Site
Strengths
- For
calculus and for pre-calculus, site strengths lies in the
detailed and wordy development of logic and algebra in chapters 1 to
25 of the misnamed book, Three
Skills for Algebra, and in an online postscript: what
is a variable. For beginning calculus and for
pre-calculus student, site strength also comes from this high
school level, why slopes are studied, geometric preview of
calculus, and in the the leading chapters in Volume 3, Why
Slopes and More Math. The preview and leading chapters put first
some ideas from the middle of calculus, ideas easier to learn and
teach. That placement makes calculus easier or less difficult -may
avoid some algebra shock in calculus. Students and
teachers: These arithmetic
skill testing questions with hints of algebra may
identifying common weaknesses.
See the strong site coverage of straight lines, polynomials,
quadratics and functions in the Analytic
Geometry/Functions site folder.
- For Engineering, Physics and Advanced Calculus
students, site strength also lies in this geometric
development of complex numbers. In a 1976
public talk before a general audience, the late Richard Feynmann,
briefly and geometrically described his subject physics as the
addition and multiplication of arrows in the plane. These operations
on arrows or points in the plane gives a base for learning about complex
numbers before the study of trigonometry on the unit circle.
That leads to easier and more accessible derivations not only of trig
identities, but also more accessible explanations of the cosine law
and of trigonometric formulas for dot and cross products in the plane.
See the easy consequences. See
the Euclidean
Geometry to Complex Number section for a simple high school
level development of the properties of 2D vectors and complex numbers
from Euclidean Geometry. The development has the same level of
empirical or applied mathematic rigour found in present day diagram
based accounts of trigonometry.
- For better work and studies skills,
adopt these ends,
values and methods. See too site logic chapters - short
versions in Volume 2, Three Skills for Algebra. Logic mastery is a
must for sharper work & study skills. Good format and good
notational habit, speeds comprehension and reduces errors. Learning
Tips: (i) If you a difficulty with a topic, you need to retreat to
review and master earlier skills.
Still More Advice and Directions:
B How to Learn
C
How to Read
D
What to do in School and Why
E.
How to Study Mathematics and Why
2. For students of reading, writing and reason site strength lies
in its wordy introduction of logic with logic puzzles and stories.
Even mathematical induction is introduced with the question of when is
In early parts of mathematics, students may gain skills and confidence
with methods that lead to results, repeatable and reproducible,
and hence verifiable. But after or besides learning to do, some
students may appreciate understanding how or why methods work -
explore site depths for that. In college oriented mathematics,
logic in the form of direct, if not indirect, deductive reason may
appear. Site logic
chapters give a base for that. The site account of Euclidean-Geometry
provides a simple example. The page ends,
values and methods for work and study finishes with a note or
suggestion that emphasizing or developing empirical problem solving
abilities with the use and combination of rules and patterns to arrive
at results provides a educational base for deductive reasoning skills.
There is a convergence.
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Calculus gives the best framework for
understanding calculations met in business, science and engineering.
Describing the same calculation without calculus is long and shallower
process. Good preparation for college mathematics (or calculus)
requires mastery of most of the topics, you meet in high school
mathematics: exact arithmetic with whole numbers and fractions, algebra,
geometry without and with coordinates, and trig.
Preparation for calculus prepares for most or all arts,
trades and disciplines involving mathematics. That is, preparing for
calculus will help you be better than you expect in accounting, art, chemistry,
drawing, English, physics, woodwork, metal work, geology, electricity,
mechanics, drafting and computing etc.
Many professions apart from mathematics, for instance
nursing and policing, require mastery of mathematics, even arithmetic, as
a sign of skill and intelligence in problem solving, or using rules and patterns
carefully and precisely in a repeatable and reproducible manner. Many
profession far from mathematics require mathematics mastery because its mastery
shows an error in one step makes it and everything after wrong.
Even for students who do not take calculus, preparation for
calculus would or should lead to the ability to apply or use rules and patterns,
one at a time and one after another, carefully with the knowledge that an error
in one step makes all that follows wrong or give its a lesser value.
