Calculus

Summary

Calculus is the study of functions and change. It is the bridge between the elementary mathematics of algebra, geometry, and trigonometry, and advanced mathematics. Knowledge of calculus is essential for those pursuing study in chemistry, engineering, medicine, and physics. Calculus is employed to solve many optimization problems. One example is the so-called least squares solution method commonly used in statistics and elsewhere. The least squares function best fits a set of data points and can then generalize or predict results based on that set.

Definition and Basic Principles

Calculus is the study of functions and their properties. Calculus investigates a function according to two essential ideas: rate of change and total change. These concepts are linked by their common use of calculus's most important tool, the limit. It is the use of this tool that distinguishes calculus from elementary branches of mathematics, such as algebra, geometry, and trigonometry. In elementary mathematics, one studies problems such as, “What is the slope of a line?” or “What is the area of a parallelogram?” or “What is the average speed of a trip that covers three hundred miles in five and a half hours?” Elementary mathematics provides methods or formulas that can be applied to find the answer to these and many other problems. However, how is the slope calculated if the line becomes a curve? What if the parallelogram becomes a shape with an irregularly curved boundary? What if one needs to know the speed at an instant and not as an average over a longer period?

Calculus answers these harder questions by using the limit. The limit is found by approximating the answer and then refining that approximation by improving it more. If there is a pattern leading to a single value in those improved approximations, the result of that pattern is called the limit. Note that the limit may not exist in some cases. The limit process is used throughout calculus to answer questions elementary mathematics cannot handle.

The derivative of a function is the limit of average slope values within an interval as the length of the interval approaches zero. The integral calculates the total change in a function based on its rate of change function.

Background and History

Many consider calculus to have been developed in the seventeenth century, but its roots were formed much earlier. In the sixteenth century, Pierre de Fermat did work closely related to calculus's differentiation (the taking of derivatives) and integration. In the seventeenth century, René Descartes founded analytic geometry, a key tool for developing calculus.

However, Sir Isaac Newton and Gottfried Wilhelm Leibniz share the credit as the (independent) creators of calculus. Newton's work came first but was not published until 1736, nine years after his death. Leibniz's work came second but was published first in 1684. Some accused him of plagiarizing Newton's work, although Leibniz arrived at his results by using different, more formal methods than Newton employed.

Both men found common rules for differentiation, but Leibniz's notation for the derivative and the integral are still in use. In the eighteenth century, the work of Jean le Rond d'Alembert and Leonhard Euler on functions and limits helped place the methods of Newton and Leibniz on a firm foundation. In the nineteenth century, Augustin-Louis Cauchy used a definition of limit to express calculus concepts in a form still familiar more than two hundred years later. German mathematician Bernhard Riemann defined the integral as a limit of a sum, the same definition learned by calculus students in the twenty-first century. At this point, calculus, as is taught in the first two years of college, reached its finished form.

How It Works

Calculus is used to solve a wide variety of problems using a common approach. First, one recognizes that the problem cannot be solved using elementary mathematics alone. This recognition is followed by an acknowledgment: there are some things known about this situation, even if they do not provide a complete basis for solution. Those known properties are then used to approximate a solution to the problem. This approximation may not be very good, so it is refined by taking a succession of better and better approximations. Finally, the limit is taken, and if the limit exists, it provides the exact answer to the original problem.

One speaks of taking a limit of a function f(x) as x approaches a particular value, for example, x = a. This means that the function is examined on an interval around, but not including x = a. Values of f(x) are taken on that interval as the varying x values get closer and closer to the target value of x = a. There is no requirement that f(a) exists, and many times it does not. Instead, the pattern of functional values is examined as x approaches a. If those values continue to approach a single target value, it is that value that is said to be equal to the limit of f(x) as x approaches a. Otherwise, the limit is said not to exist. This method is used in both differential calculus and integral calculus.

Differential Calculus. Differentiation is a term used to mean the process of finding the derivative of a function f(x). This new function, denoted f¢(x), is said to be “derived” from f(x). If it exists, f¢(x) provides the instantaneous rate of change of f(x) at x. For curves (any line other than a straight line), the calculation of this rate of change is not possible with elementary mathematics. Algebra is used to calculate that rate between two points on the graph, then those two points are brought closer and closer together until the limit determines the final value.

Shortcut methods were discovered that could speed up this limit process for functions of certain types, including products, quotients, powers, and trigonometric functions. Many of these methods go back as far as Newton and Leibniz. Using these formulas allows one to avoid the more tedious limit calculations. For example, the derivative function of sine x is proven to be cosine x. If the slope of sine x is needed at x = 4, the answer is known to be cosine 4, and much time is saved.

Integral Calculus. A natural question arises: If f¢(x) can be derived from f(x), can this process be reversed? In other words, suppose an f(x) is given. Can an F(x) be determined whose derivative is equal to f(x)? If so, the F(x) is called an antiderivative of f(x); the process of finding F(x) is called integration. In general, finding antiderivatives is a harder task than finding derivatives. One difficulty is that constant functions all have derivatives equal to zero, which means that without further information, it is impossible to determine which constant is the correct one. A bigger problem is that there are functions, such as sine (x²), whose derivatives are reasonably easy to calculate but for which no elementary function serves as an antiderivative.

The definite integral is an attempt to determine the amount of area between the graph of f(x) and the x-axis, usually between a left and right endpoint. This cannot typically be answered using elementary mathematics because the shape of the graph can vary widely. Riemann proposed approximating the area with rectangles and then improving the approximation by having the width of the rectangles used in the approximation get smaller and smaller. The limit of the total area of all rectangles would equal the area being sought. This notion gives integral calculus its name: By summing the areas of many rectangles, the many small areas are integrated into one whole area.

