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.
What You’ll Learn in Applications of the Integral
- A Real-Life Interpretation
- The Area Between Two Curves
- Integrating with Respect to y
- What Happens When the Curves Intertwine?
- Solids of Revolution
- The Disk Method
- The Washer Method
- The Method of Cross Sections
- The Method of Cylindrical Shells
- Applying the Method of Cylindrical Shells
- Shells by Revolving About the x-axis
- Definition of Arc Length
- Length of a Smooth Curve
- The Arc Length Function
- Surfaces of Revolution
- Work Done by a Constant Force
- Work Done by a Variable Force
- Hooke’s Law
- Moving Nonrigid Matter
- Work Done by an Expanding Gas
- Fluid Pressure
- Force Exerted by a Fluid
- Measures of Mass
- Center of Mass of a System on a Line
- Center of Mass of a System in the Plane
- Center of Mass of Laminas
- The Theorem of Pappus
- The Graphs of the Hyperbolic Functions
- Hyperbolic Identities
- Inverse Hyperbolic Functions
- Derivatives and Integrals of Hyperbolic Functions
- Derivatives of Inverse Hyperbolic Functions
- An Application
This course is a continuation of the theory of integrals from calculus 1. So we begin this course with several applications of the integral. First up is the area between curves. We motivate and prove theorems that give us the flexibility of finding areas from different viewpoints. In doing so, we work through several examples that demonstrate each strategy.
After finding areas, we continue with finding volumes. We begin by explaining solids of revolution and then walk through (in detail) several methods for find volumes, including using disks, washers, cross-sections, and cylindrical shells.
We then turn to arc length and its generalization: areas of surfaces of revolution. We motivate these concepts and work through several examples and applications. In detail, we examine work, fluid pressure and force, and moments and center of mass.
At the end of this course, we study hyperbolic functions and some applications. We give a thorough treatment by defining these functions, examining their graphs, exploring fundamental identities, proving derivative formulas and derivative formulas of their inverses.
What You’ll Learn in Techniques of Integration
- The Method of Integration by Parts
- Reduction Formulas
- Integrals Involving Products of Sines and Cosines
- Integrals Involving Products of Tangents and Secants
- Integrals with Products of Cotan and Cosec
- Converting to Sines and Cosines
- Different Angles
- Using Trigonometric Substitution
- More Examples
- Partial Fractions
- Case 1: Distinct Linear Factors
- Case 2: Repeated Linear Factors
- Case 3: Distinct Irreducible Quadratic Factors
- Infinite Intervals of Integration
- Improper Integrals with Infinite Limits of Integration
- Improper Integrals with Infinite Discontinuities
- Improper Integrals Whose Integrands Have Infinite Discontinuities
- A Comparison Test for Improper Integrals
After studying the integral from calculus 1 and having seen some integral applications in calculus 2, we are now ready for an in-depth look at integration techniques. We begin this course with one of the most commonly used techniques, integration by parts. In the first lesson, we derive the formula and work through several examples of using it and when to use it.
In the next lesson, we cover trigonometric integrals. These are integrals that involve the product of trigonometric functions. In particular, the product of powers of sines and powers of cosines are detailed. Guidelines for several other forms are also detailed.
After understanding trigonometric integrals, we can now look for those integral substitutions that yield a trigonometric integral. We classify several types of substitutions here in this lesson and demonstrate how to use each one.
The method of partial fractions is explored next. We break this technique down into four cases: namely, distinct linear factors, repeated linear factors, distinct irreducible quadratic factors, and repeated irreducible quadratic factors. Each case is thoroughly discussed.
After partial fractions, we study improper integrals. We examine both infinite intervals of integration and improper integrals with infinite discontinuities. We end this fun course with a comparison test for improper integrals.
What You’ll Learn in Conic Sections, Plane Curves, and Polar Coordinates
- Conic Sections
- Reflective Property of the Parabola
- Reflective Property of the Ellipse
- Eccentricity of an Ellipse
- Shifted Conics
- Why We Use Parametric Equations
- Sketching Parametric Curves
- Tangent Lines to Curves
- Horizontal and Vertical Tangents
- Higher Order Derivatives
- The Length of a Smooth Curve
- The Area of a Surface of Revolution
- The Polar Coordinate System
- Relationship Between Polar and Rectangular
- Graphs of Polar Equations
- Tangent Lines to Graphs of Polar Equations
- Areas in Polar Coordinates
- Area Bounded by Two Graphs
- Arc Length in Polar Coordinates
- Area of a Surface of Revolution
- Points of Intersections in Polar Coordinates
- Polar Equations
- Eccentricity of a Conic
- Motion of Celestial Bodies
This course provides a detailed review of conic sections. We emphasize concepts and work through examples. After this review, we then focus on plane curves and parametric equations, including sketching curves defined by parametric equations.