Students should not be surprised by the expectation that a multi-step method
leads to repeatable and reproducible results when sufficient care is taken.
| Each
of the following links except the starter and warm up lessons
corresponds to chapters in a typical North American calculus
text.
[ Starter & Warm Up Lessons ]
[ 1. Usual Review/Starter Lessons ]
[ 2. Limits [13] ]
[ 3. Differentiation Rules[28] ]
[ 4. Applications of Derivatives [5] ]
[ 5. Definite Integrals - Preview [5]]
[ 6. Integration Applications [6] ]
[ Advanced Material ]
Algebra is required at full strength in calculus.
The advice and directions in this section are aimed at easing
or avoid difficulties by providing a progressive review and
development of skills and concepts. Exploring this site
section and online volumes 2 and 3 will provide you insights
in to how to learn or teach a first course in calculus with
greater ease. Good luck.
The online book 3 .Why.Slopes.&.More.Math.1995
may be regarded as the first calculus section in this site.
This second section exists to further support calculus
learning and teaching, and to avoid overwhelming the Volume
3 section with postscripts.
If sharing ideas with fellow instructor on how to ease or
avoid difficulties in learning and teaching mathematics is
impolite, this site is extremely impolite. Oops and
Ouch.
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Starter
and Warm-up Lessons (Early Calculus or Late
Precalculus Students)
This
geometric
preview and chapters
2 to 6 in Volume 3, Why
Slopes and More Math, give a
context for the senior high school level study of
slopes and of factored polynomials. The same material
may be employed at the start of calculus to make it
easier. Calculus asks students to calculate
derivatives (slopes for straight lines) and to do
sign analysis, that is, to say identify interval
where derivatives or slopes are zero, positive or
negative. While calculus upto the calculation of
derivatives is algebraically challenging, the
sign analysis and interpretation as introduced in geometric
preview and chapters
2 to 6 is very simple. Moreover, it develops
algebraic skills in a way that makes the calculation
of derivatives and before that limits, much easier.
| Vol
2, Three
Skills for Algebra covers many
topics in algebra and logic that students
starting calculus should have mastered or will
have to master sooner or later. Also includes
arithmetic review problems to catch common
mistakes.
Vol. 3, Why
Slopes & More Maths, gives starter
lessons for differential and integral
calculus.
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Chapter 1:
High School Math Revisited - last minute
preparation for calculus. The aim here is to
catch common errors, improve reading skills and
revisit some basic concepts in algebra -
Most calculus text include a chapter reviewing high
school material starting with Functions: The
site treatment is comprehensives.
The online version of this chapter, see below,
starts before that and points students to Arithmetic
Review Problems with Hints of Algebra,
Chapter 2: Limits
of Functions - Saying how to calculate a
number directly or via the limit of approximations
defines it. In the study of derivatives, Limits
of approximations are used to provide
"official" definitions of slopes to curves y
= f(x), velocity, other instantaneous
rates of change, and acceleration. Areas and
volumes are further defined by the limit of
approximations (Riemann sum approximations) and some
of these limits may be evaluated via reversal of
slope or derivative methods. Watch out for a
twist: Limits are said to exist only if discrete
or continuous quantities approach a finite limit.
Then limits with values +oo (plus infinity) or -oo
(negative infinity) are defined but said not to exist
because they are not finite. This website
re-introduces the self-sufficient decimal viewpoint of
limits to make them accessible, or to serve as a
stepping stone to the algebraically challenging,
epsilon-delta, decimal-free, viewpoint - which almost
all do not get.
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Chapter 3: Derivatives
and Differentiation Rules - (1)
Derivatives (the slopes of a function in the preview)
at a point are defined or calculated via limits of
approximations to what the slope of a tangent line
should be. (2) The arithmetic viewpoint is easy
to follow. But the dependence of derivatives
(slopes) of y =f(x) on the x-coordinate requires the
algebraic concept of keeping x constant while an
h or dx in a secant approximation to the derivative
varies towards to zero in a limiting process.