As with derivatives, these limit calculations can be quite tedious. Methods have been discovered and proven that allow the limit process to be bypassed. The crowning achievement of the development of calculus is its fundamental theorem: The derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit; the value of a definite integral is the difference between the values of an antiderivative evaluated at the limits. If one is looking for the definite integral of a continuous f(x) between x = a and x = b, one need only find any antiderivative F(x) and calculate F(b) − F(a).

Applications and Products

Optimization. A prominent application of differential calculus is in the area of optimization, either maximization or minimization. Examples of optimization problems include: What is the surface area of a can that minimizes cost while containing a specified volume? What is the closest that a passing asteroid will come to Earth? What is the optimal height at which paintings should be hung in an art gallery? (This corresponds to maximizing the viewing angle of the patrons.) How shall a business minimize its costs or maximize its profits?

All of these can be answered using the derivative of the function in question. Fermat proved that if f(x) has a maximum or minimum value within some interval, and if the derivative function exists on that interval, then the derivative value must be zero. This is because the graph must be hitting either a peak or the bottom of a valley and has a slope of zero at its highest or lowest points. The search for optimal values then becomes the process of finding the correct function modeling the situation in question, finding its derivative, setting that derivative equal to zero, and solving. Those solutions are the only candidates for optimal values. However, they are only candidates because derivatives can sometimes equal zero even if no optimal value exists. What is certain is that if the derivative value is not zero, the value is not optimal.

The procedure discussed here can be applied in two dimensions (where there is one input variable) or three dimensions (where there are two input variables).

Surface Area and Volume. If a three-dimensional object can be expressed as a curve that has been rotated about an axis, then the surface area and volume of the object can be calculated using integrals. For example, Newton and Johannes Kepler studied the problem of calculating the volume of a wine barrel. If a function can be found that represents the curvature of the outside of the barrel, that curve can be rotated about an axis, and pi (p) times the function squared can be integrated over the length of the barrel to find its volume.

Hydrostatic Pressure and Force. The pressure exerted on, for example, the bottom of a swimming pool of uniform depth is easily calculated. The force on a dam due to hydrostatic pressure is not so easily computed because the water pushes against it at varying depths. Calculus discovers the answer by integrating a function found as a product of the pressure at any depth of the water and the area of the dam at that depth. Because the depth varies, this function involves a variable representing that depth.

Arc Length. Algebra can determine the length of a line segment. If that path is curved, whether in two or three dimensions, calculus is applied to determine its length. This is typically done by expressing the path in parametric form and integrating the function representing the length of the vector that is tangent to the path. The length of a path winding through three-dimensional space, for example, can be determined by first expressing the path in the parametric form x = f(t), y = g(t), and z = h(t), in which f, g, and h are continuous functions defined for some interval of values of t. Then, the square root of the sum of the squares of the three derivatives is integrated to find the length.

Kepler's Laws. In the early seventeenth century, Kepler formulated his three laws of planetary motion based on his analysis of the observations kept by Tycho Brahe. Later, calculus was used to prove that these laws were correct. Kepler's laws state that any planet's orbit around the sun is elliptical, with the sun at one focus of the ellipse; that the line joining the sun to the planet sweeps out equal areas in equal times; and that the square of the period of revolution is proportional to the cube of the length of the major axis of the orbit.

Probability. Accurate counting methods can be sufficient to determine many probabilities of a discrete random variable. This would be a variable whose values could be, for example, counting numbers, such as 1, 2, 3, and so on, but not numbers in between, such as 2.4571. If the random variable is continuous so that it can take on any real number within an interval, then its probability density function must be integrated over the relevant interval to determine the probability. This can occur in two or three dimensions.

One common example is determining the likelihood that a customer's wait time is longer than a specified target, such as ten minutes. If the manager knows the average wait time that a customer experiences at an establishment is, for example, six minutes, then this time can be used to determine a probability density function. This function is integrated to determine the probability that a person's wait time will be longer than ten minutes, less than three minutes, between five and thirteen minutes, or within any range of times that is desired.

Careers and Course Work

A person preparing for a career involving the use of calculus will most likely graduate from a university with a degree in mathematics, physics, actuarial science, statistics, or engineering. In most cases, engineers and actuaries can join the profession after earning their Bachelor's degree. For actuaries, passing one or more of the exams given by the Society of Actuaries or the Casualty Actuarial Society is also expected, which requires a thorough understanding of calculus. In statistics, a Master's degree is typically preferred, and to work as a physicist or mathematician, a Doctorate is the standard. In terms of calculus-related coursework, in addition to the calculus sequence, students will almost always take a course in differential equations and perhaps one or two in advanced calculus or mathematical analysis.

Social Context and Future Prospects

Calculus itself is not an industry, but it forms the foundation of other industries. In this role, it continues to power research and development in diverse fields, including those that depend on physics. Physics derives its results by way of calculus techniques. These results, in turn, enable developments in small- and large-scale areas. An example of a small-scale application is the ongoing development of semiconductor chips in the electronics field. Large-scale applications in the solar field and space physics are critical for ongoing efforts to explore the solar system and beyond. These are just two examples of calculus-based fields that will continue to have a significant impact in the twenty-first century. While calculus will undoubtedly continue to play a role in the engineering and physics fields, it can also be used to understand trends and risks in the financial markets, play a role in the medical field through its use in MRI (magnetic resonance imaging) and CT (computed tomography) scans, and support algorithms created by artificial intelligence in computer science. Calculus can also continue to aid architects in creating safe and durable structures and allow environmentalists to form accurate climate models.

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