Next, we begin a thorough exploration of the calculus of parametric equations. Finding derivatives and second derivatives, finding the length of a smooth curve, and finding the area of a surface of revolution are all detailed.
So far, in this course, we have concentrated on rectangular coordinates. Now here we examine conic sections and plane curves in polar coordinates. We explain symmetry, converting between coordinate systems, and sketching graphs. For example, tangent lines to graphs of polar equations, and areas and arc lengths in polar coordinates are covered. Kepler’s Laws are discussed at the end.
What You’ll Learn in Infinite Sequences and Series
- Introduction to Sequences
- Limit of a Sequence
- Limit Laws for Sequences
- Squeeze Theorem for Sequences
- Bounded Monotonic Sequences
- Infinite Series
- Convergence of Infinite Series
- Geometric Series
- The Harmonic Series
- The Divergence Test
- Properties of Convergent Series
- Integral Test
- The p-Series
- The Comparison Test
- The Limit Comparison Test
- The Alternating Series Test
- Absolute Convergence
- The Ratio Test
- The Root Test
- Approx the Sum of an Alternating Series
- Summary of Tests for Convergence
- Rearrangement of Series
- Introduction to Power Series
- Interval of Convergence
- Differentiation and Integration of Power Series
- The Taylor and Maclaurin Series
- Techniques for Finding Taylor Series
- The Maclaurin Polynomial
- Taylor’s Formula with Remainder
- Representing a Function by a Series
We start this interesting course off with a review of sequences. We discuss what sequences are, limits of sequences, limit laws for sequences, and the squeeze theorem for sequences. Next, we consider bounded monotonic sequences and then prove the monotone convergence theorem for sequences.
After this introduction to sequences, we begin a thorough investigation of infinite series. We discuss partial sums and convergence at length as well as geometric series and the harmonic series. Properties of convergent series and the divergence test are also detailed.
We then explore several convergence tests in detail, including the integral test, alternating series test, comparison tests, and the ratio and root tests. For each convergence test, we motivate why it is essential, how it works, and consider several examples.
After that, one of the main topics of calculus is explored: power series. We go into a great deal of attention on power series, including the Talyor and Maclaurin series. In the end, we examine approximation by Taylor polynomials, which is a generalization of linear approximation as studied in calculus 1.
What You’ll Learn in Vectors and the Geometry of Space
- Scalar Multiples
- Vector Addition: The Parallelogram Law
- Vectors in the Coordinate Plane
- Length of a Vector
- Vector Addition in the Coordinate Plane
- Properties of Vectors
- Unit Vectors
- Standard Basis Vectors
- Angular Form of the Unit Vector
- Coordinate Systems in Space
- The Distance Formula
- The Midpoint Formula
- Vectors in 3-Space
- Standard Basis Vectors in Space
- Finding the Dot Product
- The Angle Between Two Vectors
- Orthogonal Vectors
- Direction Cosines
- Vector Projections and Components
- The Cross Product of Two Vectors in Space
- Geometric Properties of the Cross Product
- Finding the Area of a Triangle
- Properties of the Cross Product
- The Scalar Triple Product
- Equations of Lines in Space
- Equations of Planes in Space
- Parallel and Orthogonal Planes
- The Angle Between Two Planes
- The Distance Between a Point and a Plane
- Quadratic Surfaces
- The Cylindrical Coordinate System
- The Spherical Coordinate System
In this course, we begin with an introduction to vectors. We discuss vectors in the plane, and we go into detail on vector addition, scalar multiplication, and the length of a vector. Many properties of vectors and unit vectors are also covered.
After this introduction to vectors, we begin working in three dimensions. Many of the concepts just introduced are implemented easily in 3-space. We describe the dot product and cross product, including vector projections, work, and finding areas. Many properties of both these operations are covered in detail with several examples.
After these studies on vectors, we discuss lines and planes in space. We thoroughly explain equations of lines in space, equations of planes in space, and several other topics such as the angle between two planes and the distance between a point and a plane.
After studying vectors, lines, and planes, we now investigate surfaces in space. We cover cylinders and various types of quadratic surfaces, including ellipsoids, hyperboloids, cones, and paraboloids.
Having explored rectangular coordinates so far in this course, we now turn our attention to other coordinate systems. The cylindrical coordinate system and the spherical coordinate system are detailed. We explain how these coordinate systems work and how to convert points and equations between them, including rectangular coordinates.
Explore Calculus 2 with these articles and videos on integration, infinite series, and other topics that explain (step-by-step) in great detail.
Calculus 2 is a mathematics course for high-school or undergraduate students. The prerequisite for this course is Calculus 1, as it is a direct continuation. This course’s main topics include applications of the definite integral, integration techniques, and a great deal about infinite sequences and series. Other topics that instructors sometimes have are analytic geometry in three dimensions with vector geometry and differential equations. This course is magnificent as students become skilled in integration and learn about improper integrals and Taylor Series.