That pattern depends on a full mastery of what is a
variable. (3) Next, there is a further
twist. Namely, differentiation rules give algebraic
methods (justified by limit consideration) for
calculating derivatives algebraically. (4) Then
rules for differentiating (obtaining slopes) for
polynomials, trig functions, logs and/or
exponentials alone, or combined as in algebraic
expressions or composed follow. Limits and properties
of these functions play key role in justifying and
implying algebraic rules. Your aim here is to master
the algebraic rules, and be well aware of how limits
were use to imply those rules. Your aim is to
also to master the chain-rule. Differentiation
rules will also be used alone and in combination. to
develop more rules. Mathematical induction will appear
in that development. There may be
variation between calculus courses. Some will postpone
the discussion of logs and exponentials to later.
(5) There may also be a discussion of implicit
differentiation - very few further topics depend on
it. (6) In this or the next chapter, you may meet
several theorems - patterns to apply when certain
conditions are met. Your aim is to understand their
statement and in that statement be aware of the
difference between saying A if B and saying A if and
only if B. That is where logic appears.
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| Chapter
4: Application
of Derivatives - The calculus starter lessons are
previews of these applications. In the previews,
formulas for slopes or derivative functions are given, the
application here require their calculation and analysis to
locate maximums and minimumss of height and also slope
functions. Exercises include graphing functions and
identifying interior and end point maximums and minimums of
functions or their derivatives. Interior maximums and
mininumss of derivatives (slopes) are called inflection
points. You will also met first (slope) and second derivative
(slope of slope) tests for interior maximums and minimums.
Graphing may also involve vertical, horizontal and slanted
asymptotes. The calculus preview included the first test.
Further application include word max-min problems in which you
will define a function y = f(x) and have find its max or min.
Here velocity, rates of changes appear as derivatives while
acceleration appears as a second derivative.
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Chapter
5: Integration:
The Riemann sum approximation (whatever that may be)
of areas under curves y = f(x) between say x = a and x =
b (b>a) in the limit, when the limit exists, leads to
a definite integral. The first fundamental theorem gives
conditions for the existence of that limit with and without
the area interpretation. Then the second fundamental
theorem of calculus says how to calculate the limit or
definite integral with the aid of functions F(x) whose
derivative or slope function is f(x). The net result is
a Riemann sum approximation and limt process that yields a
definite integral involving f(x) which can be calculated by
finding (if you can) an anti-derivative of f(x). In the
foregoing envelope, you will meet (a) summation notation
for sums, see the algebraic properties of sums, derive those
properties via mathematical induction, (b) a finer
discussion of the Riemann sum approximation process - the
requirement for the common or maximum width of rectangles in
the Riemann sum approximation process to tend to zero; and (c)
ad hoc antidifferentiation methods for finding from f(x) an
anti-derivative F(x) with the property that f(x) is the
derivative of F(x). Here all the rules for
differentiation are applied in reverse. The introduction
of indefinite integrals provides a context for this
independent of the interpretation of definite integrals as
limits of the Riemann sum approximation process.
Most likely, you will meet indefinite integrals
and anti-derivatives first, along with algebraic
properties inherited from those for differentiation. |
Chapter
6.
Integration Applications:In essence, the applications
consist of identifying physical quantities which can be
approximated by and calculated in the limit via Riemann sums
and the definite integral representation of the limit.
Area under a curve is the application met in the introduction
and motivation of the Riemann Sum Approximation and Limit
process, the process that leads to a definite integral.
This Riemann sum approximation and limit process yields
definite integral representations and even definitions
for areas between two curves via vertical, horizontal or
slanted slicing, volumes of solids via vertical,
horizontal or slanted slicing, volumes of solid of revolutions
via slicing in planes perpendicular to the axis of revolution
in the so called disk or washer methods; volumes
of solid of revolutions via slicing into cylindrical shells
around the axis of revolution. Here the convergence of the
Riemann sum implies what an area or volume should be, and so
provides a definition of area or volume for regions in 2D and
solids in 3D for which area and volume were not previously
defined. Further applications of the Riemann Sum
Approximation and Limit process yields formulas (definite
integrals) for work, fluid pressure, arc-length, moments,
center of mass and so on - quantities need in geometry of
engineering and physics - and college level statistics. |
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