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Appendix H
Integration Assertions

First Fundamental Theorem

The following theorem on the limits of Riemann sums is a version of the First Fundamental Theorem of Calculus. It guarantees the existence of area underneath a continuous curve y = f(x). The second fundamental theorem of calculus shows how this area can be computed with the aid of an anti-derivative of y = f(x), that is, by reversing slope calculations. See the chapter Slopes and Areas in this Volume 3

Theorem H.1 [Limit of Riemann Sums] Suppose f(x) is continuous at each point in the interval [a,b] with b ¹ a. Suppose
x0 = a < x1 < x2 < xj < xn < xn+1 = b.
Further suppose xj £ wj £ xj+1. For every whole number k > 0, there exists a d > 0 such that all sums of the form
n
å
j = 1 
f(wj)·(xj+1-xj)

agree to k decimal places whenever  0 < xj+1-xj  < d.

 Each finite sequence
x0 = a < x1 < x2 < xj < xn < xn+1 = b
divides (or partitions) the interval [a,b] into n subintervals [xj,xj+1] of varying width. When wj is in the interval [xj,xj+1], the term f(wj)·(xj+1-xj) represents the area of the rectangle with base given by the interval [xj,xj+1] on the horizontal axis, and its so-called top or off x-axis side (above, below or even on the x-axis) at y = f(wj). When f(wj) > 0 is positive, this area is positive. Otherwise, the area of the rectangle is taken to be negative.

In this approximation or perspective of area between a curve y = f(x) and the horizontal x axis, regions above the axis have positive areas and regions below the horizontal axis have negative areas. See a calculus text for a further discussion of this signed area between a curve y = f(x) and the horizontal x-axis.

This theorem indicates in principle how to define and directly compute the limit L of the Riemann Sums to any number k of decimal places. The limit is written as
L = ó
õ
b


a
 
f(xdx
The right hand side is read the integral from a to b of f(x). The number L gives the signed area between the curve y = f(x) and the horizontal x-axis, from x = a to x = b. If f(x) > 0 then L becomes the area under the curve y = f(x).

Both theorems in this section are consequence of the equicontinuity theorem and some very detailed accounting. (You may wish to skip over the accounting on first reading. And on a first reading of the accounting, that is the proof below, you could further keep in mind the special case where all xj are of the form xj = xm,j = a+[(j)/(2m)](b-a) for some m.) The accounting is somewhat simpler for Lipschitz continuous functions.

Thereom H.2 [Base-Two Riemann Sums] Assume f(x) is continuous at each point in the interval [a,b] with b > a. For 0 £ j £ 2m = n-1, put xm,j = a+[(j)/(2m)](b-a). Then
xm,0 = a < xm,1 < xm,2 < xm,j < xm,n < xm,n+1 = b
Further, the limit

Further the limit
L
=

lim
m-> ¥ 
2n
å
j = 1 
f(xm,j)·(xj+1-xj)
=

lim
m-> ¥ 
2n
å
j = 1 
f æ
ç
è
a+ j
2m
(b-a) ö
÷
ø
· 1
2m
exists (i.e is finite). 

Some Calculus books will use the limit
L =
lim
m-> ¥ 
2n
å
j = 1 
f æ
ç
è
a+ j
2m
(b-a) ö
÷
ø
· 1
2m
to define the (signed) area òabf(x) dx between a curve y = f(x) and the horizontal axis from x = a to x = b. Here xj = a+[(j)/(2m)](b-a) for 0 £j£n = 2m and xj+1-xj = [1/(2m)]. On a first reading of the following proof, you could assume the sequences appearing in it have this form for two different whole numbers m, say m1 and m2.

Proof of First Fundamental Theorem.

Let e = [1/4][1/(10k)][1/(|b-a|)]. By the equicontinuity theorem, there exists a d > 0 such that for every pair of numbers u and v in the interval [a,b] such that |u-v| £ d implies |f(u)-f(v)| £ e = (1/4)(10**k |b-a|)**(-1). (We are mixing here the decimal-free and decimal-perspective of continuity - a small crime. Exercise: Formulate the derivative decimal-free version of this theorem.)

