Master of Science in Mathematics
Lecture Notes. Accessed on: 2019-10-21 17:53:13
Calculus In Its Simplest Terms: Introduction
Calculus might simply be described as a set of clever tricks. That may seem a bit glib, but it’s not far off! You might lose your money on the street when the person operating the shell game shows you the impossible. The ball appears under a different shell, and you have fallen for a clever trick. Is it any less impossible to describe how something is changing. . . during an instant? Is it any less clever to effortlessly add up an infinite number of things?
Calculus addresses these seemingly impossible tasks. The way it was discovered was nothing short of clever! From the beginning, mathematicians used infinitesimally small quantities to work through the two major problems above. Differential calculus, which studies changes, deals with the first problem. Integral calculus, on the other hand, works with the latter problem, the addition of infinite things.
We will also use this article to look at the history of discoveries in calculus. If we stick with the magic analogy, calculus moved on from the shell game to tricks like sawing a person in half, or making planes disappear. In calculus, as with magic, the later tricks operate on many of the same principles as the very first ones. Discoveries came from necessity, and sometimes from sheer imagination. Nonetheless, all of them come back to the integral, and the derivative.
What is Differential Calculus?
Differential calculus is the study of change. Many systems have measurable ways in which they change over time. For example, we could study populations in an ecosystem, the stress on a baseball bat as it hits a ball, or the motion of a comet through space. Differential calculus gives us tools to study how any of those change over time.
We could graph any of the quantities above. The graph could show us the value of any of the variables at any time (populations, stress, motion). The change in those variables is just the slope of the graph. Differentiation allows us to take the equations for these relationships, and to find an equation for their slopes.
Sometimes we have the laws in place to predict those changes. Gravity affects a comet in a very predictable way, for example. Calculus gives us ways to understand how the speed, position, and acceleration of the comet change as it orbits.
The stress on a baseball bat is more complex. Equations still exist which help us understand how things might look at any moment. They may depend on more than just time. The hand placement on the bat, the speed of the ball, and the contact point on the bat all have their own effect on the stress in the bat. Different systems have different equations to describe them.
Much of the work of calculus in the last 300-400 years has been the search for new methods to differentiate more and more challenging equations. Today scientists, engineers, economists, and mathematicians have a wide range of tools at their disposal for studying and predicting change. There are times, however, when the challenge is simply to find the equations to describe their observations. At times like these, they turn to integral calculus.
What is Integral Calculus?
Much like addition and subtraction achieve opposite goals, integrating and differentiating are opposites. If you find the derivative of a function, you get a new function. Integrate the new function, and you are back at your original function. There’s a disclaimer on that last sentence, but it is close enough to true! This means that there might be times where the way something changes may be the key to finding the equation that describes it.
More often, integral calculus is used to add things up. Total distance traveled, total money earned, or even simple things like area. Each of these can be viewed as being made up of tiny little pieces. Integral calculus gives us the tools for adding up those pieces.
Historically, this seemed impossible, as the prospect of adding up an infinite number of pieces seemed daunting to say the least. As with differential calculus, many methods have been found for integrating a wide array of complex functions.
Countless Professions Every Day
Some of those applications are purely geometrical. For example, we may want to find the surface area of an object with curved faces. Perhaps we instead want the volume of a bowl that is a section of a hemisphere, or the area between two curves. These are all classic integration problems. Not surprisingly, they spring up in countless professions every day.
In many situations, the area under a curve has some substantial meaning, and integral calculus gives us a way to find it. For velocity versus time graphs, this area is equal to the distance traveled. It may be easiest to see this with a constant velocity, where the graph just forms a horizontal line.
If we look at the graph of a during the first ten seconds, the area is equal to ten times the velocity, $A=vt$. That area is equal to the distance traveled during those ten seconds. With more complex functions, the math is more complex, but the meaning of the area under the curve remains the same.
Brief History of Calculus
The 17th century gave birth to calculus when mathematicians hit a brick wall while trying to solve problems in geometry, astronomy, and physics. Two names in particular arise when we talk about the new methods for solving those problems, Newton and Leibniz.
These contemporaries were the center of a heated debate, as others worked to decide who deserved credit for the achievement. Newton laid out the foundations of calculus first, but did so in notebooks that would not be published for many years. Leibniz independently matched Newton’s feat, and published his work in 1684.
The heat of this debate matched the usefulness and beautiful simplicity of these new methods. The motions of planets, and the laws that describe them had taken Kepler 20 years to determine. They now resided in an appendix in Newton’s Principia Mathematica.
