Calculus and calculus education

Summary: Once reserved for upper-level majors, the study of derivatives and integrals of functions has been mainstreamed by its applications.

Calculus, which takes its name from the Latin word for “pebble,” is one of the most important branches of mathematics and one of the cornerstones of mathematics education. In ancient history, pebbles were used for counting, and “calculus” initially referred to that. The word now represents the method of calculation linked often with the study of change attempting to investigate motions and rates of change. From its mathematical development to the philosophy of calculus education, calculus has been fraught with rigorous debate and change.

Its appreciation and deeper understanding is a fundamental requirement in order to proceed toward a more advanced mathematical education and be involved with topics such as mathematical analysis. The topic finds its use and application in a vast number of different applied disciplines, such as biology, engineering, physics, population dynamics, statistics, and, in general, any scientific area that involves the study of instantaneous change.

Calculus education has a rich and varied history. Takakazu Seki is remembered as an influential teacher who passed his form of calculus on to his students. However, during the seventeenth century, secrecy surrounded rival schools in Japan, so it is difficult to determine his exact contributions.Successful calculus textbooks date back to at least Maria Gaetana Agnesi in the eighteenth century. She wrote Analytical Institutions, probably as a textbook for her brothers. She mastered many languages, which were useful when she integrated the knowledge of the time. She also added her own examples and expositions. Her book was widely translated and used all over the world, making the concepts of calculus more accessible.

Calculus education underwent many changes in the twentieth century. Early on, calculus was often an upper-division college course in North America while it was a pre-college course in France. U.S. President John F. Kennedy’s race to the moon impacted calculus education in the United States. More engineering students were recruited, and as a result, calculus shifted earlier in the college curriculum. Another change was an emphasis on set theory in such texts as Tom Apostol’s Calculus. Beginning in the latter half of the twentieth century, high schools offered AP Calculus. The shift of calculus to lower-level students also occurred in other countries, such as in Japan.

However, students who did not have the aptitude to succeed in competitive programs were filtered out in lower-level college courses, and educators debated this problem internationally. A calculus reform movement in the United Statesoriginated in the late 1980s, epitomized by the slogan “Calculus should be a pump, not a filter.” Teachers debated the roles of lectures, technology, and rigor. With minimal theoretical support for the choice of teaching strategies, mathematicians relied on empirical studies todetermine what would help calculus students succeed. Educators tested many different approaches, such as those emphasizing active learning, graphing calculators, computer algebra software, historical sources, writing, humanistic perspectives, real-life applications, distance education, or calculus as a laboratory science. New teaching approaches were met with widespread acceptance on some campuses and rejection and backlash on others. Some schools reported a decline in the number of students failing calculus. In the early twenty-first century, mathematicians continue to discuss and refine the calculus course.

Calculus—A Journey Through History

Even though counting as a process appears from the very first stages of humanity and its traces are lost in history of various civilizations, calculus was officially introduced as a realization of the deeper need to set rules and construct generally approved techniques that would assist toward quantification of any kind of change in time or space.

It could also perform modeling of systems that continuously evolve, and hence aid the interpretation and deduction of the consequences of the existence of such systems. Basic ideas of calculus involve limits, continuity, derivatives, and integrals.

Archimedes is one of the main scholars of ancient history who is linked with the ideas of calculus (c. second century b.c.e.). However, two important scholars of the seventeenth century made significant contributions to the introduction and the establishment of calculus as a quantitative language. Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) are both recognized as fathers of modern calculus. Even though they worked independently and were influenced by two different areas—Newton by physics and Leibniz by geometry—they both reached into discovering the same fundamental ideas of calculus.

Differential Calculus

If x is a variable that changes with time (for example, x=x(t) is a function of time t) then one denotes

the first derivative of the function x, which represents the rate of change of x(t) with respect to time t. Newton used the notation ẋ while Leibniz used

In case of a moving object in one dimension, x(t) represents the position of the object and ẋ(t) its velocity.

