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Calculus 1 Problems & Solutions – Chapter 2 – Section 2.6 |
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2.6
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Review |
1. Infinitesimals |
Recall that the derivative of y = f(x) at any point x is:
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2. The Differentials |
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Fig. 2.1
dy = f '(a) dx. |
At a particular point x = a in dom( f ), the
function dy
= f '(a) dx is called
the differential of f at x = a. Each point
of
dom( f ) generates
the differential of f
at that point. The set of all points of dom( f ) generates the set of all the
differentials of f
at those points. This set of differentials is called the differential of
f.
Formally the differential of f is
defined as the operation that associates or maps dom( f ) to this set of differentials.
Definitions 2.1
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i. The differential of x is a change in x, denoted dx.
ii. Let the
function y
= f(x) be
differentiable at x
= a.
The differential of f at a is the function
defined by dy =
iii. Let the function y = f(x) be
differentiable at all points of dom( f ).The differential of f
is the operation that
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Remarks 2.1
also. This has led us to denote
differentials by the letter “d”,
like dx
or dy
or dt,
the same as the notation for
infinitesimals.
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3. Differentials Vs Infinitesimals |
Let y = f(x). The differential dy of f at an
arbitrary point x
in dom ( f ) is dy = f '(x) dx, where dx is the differential of
x.
So dy/dx = f '(x). That is,
the ratio dy/dx, where dy and dx are
differentials, is the derivative of y with respect to x.
This is in accordance with the Leibniz notation dy/dx of the derivative of y with
respect to x,
where dy
and dx
are
infinitesimals.
When Leibniz introduced the
concept of infinitesimals, he was roundly criticized for inventing such
infinitesimals out of thin
air and attempting to do mathematics with them. It's only during the last 60
years or so that mathematicians have put his
view on a firm mathematical foundation by showing that it's possible to extend
the set of real numbers to include such
infinitesimals.
Remember, there are 2 interpretations for the set of the
quantities dx
and dy:
differentials and infinitesimals. Calculus
with dx
and dy
interpreted as differentials is called standard analysis. Calculus with dx and dy
interpreted as
infinitesimals and the set of real numbers extended to include them is called non-standard
analysis.
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4. dy/dx As A Normal Fraction |
If dx and dy are interpreted as
differentials, the quotient dy/dx is a
normal fraction because dx
and dy
are each a real
number. If dx
and dy
are interpreted as infinitesimals, the quotient dy/dx is a normal fraction because dx and dy are
each a real number in the extended set of real numbers. Whether dx and dy are
interpreted as differentials or as
infinitesimals, the ratio:
where the du's appear to
cancel out, like in normal fractions. We'll also see times and again later on
that the notation
dy/dx indeed
behaves like a normal fraction.
1. Find the differential of y = f(x) = x2 – 3x + 1 at x = 5.
Solution
f(x) = x2 – 3x + 1,
f '(x) = 2x – 3,
f '(5) = 2(5) – 3 = 7,
dy = f '(5) dx = 7 dx.
2. Find the differential of y = (x + 1)2/(x2 – 2)3 at x = 0.
Solution
3. Find the differential of the function y = x/(x + 1).
Solution
4. Find the differential of the function s = etan 3t , Use the formulas (d/dx) ex = ex and (d/dx) tan x = sec2 x.
Solution
5. Let y = f(x) be a
differentiable function. See the figure below. Let a be a point in dom( f ) and dx a
differential of x.
We
claim that this error E(dx) is small
(close to 0) compared with the size of dx if dx itself is sufficiently small.
Prove this claim by proving that:
Solution
= f '(a) – f '(a)
= 0.
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