Chapter 14
Limits, Error Control and Continuity
- [Play
Video] 4½ minutes: Algebraic View of
Limits. Example involving sums and quotients.
- [Play
Video] 2½ minutes: Algebraic
Properties of Limits I.
- [Play
Video] 2¼ minutes: Algebraic Properties of
Limits II.
- [Play
Video] 5½ minutes: Limits and Error
Control for Linear Expressions
- [Play
Video] 2¾ minutes: Error Control to N
decimal Places, say 5 or 10.
- [Play
Video] 3¼ minutes: Limits as Error
Control for an unlimited number of decimal places.
Error control for the evaluation of functions y = f(x)
provides a simple context and motivation for continuity and convergence.
Continuity at Point
To explain the idea of continuity of a function y = f(x) at
a point x = a, we ask the following error-control question with b
= f(a): to what number m of places should the decimal
expansions of x and a agree, for the decimal expansion of the
number f(x) to agree with that of b = f(a) to
n-decimal places? That is, given a whole number n, is there an m
such that
| |x-a|
< d = |
1
2
|
· |
1
10m
|
implies |f(x)-f(a)|
< e = |
1
2
|
· |
1
10n
|
(?) |
|
An affirmative answer requires that agreement of x
with a to m decimal places implies the agreement of f(x)
with f(a) to n decimal places. An affirmative answer says
unlimited accuracy and error control is possible at x = a.
The Greek letters d (delta) and e
(epsilon) above are employed here in accordance with tradition of some (not all)
calculus texts. For simplicity, the error control tolerances e
and d in the first instance here and below, may be
restricted to be numbers of the form [1/2] ·10-k
= [1/2] [1/(10k)]. The decimal free discussion of error
control and continuity dispenses with this requirement.
We say a function f(x) is continuous at a point x = a
if and only if unlimited error control is possible there. More formally, we
state the following definition.
Theorem 14.1 [Continuity at a Point] If f(x) is a
real-valued function of a real number x in an interval [c,d],
and a is a number in the interval [c,d] then the function f
is said to be continuous at the number x = a if and only if the
following holds. If for every n, there exist an m such that
| |x-a|
< d = |
1
2 |
· |
1
10m |
implies |f(x)-f(a)|
< e = |
1
2 |
· |
1
10n |
· |
|
Decimal-Free Form
The decimal-free description or definition of continuity at a point x = a
is as follows.
[Continuity at Point] If f(x) is a real-valued function of a
real number x in an interval [c,d], and a is a point
in the interval [c,d] then the function f is said to be
continuous at x = a if and only if the following holds: For every e1
> 0, there exist a d1 > 0 such that
| |x-a|
< d1
implies |f(x)-f(a)|
< e1 |
|
It is easily shown that the decimal-free and decimal-based definitions are
equivalent. The proof of equivalence, better left to a second reading of this
work, follows.
Proof of Equivalence.
To show the decimal-based description implies the decimal-free description
of continuity, observe the following. First given e1
> 0, there is an n > 0 such that e1
> [1/2]·[1/(10n)] = e. The
decimal-based requirement for continuity now is satisfied for some d
= [1/2] ·[1/(10m)]. So the decimal-free version holds with d1
= d = [1/2]·[1/(10m)].
Conversely, the other way that is, to show the latter decimal-free form
implies the decimal-based description of continuity, observe the following.
Given m > 0, let e1 = e
= [1/2] ·[1/(10m)]. Then choose d1
> 0 so that the decimal-free requirement is satisfied. The decimal-based
version is then satisfied if m > 0 is selected so that d1
³ d = [1/2] ·[1/(10m)].
| |
Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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