Calculus (
Latin,
calculus, a small stone used for
counting) is a branch in
mathematics
focused on
limits,
functions,
derivatives,
integrals,
and
infinite series. This subject
constitutes a major part of modern mathematics education. It has
two major branches,
differential calculus and
integral
calculus, which are related by the
fundamental theorem of
calculus. Calculus is the study of change, in the same way that
geometry is the study of shape and
algebra is the study of operations and their
application to solving equations. A course in calculus is a gateway
to other, more advanced courses in mathematics devoted to the study
of functions and limits, broadly called
mathematical analysis. Calculus has
widespread applications in
science,
economics, and
engineering and can solve many problems for
which
algebra alone is
insufficient.
Historically, calculus was called "the calculus of
infinitesimals", or "
infinitesimal calculus". More
generally,
calculus (plural
calculi) may refer to
any method or system of calculation guided by the symbolic
manipulation of expressions. Some examples of other well-known
calculi are
propositional
calculus,
variational
calculus,
lambda calculus,
pi calculus and
join calculus.
History
Ancient
The ancient period introduced some of the ideas of
integral calculus, but does not seem to have
developed these ideas in a rigorous or systematic way. Calculating
volumes and areas, the basic function of integral calculus, can be
traced back to the
Egyptian
Moscow papyrus (c. 1820
BC), in which an Egyptian successfully calculated the
volume of a
pyramidal frustum.
From the school of
Greek
mathematics,
Eudoxus (c.
408−355 BC) used the
method of
exhaustion, which prefigures the concept of the limit, to
calculate areas and volumes while
Archimedes (c. 287−212 BC)
developed this idea
further, inventing
heuristics which
resemble integral calculus. The
method of exhaustion was later used in
China by
Liu
Hui in the 3rd century AD in order to find the area of a
circle. In the 5th century AD,
Zu
Chongzhi used what would later be called
Cavalieri's principle to find the
volume of a
sphere.
Medieval
Around AD 1000, the
Islamic mathematician Ibn al-Haytham (Alhacen) was the first to
derive the formula for the sum of the fourth powers of an
arithmetic progression, using a
method that is readily generalizable to finding the formula for the
sum of any higher integral powers, which he used to perform an
integration. In the 11th century, the Chinese
polymath Shen Kuo developed
'packing' equations that dealt with integration. In the 12th
century, the
Indian
mathematician,
Bhāskara II,
developed an early
derivative
representing infinitesimal change, and he described an early form
of
Rolle's theorem. Also in the 12th
century, the
Persian mathematician
Sharaf al-Dīn
al-Tūsī discovered the
derivative of
cubic polynomials, an important
result in differential calculus. In the 14th century,
Madhava of Sangamagrama, along with
other mathematician-astronomers of the
Kerala school of
astronomy and mathematics, described special cases of
Taylor series, which are treated in the text
Yuktibhasa.
Modern
In the modern period, independent discoveries relating to calculus
were being made in early 17th century
Japan, by mathematicians such as
Seki Kowa, who expanded upon the
method of exhaustion.
In Europe, the foundational work was a treatise due to
Bonaventura Cavalieri, who argued that
volumes and areas should be computed as the sums of the volumes and
areas of infinitesimal thin cross-sections. The ideas were similar
to Archimedes' in
The
Method, but this treatise was lost until the early part of the
twentieth century. Cavalieri's work was not well respected since
his methods can lead to erroneous results, and the infinitesimal
quantities he introduced were disreputable at first.
The formal study of calculus combined Cavalieri's infinitesimals
with the
calculus of
finite differences developed in Europe at around the same time.
The combination was achieved by
John
Wallis,
Isaac Barrow, and
James Gregory,
the latter two proving the
second fundamental theorem of
calculus around 1675.
The
product rule and
chain rule, the notion of
higher derivatives,
Taylor series, and
analytical functions were introduced by
Isaac Newton in an idiosyncratic
notation which he used to solve problems of
mathematical physics. In his
publications, Newton rephrased his ideas to suit the mathematical
idiom of the time, replacing calculations with infinitesimals by
equivalent geometrical arguments which were considered beyond
reproach. He used the methods of calculus to solve the problem of
planetary motion, the shape of the surface of a rotating fluid, the
oblateness of the earth, the motion of a weight sliding on a
cycloid, and many other problems discussed in his
Principia
Mathematica. In other work, he developed series expansions for
functions, including fractional and irrational powers, and it was
clear that he understood the principles of the
Taylor series. He did not publish all these
discoveries, and at this time infinitesimal methods were still
considered disreputable.