At first, the applications appeared only in those fields that pushed these pioneers to develop calculus. Astronomy, physics, and pure mathematics advanced by leaps and bounds. This changed quickly. From the late 17th century through the early 19th century, Euler, Lagrange, Legendre, and several members of the prodigious Bernoulli family found new methods and new applications in calculus. Fluid mechanics, engineering, acoustics, and various optimizing problems all saw new discoveries in this period, thanks to calculus
The 19th century saw increased formalization of calculus, in its methods, definitions, and proofs of its theorems. Advances continued, with one of the greatest coming in physics. Using calculus, James Clerk Maxwell elegantly expressed the laws governing electricity and magnetism in 1861 and 1862.
The application of calculus to the fundamental forces of nature continued through the 20th century, where it was the underlying language for quantum theory. Today, it is almost impossible to think of new discoveries happening in the world of physics without calculus.
What Is the Purpose of Calculus?
For the past half-century, these applications have become commonplace fixtures in numerous fields of work. The ability to produce models gives calculus its greatest value. There are times where things may be too large, too small, too far away, or otherwise inaccessible to study directly. Being able to produce models of these kinds of systems helps us to study and understand them. At other times, mathematical modeling can save enormous research costs. We find calculus at the center of many of these models.
Civil Engineers use calculus as they model a bridge design. The distribution of weight, considerations with traffic and temperatures, and stability in windy conditions all have to be considered. Optimizing those needs, while also considering material costs can quickly become a complex problem where calculus is at the heart of the solution.
As another example, aeronautical engineers rely on calculus-based models as they plot out the course for spacecraft headed to Mars. They program for changes in flight trajectories, making adjustments to unimaginable degrees of precision.
Economists also put calculus to use in ways that Newton and Leibniz may not have foreseen. The day to day applications, however, often fall right in with the opening chapters of a calculus text. How much would it cost to make one more of a product? How about one more after that? It can again become an optimizing problem, where they are trying to minimize costs, and maximize profits.
Why Is Calculus Important in Modern Life?
Some of the most interesting applications of calculus might not be laid out in full in your calculus text. Not yet! Here we will take a peek at some of the surprises that have come from more recent decades.
Calculus was, of course, used in programs that sent exploratory spacecraft to Mars. The surprise here, however, lies with where the engineers applied calculus. For the Insight lander, one of the decisions to be made was where to have the spacecraft land. You could easily imagine how interesting it would be to sit in on a discussion of that question.
Interestingly, one of the biggest contributors to this conversation was not carbon-based, but silicon-based! Computer programs were designed to consider various parameters. Paths to points of interest were mapped out, with line integrals determining the amount of energy needed to travel from point to point over the varied Martian landscape.
Programs quantified safety considerations, mapping them out. Then, just as Newton and others used calculus to find the optimal firing angle of a cannon, coders used calculus to provide a small number of optimal landing sites. People traditionally tackled the early stages of this decision-making before moving on to computer models. By contrast, they now apply calculus in models at the front end of the decision-making.
Search engines use a wide range of mathematical methods to find the page you want. The Internet itself, as large as it is, presents only part of the challenge. The search word combinations are infinite, or nearly so. Nonetheless, search engines find our destinations in a flash.
Consider “calculus” as a simple example. At what point did your search engine decide that you were more likely to need this site than a dental exam? Surprisingly, search engines use vector calculus to navigate you through the web.
Calculus Problems: Calculus Skill is Desired
How many of today’s composers spent time playing scales? How many of today’s great basketball players spend time perfecting their free throw? The work of attacking problem sets and developing skill is just as vital to the mathematician as the practice done by these other professionals.
Knowledge of calculus, the definitions, laws, theorems, and methods is only a starting point. Practice with problems can help you to move from memorizing a method to mastering it. Additionally, occasional failure inspires innovation and persistence. Michael Jordan bragged about the number of free throws he missed in his career. Why not roll up your sleeves and miss some problems?
Over time, just as a quarterback recognizes defensive schemes, and decides on a course of action, you’ll learn to recognize the different kinds of problems and the methods for solving them that are most successful. In that sense, these skills should not be the end goal of a calculus course. There is value in developing the insight needed to decide which skills to use in novel situations.
Even beyond this stage of comfortable, fluid application of calculus, there is the need for innovation. With an understanding of what can be done with calculus, the world needs creative minds hard at work finding new applications! As stated above, all of this only becomes possible with practice.
More than Equations
Calculus is extremely practical. We rely on it in our day-to-day life more than, say, literature, or art. It’s easy to lose sight of it as a search for beauty and truth in the world. This might be especially true during your nth hour working through a problem set! But beauty and truth are certainly found in all of mathematics.
A short way into a calculus course, you’ll make a realization. A vast majority of these methods take their names from mathematicians. Naturally, this reminds us that each new method shares a truth discovered by real people. It’s more than words and symbols in a book. We hope that this introduction helps you connect to this rich history.