The term “function” was first introduced by Leibniz and is one of the fundamental terms of mathematics. In practice, a quantity y is defined as a function of another quantity t if there is a rule (method or process) in which a unique y will be assigned to any t. Leonhard Euler (1707–1783) introduced the notation y=f(t) to identify a function f.

“Method of fluxions” is the term Newton used for his set of techniques to study the continuous flow of change. The process of finding a formula for the function x, given the formula for the function of x, is known as “differentiation” and the methods used for this belong to the field of differential calculus.

Rate of Change

There is a particular interest in studying the change of a quantity and by extension the function rate of change of a quantity as another quantity changes in a very small amount. As Newton and Leibniz were developing calculus, they both used “infinitesimals” in order to emphasize the idea of such a small quantity that is not zero and that cannot be measured (“infinitesimal calculus” or “calculus of infinitesimals”). Hence, the infinitesimal number dx was considered to be different from zero and less than any positive real number.

Their approach raised criticism among other well-known scholars such as George Berkeley (1685–1783), and the idea of using infinitesimals became gradually unpopular. The introduction of “limits” from Augustin Louis Cauchy (1789–1857), Karl Theodor Wilhelm Weierstrass (1815–1897), and Georg Friedrich Bernhard Riemann (1826–1866) led in a better realization of the fundamental ideas of calculus and reestablished the topic within a more sound framework. However, nonstandard analysis (Abraham Robinson, 1960) and smooth infinitesimal analysis as introduced in the twentieth century have brought back into use the idea of infinitesimals.

Limit

The definition of limit is a cornerstone for advanced mathematics and especially for mathematical analysis. Limit is what distinguishes calculus from other areas of mathematics, such as algebra, geometry, and number theory. Even though mathematics has a history of more than 3000 years, limits were treated as a special area of mathematics only from 1823 c.e. when the concept was published for the first time in Cauchy’s book Résumé of Lessons of Infinitesimal Calculus. The first appearance of the term belongs to the Greek mathematician and philosopher Zeno of Elea (495–435 c.e.). However, the definition that was finally accepted and used by the mathematical community is the (ɛ, δ) definition as stated by Weierstrass.

Weierstrass Definition

Assume that L is a real number and that f(x) is defined in an open interval where x₀ belongs. Then the limit of f(x) as x tends to x₀ is equal to L and is denoted with

if the following is true: for any real number ɛ, there exists another real number δ such that for all x in the interval x₀-δ≤x≤x₀+δ the value f(x) of f lies within the range from L- ɛ to L+ ɛ.

In terms of infinitesimals, the limit is defined as follows: L is the limit of f(x) as x tends to x₀ and is denoted as

if the following is true: for any infinitesimal number dx, the value of f(a+dx) is finite, and the standard part of f(a + dx) equals to L.

Equation of Tangent

The term “derivative” as introduced from Newton and Leibniz signified a new era in mathematics. The term assisted mathematicians in finally solving rigorously the problem of constructing a unique tangent passing from a point of a curve. Historically, mathematicians since Archimedes’ period were constantly trying to solve the problem of a unique tangent on a point of a curve. Ancient Greeks believed first that the tangent at a point of a circle should be the line that passes from the particular point and is vertical to the radius of the circle. Archimedes devoted a significant part of one of his books to this specific problem, which is known as “Archimedean spiral.” However, it was because of the introduction of the first derivative that the researchers could actually provide the equation for the tangent line of a curve C:y=f(x) at a point (x₀, f(x₀)) as

is the rate of change of the function at that point

corresponds and defines the slope of the line tangent to the curve C at point

if and only if the limit exists.

Higher Order Derivatives

Thinking of the example of a moving object in time, it can be easily identified that there is a need for estimating the acceleration of the object. Acceleration is the rate at which an object changes its velocity. Therefore, acceleration in mathematical terminology is nothing else but the derivative of the derivative of x denoted

and called “second derivative.” Since the first derivative provides information on the rate of change of a function, the second derivative refers to the rate of change of the rate of change. In general, a higher order derivative is denoted as

In a more geometric framework, the first and second derivatives can be used to determine the concavity; in other words, the way that the slopes of tangent lines of a curve y=f(x) change as x changes in an interval (a,b).