These ideas were systematized into a true calculus of
infinitesimals by
Gottfried
Wilhelm Leibniz, who was originally accused of
plagiarism by Newton. He is now regarded as an
independent inventor of and contributor to calculus. His
contribution was to provide a clear set of rules for manipulating
infinitesimal quantities, allowing the computation of second and
higher derivatives, and providing the
product rule and
chain
rule, in their differential and integral forms. Unlike Newton,
Leibniz paid a lot of attention to the formalism – he often spent
days determining appropriate symbols for concepts.
Leibniz and
Newton are usually both credited with the
invention of calculus. Newton was the first to apply calculus to
general
physics and Leibniz developed much
of the notation used in calculus today. The basic insights that
both Newton and Leibniz provided were the laws of differentiation
and integration, second and higher derivatives, and the notion of
an approximating polynomial series. By Newton's time, the
fundamental theorem of calculus was known.
When Newton and Leibniz first published their results, there was
great
controversy over which mathematician (and therefore which
country) deserved credit. Newton derived his results first, but
Leibniz published first. Newton claimed Leibniz stole ideas from
his unpublished notes, which Newton had shared with a few members
of the
Royal Society. This controversy
divided English-speaking mathematicians from continental
mathematicians for many years, to the detriment of English
mathematics. A careful examination of the papers of Leibniz and
Newton shows that they arrived at their results independently, with
Leibniz starting first with integration and Newton with
differentiation. Today, both Newton and Leibniz are given credit
for developing calculus independently. It is Leibniz, however, who
gave the new discipline its name. Newton called his calculus
"
the science of fluxions".
Since the time of Leibniz and Newton, many mathematicians have
contributed to the continuing development of calculus. In the 19th
century, calculus was put on a much more rigorous footing by
mathematicians such as
Cauchy,
Riemann, and
Weierstrass
(see
-definition of
limit). It was also during this period that the ideas of
calculus were generalized to
Euclidean
space and the
complex plane.
Lebesgue generalized the notion of the
integral so that virtually any function has an integral, while
Laurent Schwartz extended
differentiation in much the same way.
Calculus is a ubiquitous topic in most modern high schools and
universities around the world.
Significance
While some of the ideas of calculus were developed earlier in
Greece,
China,
India,
Iraq, Persia, and
Japan, the modern use of calculus began
in
Europe, during the 17th century, when
Isaac Newton and
Gottfried Wilhelm Leibniz built on
the work of earlier mathematicians to introduce its basic
principles. The development of calculus was built on earlier
concepts of instantaneous motion and area underneath curves.
Applications of differential calculus include computations
involving
velocity and
acceleration, the
slope of
a curve, and
optimization. Applications of
integral calculus include computations involving
area,
volume,
arc length,
center of
mass,
work, and
pressure. More advanced applications include
power series and
Fourier series. Calculus can be used to
compute the trajectory of a shuttle docking at a space station or
the amount of snow in a driveway.
Calculus is also used to gain a more precise understanding of the
nature of space, time, and motion. For centuries, mathematicians
and philosophers wrestled with paradoxes involving
division by zero or sums of infinitely many
numbers. These questions arise in the study of
motion and
area. The
ancient Greek philosopher Zeno
gave several famous examples of such
paradoxes. Calculus provides tools,
especially the
limit and the
infinite series, which resolve the
paradoxes.
Foundations
In mathematics,
foundations refers to the
rigorous development of a
subject from precise axioms and definitions. Working out a rigorous
foundation for calculus occupied mathematicians for much of the
century following Newton and Leibniz and is still to some extent an
active area of research today.
There is more than one rigorous approach to the foundation of
calculus. The usual one today is via the concept of
limits defined on the
continuum of
real numbers. An alternative is
nonstandard analysis, in which the real
number system is augmented with
infinitesimal and
infinite numbers, as in the original Newton-Leibniz
conception. The foundations of calculus are included in the field
of
real analysis, which contains full
definitions and
proof of the
theorems of calculus as well as generalizations such as
measure theory and
distribution theory.