Let a denote a positive whole number. Suppose the sequence
r0 = a < r1 < r2 < ¼ < ra < ra+1 = b
of a numbers satisfies 0 <rj+1-rj < d. Assume rj <r*j < rj+1. Further suppose the finite sequence of  a numbers satisfies 0 < rq+1-rq < d.

for 0 < q <     Assume rq < r*a < rq+1 for 0 < q <   a. Further suppose the finite sequence

t0 = a < t1 < t2 < ¼ < tb < tb+1 = b
satisfies 0 <tj+1-tj < d as well. Assume tj < t*j < tj+1. Our aim is to show that the two Riemann sums
n
å
q = 0 
f(r*q)·(rq+1-rq)
b
å
p = 0 
f(t*p)·(tp+1-tp)
 agree to k  decimal places. To do this, we will introduce another sequence  sj by combining the above two sequences into one. How follows.

Take the union U of the set of points provided by the rq and tp. In this union U, let s0 = a and let sj be the j+1st element. Then for some b > 0,
r0 = a < s1 < s2 < ¼ < sn < sn+1 = b
 where n is the number of distinct points in the union U. (Vocabulary: The union may be called a refinement of both sequences.) We will show that one (and hence both) of the expressions
a
å
q = 0 
f(r*q)·(rq+1-rq)
b
å
p = 0 
f(t*p)·(tp+1-tp)

differ from
n
å
q = 0 
f(sq)·(sq+1-sq)
by at most (1/4)(10**k |b-a|)**(-1) and hence to each other. In particular, we will show that åp = 0bf(t*p)·(tp+1-tp) and åq = 0nf(sq)·(sq+1-sq) differ by at most [1/4][1/(10k)][1/(|b-a|)] in absolute value. The difference will be denote by ±D.

by at most (1/4)(10**k |b-a|)**(-1) and hence to each other. In particular, we will show that 
b
å
p = 0 
f(t*p)·(tp+1-tp)

 and 
n
å
q = 0 
f(sq)·(sq+1-sq)

 differ by at most (1/4)(10**k |b-a|)**(-1) in absolute value. The difference will be denote by ± D

Now each tp = sq for some q = g(p) uniquely determined by p since tp belongs to the union U of the initial two sequences. tp+1 = sq if q = g(p+1). Let h(p) = g(p+1)-1. Then
tp+1-tp = sg(p+1)-sg(p) = h(p)
å
q = g(p
(sq+1-sq)

since by induction on whole numbers m > n+1
m-1
å
j = n 
aj+1-aj = am-an
Equalities such as this will be used often below.  Now
b
å
p = 0 
f(t*p)·(tp+1-tp)
=
b
å
p = 0 
f(t*p h(p)
å
q = g(p
(sq+1-sq)
=
b
å
p = 0 
h(p)
å
q = g(p
f(t*p)·(sr+1-sr)
and
n
å
q = 0 
f(sq)·(sq+1-sq) = b
å
p = 0 
h(p)
å
q = g(p
f(sq)·(sq+1-sq)

Here g(p) < q < g(p+1) implies
tp = sg(p) £ sp £ sq £ sq+1 £ sg(p+1) = tp+1.