If f is a differentiable function and f^′ is increasing on (a,b), then f is concave up on (a,b). The slopes of the tangent lines of the graph of f increase as x increases over (a,b); a concave up graph looks like a right-side up bowl.

If f^′ is decreasing on (a,b), then f is concave down on (a,b). The slopes of the tangent lines of the graph of f decrease as x increases over (a,b); a concave down graph looks like an upside-down bowl.

Points where the concavity changes are known as inflection points of f. Given that a function is increasing throughout an interval, if its first derivative is positive throughout the interval and vice versa, and given that f^′ is differentiable, then the following can be obtained: If

for all x in (a,b), then f^′ is increasing on (a,b) and therefore f is concave up on (a,b). If

for all x in (a,b), then f^′ is decreasing on (a,b) and therefore f is concave down on (a,b). A natural application of this concept is to find the maximum or the minimum of a function in a case in which the function is concave down or concave up throughout the whole domain respectively. This can be used further to solve problems where an optimal solution is requested.

According to Hans Hahn (1879–1934), the fundamental problem of differentiation can be expressed by two problems: (1) if the path of a moving object is known, estimate its velocity, and (2) given the existence of a curve, estimate its slope. Therefore, the inverse of these problems are (1) if the velocity of a moving object at every instance is known, estimate its path, and (2) if the slope of a curve is known, find the curve.

Integral Calculus

Generally, the process to find a formula for a function of x given the formula for the derivative of the function of x is known as “integration” and the methods used to find the formula belong to the field of integral calculus.

Historically, integral calculus was motivated by the geometric problem of estimating the area of a region inxy-plane bounded by the graph of f, the x-axis, and the vertical linesx=a and x=b. The solution of this problem came as a realization for the need of integral calculus and is linked with

which is known as the “definite integral.”

It is not known exactly for how long the aforementioned problem troubled the mathematical world. In 1858, Alexander Henry Rind (1833–1863), an Egyptologist from Scotland, discovered parts of a handwritten papyrus document that is considered to have been written in 1650 b.c.e.The Rind Papyrus, as it is known today, consisted of 85 problems by the Egyptian scribe Ahmes, who claimed that he had copied these problems from an older document. Problem number 50 indicates that before 4000 c.e., Egyptians knew how to compute the area of a circle by using the formula Area=3.16×radius ².

Eminent interest toward computations of areas of regions bounded by different kinds of curves is also seen in ancient Greece. Archimedes, whom several scholars consider as the “father of integral calculus” because of his method to estimate that the area bounded from the parabola y=x² and the rectangular lines x=1 and y=0 would be equal to 1/3. His method, which is known as the “method of exhaustion,” was an attempt to approximate the area of a curve by inscribing first in it a sequence of polygons and computing afterward their area, which must converge to the area of the containing curve. However, this method was first developed by Eudoxus; Archimedes just applied this method in order to establish the said area. This method was later generalized in what is known now as “integral calculus.”

The fundamental problem of integration focuses on finding the actual function (or, equivalently, its indefinite integral) if the derivative of the function is known.

Assume that function f exists. If there is a function F: y = F(x) such that

then f is called the indefinite integral or antiderivative of f and it is denoted as

where I stands for the first letter of the word “integral.”

Cauchy was most probably the first mathematician who provided a rigorous definition for the integral by using the limit of a sum. Riemann, later on, influenced by the theory of trigonometric series of the form

continued Cauchy’s work and defined the integral in a similar way, with the only difference that he studied the whole family of functions that can be integrated—functions for which the integral exists. During Cauchy and Riemann’s period, mathematicians were mainly concerned with integrating bounded functions. However, the need for integrating functions that cannot be bounded was soon apparent. Carl Gustav Axel Harnack (1883) and Charles De La Vallée-Poussin (1894) were among the first mathematicians to be occupied with such a problem. However, Henri Léon Lebesgue (1875–1941) is the one who, with his Ph.D. thesis titled “Integral, Length and Area” published in 1902, brought integral calculus into a new level. He defined the Lebesgue integral, which is a generalization of the Riemann integral, and defined a new measure known today as the Lebesgue Measure, which extends the idea of length from intervals to a large class of sets.

