = x – 6x, find (a) Ay, (b) dy, and (c) Ay – dy. (a) Ay = f(x + Ax) – f(x) = {(x + Ax)³ –6(x + Ax)} – {x³ – 6x} =x + 3x Ax + 3x(Ax)² + (Ax)³ – 6r – 6Ax – x+ 6x = (3x – 6) Ax + 3x(Ax)² + (Ax)³ (b) dy = principal part of Ay = (3x – 6)Ax = (3x - 6)dx, since by definition Ax = dx. Note that f'(x) = 3x² - 6 and dy = (3x – 6)dx, i.e.; dy/dx = 3x? - 6. It must be emphasized that dy and dx are not necessarily small. (c) From (a) and (b), Ay – dy = 3x(Ax)² + (Ax)³ = €Ax, where e = 3xAx + (Ax)².
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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4.10) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
Transcribed Image Text:Differentials
4.10.
If y = f(x) = x – 6x, find (a) Ay, (b) dy, and (c) Ay – dy.
(a) Ay = f(x + Ax) – f(x) = {(x + Ax)³ –6(x + Ax)} – {x³ – 6x}
=x' + 3x Ax + 3x(Ax)² + (Ax)³ – 6x – 6Ax – x² + 6x
= (3x – 6) Ax + 3x(Ar)² + (Ax)³
(b) dy = principal part of Ay = (3x² – 6)Ar = (3x² – 6)dx, since by definition Ax = dx.
Note that f'(x) = 3x – 6 and dy = (3x – 6)dx, i.e.; dy/dx = 3x² – 6. It must be emphasized that dy and dx
are not necessarily small.
(c) From (a) and (b), Ay – dy = 3x(Ax)² + (Ax)³ = €Ax, where e = 3xAx + (Ax)'.
Ду — dy
Note that e → 0 as Ar → 0; i.e.,
→0 as Ar → 0. Hence, Ay – dy is an infinitesimal of higher
Ar
order than Ax (see Problem 4.83).
In case Ax is small, dy and Ay are approximately equal.
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