Principles
Limits and infinitesimals
Calculus is usually developed by manipulating very small
quantities. Historically, the first method of doing so was by
infinitesimals. These are objects
which can be treated like numbers but which are, in some sense,
"infinitely small". An infinitesimal number
dx could be
greater than 0, but less than any number in the sequence 1, ½, ⅓,
... and less than any positive real number. Any integer multiple of
an infinitesimal is still infinitely small, i.e., infinitesimals do
not satisfy the
Archimedean
property. From this point of view, calculus is a collection of
techniques for manipulating infinitesimals. This approach fell out
of favor in the 19th century because it was difficult to make the
notion of an infinitesimal precise. However, the concept was
revived in the 20th century with the introduction of
non-standard analysis and
smooth infinitesimal analysis,
which provided solid foundations for the manipulation of
infinitesimals.
In the 19th century, infinitesimals were replaced by
limit. Limits describe the value of a
function at a certain input
in terms of its values at nearby input. They capture small-scale
behavior, just like infinitesimals, but use the ordinary
real number system. In this treatment, calculus
is a collection of techniques for manipulating certain limits.
Infinitesimals get replaced by very small numbers, and the
infinitely small behavior of the function is found by taking the
limiting behavior for smaller and smaller numbers. Limits are the
easiest way to provide rigorous foundations for calculus, and for
this reason they are the standard approach.
Differential calculus
Tangent line at (
x,
f(
x)).
The derivative f′(x) of a curve at a point
is the slope (rise over run) of the line tangent to that curve at
that point.
Differential calculus is the study of the definition, properties,
and applications of the
derivative of a
function. The process of finding the derivative is called
differentiation. Given a function and a point in the
domain, the derivative at that point is a way of encoding the
small-scale behavior of the function near that point. By finding
the derivative of a function at every point in its domain, it is
possible to produce a new function, called the
derivative
function or just the
derivative of the original
function. In mathematical jargon, the derivative is a
linear operator which inputs a function and
outputs a second function. This is more abstract than many of the
processes studied in elementary algebra, where functions usually
input a number and output another number. For example, if the
doubling function is given the input three, then it outputs six,
and if the squaring function is given the input three, then it
outputs nine. The derivative, however, can take the squaring
function as an input. This means that the derivative takes all the
information of the squaring function—such as that two is sent to
four, three is sent to nine, four is sent to sixteen, and so on—and
uses this information to produce another function. (The function it
produces turns out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark
called
prime. Thus, the derivative of
the function of
f is
f′, pronounced "f prime."
For instance, if
f(
x) =
x^{2} is
the squaring function, then
f′(
x) = 2
x
is the doubling function.
If the input of the function represents time, then the derivative
represents change with respect to time. For example, if
f
is a function that takes a time as input and gives the position of
a ball at that time as output, then the derivative of
f is
how the position is changing in time, that is, it is the
velocity of the ball.
If a function is
linear (that is, if
the
graph of the function is a
straight line), then the function can be written
y =
mx +
b, where:
- m= \frac{\mbox{rise}}{\mbox{run}}= {\mbox{change in } y \over
\mbox{change in } x} = {\Delta y \over{\Delta x}}.
This gives an exact value for the slope of a straight line. If the
graph of the function is not a straight line, however, then the
change in
y divided by the change in
x varies.
Derivatives give an exact meaning to the notion of change in output
with respect to change in input. To be concrete, let
f be
a function, and fix a point
a in the domain of
f.
(
a,
f(
a)) is a point on the graph of the
function. If
h is a number close to zero, then
a
+
h is a number close to
a. Therefore (
a
+
h,
f(
a +
h)) is close to
(
a,
f(
a)). The slope between these two
points is
- m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) -
f(a)}{h}.
This expression is called a
difference quotient. A line
through two points on a curve is called a
secant line, so
m is the slope of the secant line between (
a,
f(
a)) and (
a +
h,
f(
a +
h)). The secant line is only an
approximation to the behavior of the function at the point
a because it does not account for what happens between
a and
a +
h. It is not possible to
discover the behavior at
a by setting
h to zero
because this would require dividing by zero, which is impossible.
The derivative is defined by taking the
limit as
h tends to zero,
meaning that it considers the behavior of
f for all small
values of
h and extracts a consistent value for the case
when
h equals zero:
- \lim_{h \to 0}{f(a+h) - f(a)\over{h}}.
Geometrically, the derivative is the slope of the
tangent line to the graph of
f at
a. The tangent line is a limit of secant lines just as the
derivative is a limit of difference quotients. For this reason, the
derivative is sometimes called the slope of the function
f.