Other important scholars whose names are tightly linked with the development of modern calculus are Frigyes Riesz (1880–1956), Johann Radon (1887–1956), Kazimierz Kuratowski (1896–1980), and Constantin Caratheodori (1873–1950). They succeeded in generalizing and extending even further Lebesgue’s work.

The symbol ʃ, which is used for integration, is a big S (the first letter of the German word summe, meaning “sum”) and was used for the first time by Leibniz. There are several theories regarding the origin of the symbol. F. M. Turrell has supported the theory that almost every botanist knows that if an apple is peeled by hand, and, with the help of a knife, starting from the stem and continuing in circles around the central axis without cutting off the apple skin until the opposite end is reached, then a spiral is produced that creates an extended S once placed on the top of a horizontal surface with the inner part of the skin facing upward. This observation, according to Turrell, could possibly explain the symbol of integration. Finally, the Greek letter Ʃ is strongly linked with ʃ as Euler used it to denote a sum.

The fundamental theorem of calculus asserts that differentiation and integration are inverse problems. 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

This realization has proved to be a very useful technique to estimate definite integrals in an algebraic way. Isaac Barrow, Newton, Leibniz, and Cauchy worked on the concepts and early proofs, and Riemann and Vito Volterra explored what conditions on functions were necessary in the theorem. Lebesgue’s definition of integrals avoided some of the previous problems.

Probability theory and statistics are disciplines that use calculus. A valuable application is to determine the probability of a continuous random variable from an assumed density function and define the average of the variable and a range of variation around it. The basic method used to approach the underpinning problems is to find the area under the corresponding curves (compute an integral).

For the study of joint distributions of several random variables (multivariate distributions), students and researchers need to be familiar with the fundamental ideas of multidimensional calculus. Optimization in statistics is another area where calculus is significant; when, for instance, there is a demand to find an estimator of an unknown parameter that satisfies an optimality criterion, such as minimum variance.

Other Types of Calculus

Other calculi that are linked strongly with the undergraduate and postgraduate curriculum, indicating the broadness of the topic, are vector calculus and calculus of variations.

• • Stochastic calculus is tightly linked with financial calculus. It is mostly found in higher levels of mathematical education as it requires knowledge of measure theory, functional analysis, and theory of stochastic processes.
• • Malliavin calculus or stochastic calculus of variations was initiated by Norbert Wiener (1894–1964) in an attempt to provide a probabilistic proof of Hörmander’s “sum of squares” theorem. It is an infinite-dimensional differential calculus on a Gaussian space with features that can be applied in a wide variety of advanced topics of stochastic analysis. Its development has enormously facilitated the study of stochastic differential equations where the solution is not adapted to the Brownian filtration.
• • Quantum and quantum stochastic calculus, which have gained the interest of quantum mechanics specialists, use infinitesimals rather than limits.
• • π-calculus and λ-calculus offer a simpler syntax, which is highly appreciated by those in computing, offering an easier development of the theory of programming languages: network calculus and operational calculus.

Calculus, with its all variations, can be characterized as the mathematical language that unifies science by linking different disciplines together; this is why it plays a central role in the mathematical curriculum with students exposed to its basic ideas from the high school level.

The appreciation of the influence of calculus upon the vast majority of disciplines promotes a simultaneous intuitive approach by providing sufficient examples that illustrate the applicability of the topic. Modern technology in the form of computers and graphical calculators provides the tools that can assist not only in applying the mathematical techniques but also in a smooth transmission of the scientific ideas and basic mathematical concepts.

Bibliography

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Ganter, S. L. Changing Calculus. Washington, DC: Mathematical Association of America Notes #56, 2001.

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Rassias, John M. Geometry, Analysis and Mechanics. Singapore: World Scientific Publishing, 1994.

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