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Welcome. This second site calculus section offers advice, directions and some worked
examples for a first course in calculus. 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.
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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.
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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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Professor
Whyslopes
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says
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Site value lies in the
difference between its ideas and yours.
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If one site explanation is too full for
your liking, try another.
Bon
Appetit
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Two common gaps to fill or avoid
- The Old Algebra Gap: Algebra appears
with too few words of explanation in high school and college mathematics.
Chapters
8 to 12 in online Volume 2 put more words into the explanation
and comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and indirect
use a formulas identifies a unifying theme for mathematics and logic - all
rules and patterns will be used forward and backwards. Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school exposition
of circular trig functions upto to formulas in the plane for vectors
dot and cross-products. The lesson provides the route that would
have been taken in course design if the key element of the lesson, a
December 2009 invention, had been available in the 1950s. For
further algebra skill development. See the site coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient mastery
of arithmetic with decimals and fractions is best (required) for the
high level study of mathematics alone and in science, technology and
business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That exact
and efficient command should be obtained in the last years of primary
school and the first years of secondary school.
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Would you believe: Skill mastery in mathematics has to be
seen to believed. To that end, learn or teach how-to write and draw the steps in
mathematical figuring or reasoning clearly. Do not try to save
space by doing a sequence of step in one place. Instead, do or record the
steps in sequence on a separate lines to make each step obvious and
verifiable.
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