Here is a particular example, the derivative of the squaring
function at the input 3. Let
f(
x) =
x^{2} be the squaring function.
[[File:Sec2tan.gif|thumb|300px|The derivative
f′(
x) of a curve at a point is the slope of the
line tangent to that curve at that point. This slope is determined
by considering the limiting valueof the slopes of secant lines.
Here the function involved (drawn in red) is
f(
x)
=
x^{3} −
x. The tangentline (in green)
which passes through the point (−3/2, −15/8) has a slopeof 23/4.
Note that the vertical and horizontal scales in this image are
different.]]
- \begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 9\over{h}}
\\
&=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\&=\lim_{h \to
0}{6h + h^2\over{h}} \\&=\lim_{h \to 0} (6 + h) \\&=
6.\end{align}
The slope of tangent line to the squaring function at the point
(3,9) is 6, that is to say, it is going up six times as fast as it
is going to the right. The limit process just described can be
performed for any point in the domain of the squaring function.
This defines the
derivative function of the squaring
function, or just the
derivative of the squaring function
for short. A similar computation to the one above shows that the
derivative of the squaring function is the doubling function.
Leibniz notation
A common notation, introduced by Leibniz, for the derivative in the
example above is\begin{align}y=x^2 \\\frac{dy}{dx}=2x.\end{align}In
an approach based on limits, the symbol
dy/dx is to be
interpreted not as the quotient of two numbers but as a shorthand
for the limit computed above. Leibniz, however, did intend it to
represent the quotient of two infinitesimally small numbers,
dy being the infinitesimally small change in
y
caused by an infinitesimally small change
dx applied to
x. We can also think of
d/dx as a differentiation
operator, which takes a function as an input and gives another
function, the derivative, as the output. For
example:\frac{d}{dx}(x^2)=2x.In this usage, the
dx in the
denominator is read as "with respect to x." Even when calculus is
developed using limits rather than infinitesimals, it is common to
manipulate symbols like
dx and
dy as if they were
real numbers; although it is possible to avoid such manipulations,
they are sometimes notationally convenient in expressing operations
such as the
total derivative.
Integral calculus
Integral calculus is the study of the definitions,
properties, and applications of two related concepts, the
indefinite integral and the
definite integral.
The process of finding the value of an integral is called
integration. In technical language, integral calculus
studies two related
linear
operators.
The
indefinite integral is the
antiderivative, the inverse operation to
the derivative.
F is an indefinite integral of
f
when
f is a derivative of
F. (This use of upper-
and lower-case letters for a function and its indefinite integral
is common in calculus.)
The
definite integral inputs a function and
outputs a number, which gives the area between the graph of the
input and the
x-axis. The technical
definition of the definite integral is the
limit of a sum of areas of rectangles,
called a
Riemann sum.
A motivating example is the distances traveled in a given
time.
- \mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time}
If the speed is constant, only multiplication is needed, but if the
speed changes, then we need a more powerful method of finding the
distance. One such method is to approximate the distance traveled
by breaking up the time into many short intervals of time, then
multiplying the time elapsed in each interval by one of the speeds
in that interval, and then taking the sum (a
Riemann sum) of the approximate distance
traveled in each interval. The basic idea is that if only a short
time elapses, then the speed will stay more or less the same.
However, a Riemann sum only gives an approximation of the distance
traveled. We must take the limit of all such Riemann sums to find
the exact distance traveled.
Integration can be thought of as
measuring the area under a curve, defined by
f(
x), between two points (here
a and
b).
If
f(x) in the diagram on the left represents speed as it
varies over time, the distance traveled (between the times
represented by
a and
b) is the area of the shaded
region
s.
To approximate that area, an intuitive method would be to divide up
the distance between
a and
b into a number of
equal segments, the length of each segment represented by the
symbol
Δx. For each small segment, we can choose one value
of the function
f(
x). Call that value
h.
Then the area of the rectangle with base
Δx and height
h gives the distance (time
Δx multiplied by speed
h) traveled in that segment. Associated with each segment
is the average value of the function above it,
f(x)=h. The
sum of all such rectangles gives an approximation of the area
between the axis and the curve, which is an approximation of the
total distance traveled. A smaller value for
Δx will give
more rectangles and in most cases a better approximation, but for
an exact answer we need to take a limit as
Δx approaches
zero.