Therefore sq  and t*p  both belong to the interval [tp, tp+1].  Since the length of this interval is less than d, we observe  |sq-s*p | < d  and therefore  |f(sq)-f(t*p)| £ e. Therefore the difference
D = b
å
p = 0 
f(t*p)·(tp+1-tp)- n
å
q = 0 
f(sq)·(sq+1-sq)
=
b
å
p = 0 
h(p)
å
q = g(p
f(t*p)·(sq+1-sq)- b
å
p = 0 
h(p)
å
q = g(p
f(sq))·(sq+1-sq)
=
b
å
p = 0 
h(p)
å
q = g(p
[f(t*p)-f(sq)]·(sq+1-sq)
Therefore
|D|
£
ê
ê
b
å
p = 0 
h(p)
å
q = g(p
[f(t*p)-f(sq)]·(sq+1-sq) ê
ê
£
b
å
p = 0 
ê
ê
h(p)
å
q = g(p
[f(t*p)-f(sq)]·(sq+1-sq) ê
ê
= Q
by the generalized triangle inequality applied to the outer sum. Now the generalized triangle inequality applied to the inner sums yields 
Q
£
b
å
p = 0 
h(p)
å
q = g(p
|[f(t*p)-f(sq)]·(sq+1-sq)|
£
b
å
p = 0 
h(p)
å
q = g(p
|f(t*p)-f(sq)|·|sq+1-sq|
=
b
å
p = 0 
h(p)
å
q = g(p
|f(t*p)-f(sq)|·(sq+1-sq)
=
b
å
p = 0 
h(p)
å
q = g(p
e·(sq+1-sq)
=
q = b
å
q = 0 
e·(sq+1-sq)
=
e(sb+1-sb) = e(b-a)
since |  But e =  Therefore the absolute value of the difference  
|D| £ e(b-a) £ 1
4
1
10k

This says the two sums and differ by at most as required.

Continuity and Lipschitz Continuity

Theorem H.3 [Riemann Sums & Lipschitz Functions] Suppose f(x) is defined on an interval [a,b] and differentiable when a < x < b. Further suppose for some K > 0 that
|f(x2)-f(x1)| £ K·|x2-x1|
whenever x1 and x2 are both in the interval [a,b]. Suppose

whenever x1 and x2 are both in the interval  [a,b] Suppose
x0 = a < x1 < x2 < xj < xn < xn+1 = b.

Further suppose xj < wn < xj+1 . Then  all sums of the form
n
å
j = 0 
f(wj)·(xj+1-xj)
where 0 < xj+1-xj  < d  differ by at most e = K(b-a)d   Note if e = (1/2)10-k   given first, put d  = [e/((b-a)K)].

Proof of Theorem. Let K > 0 be a Lipschitz constant for f(x) on the interval [a,b]. Let e = [1/4][1/(10k)][1/(|b-a|)]. Put d = [(e)/((b-a)K)]. Then for every pair of numbers u and v in the interval [a,b], the inequality |u-v| £ d implies |f(u)-f(v)| £ . The rest of the proof is exactly the same as the previous one.

Remark. Most piecewise continuous functions met in practice, that is, in calculus courses, are Lipschitz continuous on each interval [a,b] which does not include a singularity - a point where division by zero occurs. But in principle, the exceptions are more frequent. This is analogous to the situation with real numbers, where, in every-day practice and computation, most people and computing machines handle fractions and finite decimal or binary expansions, but where, in principle, there are more irrationals than rationals among the real numbers. In any event, Lipschitz continuous functions and criteria which identify them provide further examples of continuous functions and another link to error control or error bounding in computations. This author is undecided as to whether or not Lipschitz continuity should be emphasized in first courses on calculus.

the Real Analysis appendices of

Why Slopes
and
More Math

Volume 3

Printed in Canada
ISBN 0-9697564-3-7

These  Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson

Section Entrance
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

Proofs of  one-sided theorems could be of interest in the study of 2D topology.

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/

||
 -/[]\- 
||
   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.
Why Slopes & More.Math 1995
Tour their 
forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing Preparing for Scienceends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations
 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5
wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter   
 

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s

3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory
6.    Solving Linear Equations  
Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19
Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21.
Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
-
Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.


 


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Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep

A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition
   (8 lessons)
Comparison to Subtraction
  (9 lessons)
Multiplication
( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples.
 
-Prime Factorization Aids 
(Learn how to find factors quickly)
-Prime Factorization Examples
 
-Counting & Generating. Factors

-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences
  
Units in calculations
Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way
 & with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for AlgebraSecond, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for AlgebraWhere does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on
Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: Operations on Polynomials:
Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula  The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills. 
Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check

Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in 
Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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