The symbol of integration is \int \,, an elongated
S (the
S stands for "sum"). The definite integral is written as:
- \int_a^b f(x)\, dx.
and is read "the integral from
a to
b of
f-of-
x with respect to
x." The Leibniz
notation
dx is intended to suggest dividing the area under
the curve into an infinite number of rectangles, so that their
width
Δx becomes the infinitesimally small
dx. In
a formulation of the calculus based on limits, the notation
\int_a^b \ldots\, dx is to be understood as an operator that takes
a function as an input and gives a number, the area, as an output;
dx is not a number, and is not being multiplied by
f(x).
The indefinite integral, or antiderivative, is written:
- \int f(x)\, dx.
Functions differing by only a constant have the same derivative,
and therefore the antiderivative of a given function is actually a
family of functions differing only by a constant. Since the
derivative of the function
y =
x² +
C,
where
C is any constant, is
y′ = 2
x, the
antiderivative of the latter is given by:
- \int 2x\, dx = x^2 + C.
An undetermined constant like
C in the antiderivative is
known as a
constant of
integration.
Fundamental theorem
The
fundamental theorem
of calculus states that differentiation and integration are
inverse operations. More precisely, it relates the values of
antiderivatives to definite integrals. Because it is usually easier
to compute an antiderivative than to apply the definition of a
definite integral, the Fundamental Theorem of Calculus provides a
practical way of computing definite integrals. It can also be
interpreted as a precise statement of the fact that differentiation
is the inverse of integration.
The Fundamental Theorem of Calculus states: If a function
f is
continuous on the
interval [
a,
b] and if
F is a function
whose derivative is
f on the interval (
a,
b), then
- \int_{a}^{b} f(x)\,dx = F(b) - F(a).
Furthermore, for every
x in the interval (
a,
b),
- \frac{d}{dx}\int_a^x f(t)\, dt = f(x).
This realization, made by both
Newton
and
Leibniz, who based their
results on earlier work by
Isaac
Barrow, was key to the massive proliferation of analytic
results after their work became known. The fundamental theorem
provides an algebraic method of computing many definite
integrals—without performing limit processes—by finding formulas
for
antiderivatives. It is also a
prototype solution of a
differential equation. Differential
equations relate an unknown function to its derivatives, and are
ubiquitous in the sciences.
Applications
Calculus is used in every branch of the
physical sciences,
actuarial science,
computer science,
statistics,
engineering,
economics,
business,
medicine,
demography, and in other fields wherever
a problem can be
mathematically
modeled and an
optimal solution is
desired.
Physics makes particular use of calculus;
all concepts in
classical
mechanics are interrelated through calculus. The
mass of an object of known
density, the
moment of
inertia of objects, as well as the total energy of an object
within a conservative field can be found by the use of calculus. In
the subfields of
electricity and
magnetism calculus can be used to find the
total
flux of electromagnetic fields. A more
historical example of the use of calculus in physics is
Newton's second law of motion, it
expressly uses the term "rate of change" which refers to the
derivative:
The rate of change of
momentum of a body is equal to the resultant force acting on the
body and is in the same direction. Even the common expression
of Newton's second law as
Force = Mass × Acceleration involves
differential calculus because acceleration can be expressed as the
derivative of velocity. Maxwell's theory of
electromagnetism and
Einstein's theory of
general relativity are also expressed in
the language of differential calculus. Chemistry also uses calculus
in determining reaction rates and radioactive decay.
Calculus can be used in conjunction with other mathematical
disciplines. For example, it can be used with
linear algebra to find the "best fit" linear
approximation for a set of points in a domain. Or it can be used in
probability theory to determine
the probability of a continuous random variable from an assumed
density function.
Green's Theorem, which gives the
relationship between a line integral around a simple closed curve C
and a double integral over the plane region D bounded by C, is
applied in an instrument known as a
planimeter which is used to calculate the area of
a flat surface on a drawing. For example, it can be used to
calculate the amount of area taken up by an irregularly shaped
flower bed or swimming pool when designing the layout of a piece of
property.
In the realm of medicine, calculus can be used to find the optimal
branching angle of a blood vessel so as to maximize flow.
In
analytic geometry, the study of
graphs of functions, calculus is used to find high points and low
points (maxima and minima), slope,
concavity and
inflection points.
In economics, calculus allows for the determination of maximal
profit by providing a way to easily calculate both
marginal cost and
marginal revenue.
Calculus can be used to find approximate solutions to equations, in
methods such as
Newton's method,
fixed point iteration, and
linear approximation. For
instance, spacecraft use a variation of the
Euler method to approximate curved courses
within zero gravity environments.
See also
Lists
Related topics
References
Notes
Books
- Larson, Ron, Bruce H.
Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning.
ISBN 9780547167022
- McQuarrie, Donald A. (2003). Mathematical Methods for
Scientists and Engineers, University Science Books. ISBN
9781891389245
- Stewart, James
(2008). Calculus: Early Transcendentals, 6th ed., Brooks
Cole Cengage Learning. ISBN 9780495011668
- Thomas, George B., Maurice D.
Weir, Joel Hass, Frank R. Giordano (2008), "Calculus", 11th ed.,
Addison-Wesley. ISBN 0-321-48987-X
Other resources
Further reading
- Courant, Richard ISBN
978-3540650584 Introduction to calculus and analysis
1.
- Edmund Landau. ISBN 0-8218-2830-4
Differential and Integral Calculus, American Mathematical
Society.
- Robert A. Adams. (1999). ISBN 978-0-201-39607-2 Calculus: A
complete course.
- Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden,
ed. (1986) Undergraduate Programs in the Mathematics and
Computer Sciences: The 1985-1986 Survey, Mathematical
Association of America No. 7.
- John L. Bell: A Primer of Infinitesimal Analysis,
Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses
synthetic differential
geometry and nilpotent infinitesimals.
- Florian Cajori, "The History of
Notations of the Calculus." Annals of Mathematics, 2nd
Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46.
- Leonid P. Lebedev and Michael J. Cloud: "Approximating
Perfection: a Mathematician's Journey into the World of Mechanics,
Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004.
- Cliff Pickover. (2003). ISBN
978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the
Hungry Mind.
- Michael Spivak. (September 1994).
ISBN 978-0-914098-89-8 Calculus. Publish or Perish
publishing.
- Tom M. Apostol. (1967). ISBN 9780471000051
Calculus, Volume 1, One-Variable Calculus with an Introduction
to Linear Algebra. Wiley.
- Tom M. Apostol. (1969). ISBN 9780471000075
Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra
with Applications. Wiley.
- Silvanus P. Thompson and Martin Gardner. (1998). ISBN
978-0-312-18548-0 Calculus Made Easy.
- Mathematical
Association of America. (1988). Calculus for a New Century;
A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300
252.
- Thomas/Finney. (1996). ISBN 978-0-201-53174-9 Calculus and
Analytic geometry 9th, Addison Wesley.
- Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From
MathWorld—A Wolfram Web Resource.
Online books
- Crowell, B. (2003). "Calculus" Light and Matter,
Fullerton. Retrieved 6 May 2007 from http://www.lightandmatter.com/calc/calc.pdf
- Garrett, P. (2006). "Notes on first year calculus"
University of Minnesota. Retrieved 6 May
2007 from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf]
* Faraz, H. (2006). "''Understanding Calculus''" Retrieved [[6
May]] [[2007]] from Understanding Calculus, URL
[http://www.understandingcalculus.com/
http://www.understandingcalculus.com/] (HTML only) * Keisler, H. J.
(2000). "''Elementary Calculus: An Approach Using Infinitesimals''"
Retrieved [[6 May]] [[2007]] from
[http://www.math.wisc.edu/~keisler/keislercalc1.pdf
http://www.math.wisc.edu/~keisler/keislercalc1.pdf] * Mauch, S.
(2004). "''Sean's Applied Math Book''" California Institute of
Technology. Retrieved [[6 May]] [[2007]] from
[http://www.cacr.caltech.edu/~sean/applied_math.pdf
http://www.cacr.caltech.edu/~sean/applied_math.pdf
- Sloughter, Dan (2000). "Difference Equations to
Differential Equations: An introduction to calculus".
Retrieved 17 March 2009
from http://synechism.org/drupal/de2de/
- Stroyan, K.D. (2004). "A brief introduction to
infinitesimal calculus" University of Iowa. Retrieved 6 May 2007 from http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm
(HTML only)
- Strang, G. (1991). "Calculus" Massachusetts Institute
of Technology. Retrieved 6 May 2007 from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
- Smith, William V. (2001). "The Calculus" Retrieved
4 July 2008 [453] (HTML only).
Web pages