Calculus – the epitome of math anxiety, distaste and fear – well at least for the majority of students entering senior year in high school or the fresh college admits. The truth is, most of us develop a strong phobia of calculus even before we’ve come to know it. Such is derived from a variety of misinformed lessons that have clouded our ability to view calculus objectively without breaking a sweat. To reverse such a student’s state to Calculus, there are timeless techniques to approach this such as these 5 Simple Tricks Parents Use To Ease Their Kid’s Maths Anxiety.
One classroom failure that has perpetuated premature negativity towards calculus is the method math has been taught. While educational reform has attempted to redeem students from the clutches of mathematical boredom through active learning strategies and the introduction of technology in math learning, math lessons in modern classrooms have barely progressed past traditional drills of worksheets, endless textbook exercises, chapter after chapter. The lack of excitement and mundane number crunching quickly spirals into a slippery slope, whereby the mind-numbering process results in the objectification of numbers as nothing more than mere numbers (which is true, but not definitive). Sadly by the time we’ve arrived at a place as exciting as calculus, there are no eager learners. Contemporary math teaching methods should include new alternative techniques as described in these 3 Ways Math Makes Games More Fun!
Another misconception that has festered student’s mind-sets from the beginning has its roots in a misconstrued and oversimplified psychology lesson. Nowadays it is far too common for students with an aversion for math to blame it the orientation of their brain – more specifically those that have more of a right brained orientation are quick to use this card in response to poor math grades, boredom and their career choices. While there certainly exists natural talent and specific inclinations, answering 31 questions and coming to a fixed conclusion regarding the inclination of your brain when you are 16 is a poor means to justify why you may not be good at math. This kind of wrongful conditioning however can really cripple one’s motivation, attitude, and ability to perform well in calculus.
With no guide to shed light on how the pages of mathematical textbooks translate outside the walls of a classroom, students leave, graduate and can even do exceedingly well without ever having a sense of wonder. Without an inkling of appreciation it is no wonder that when students come face to face with calculus, they cringe and/or crumble. While it is no secret that calculus will test the limits of any student, there is a fine line between hard and challenging. As a higher-level mathematical course that applies knowledge from all other foundational math topics (e.g. algebra, trigonometry, functions, matrices, statistics and probability), this amalgamation is bound to be both challenging and difficult. Such should be anticipated, and be met with the right expectations and work ethic. That being said, we’re not here to give you another lecture, we’re here to see why you should start caring more about calculus. Let us enlighten you about the utility and ubiquity of calculus and help you embrace it with more curiosity and confidence than ever before.
What is Calculus?
Over the course of centuries, mathematicians have investigated and refined calculus to what we now learn today. For the more acquainted student, calculus can be defined as the study of limits, derivatives, and integrals of functions. Broadly speaking, this encompasses the study of rates of change and following consequences that are brought about in a system at an infinitesimal scale. In fact, the term calculus is latin for “small pebble used for counting”, highlighting the approach this branch of mathematics, studies the real world.
Calculus can be divided into two major arms – differential calculus and integral calculus. Differential calculus focuses on solving derivatives of functions to discover the rate of change and behaviour of a system by studying changes on an infinitesimal scale. On the flipside, integral calculus studies the entirety of a system, using anti-differentiation methods to collate infinitesimal quantities, this allows the system to be viewed as a whole so that changes to the system can be analysed even when infinitesimal changes are made.
Simply put, calculus is the mathematical skill that enables one to analyse changes to a system. By studying these changes, we can make predictions and pre-emptively anticipate and control both desirable and undesirable changes to a system. With the power of calculus, mathematical modelling, and computing advancements, optimal models of real world systems can be fashioned and tested so that we can sift through the endless possibilities and spend more time and resources on the functional systems.
Where do we see Calculus?
Calculus is a universal language that can be applied to a multitude of disciplines. But before we blow your minds, lets tackle some of the applications you may have covered in school because the truth is there is an obligation for most curricula to develop lessons of calculus that are focused purely on applications of calculus.
One example of this is the study of position, velocity and acceleration. Some of you may recall this from calculus 1 when dealing with derivatives of functions to calculate the slope and/or instantaneous rate of change. Taking this one step further, imagine if we were apply thing this to a function that mirrored the position of a moving object? The derivative would give us information on the velocity of the moving object and the consequent derivative would tell us the acceleration of an object. Now while you may recall this application as a random dot moving across a straight line, imagine the utility of this when applied to NASCAR, Formula 1, a prototype aircraft or testing of automobile safety.
Another application you may be familiar with is the use of differential calculus in optimization. Using critical numbers, maximum and minimum numbers, we can determine the optimum numbers to maximize profits while minimizing costs. Whether this is in the form of meeting sales targets of an independent café to break even, maximizing the area required for the construction of a new enclosure for pandas, or the redesign of a Toyota corolla in 2018 using the same material budgets as 2017, this calculus concept is incredibly valuable for many who harness it.
While these concepts may sound familiar to you, some of these examples may sound foreign to you. We don’t blame you – it was unlikely that they were framed in such a practical and interesting manner. Often times the theory is emphasized but not the practical aspects. Most practical applications of this may have manifested in an ugly exam question, or a passing homework assignment. Furthermore none of these examples may be particularly relevant to you, so we get it if this fails to turn heads. So lets examine how calculus may actually play out in your life and future career –
Working with your enemy
Physicists are constantly using math to help others. A variety of formulas in fact are solved by way of calculus. Some examples include calculating the centre of mass, centre of gravity, and mass moment of inertia. These are all important variables not just to physicists, but also to numerous disciplines including auto-engineers, astronauts, and military operatives. Other fundamental concepts including the theory of relativity, harmonics and acoustics of sound, laws of light, momentum, gravity, electromagnetism, trajectory, position, velocity and acceleration, are best understood with the help of calculus.
Astronomy and Aerospace
The study of stars and intergalactic exploration would be impossible without the aid of calculus. Astronomers have leveraged calculus to determine the distance, velocity and gravitational force of stars, moons, planets, solar systems and asteroid belts. These calculations have helped create maps that have assisted the navigation of astronauts, informed flight trajectories and rocketship development, launch of satellites, and the unmanned missions to the nether regions of our galaxy. Intergalactic motion of planets and theories of universe expansion are also posited with the help of calculus. Closer to home, the latest and greatest commercial and military plane designs also find their answers in calculus. With mathematical modelling, hundreds and thousands of plane designs are filtered through with the help of computing power and algorithms that help minimize costs and time, while improving overall manufacturing/design efficiency.
Engineering
Engineers use calculus more often than other professions in the industry. Electrical engineers for example, rely on calculus to build and connect electrical towers and substations. Due to the curvature of hanging wires and the large distances these wires travel, calculus ensures there is precision in the development and construction and connection of these electrical infrastructures.
Civil engineers utilize calculus to determine fluid mechanics of systems. Some examples include storm basins, dams, and drainage waterways. Using calculus the rate of flow and total volume capacity can be determined. Such is important to prevent the potential damages of flooding, dam failures and the hazard of water contamination and shortage.
Mechanical engineers are often tasked with using calculus to design and create low-friction mechanical systems. Whether this is in the shape and form of combustion engines in vehicles, gas turbines, appliances such as refrigerators, escalators, elevators, robotics and other automated assisted technology (e.g. conveyor belt),
Athletics
While using math in sports may seem like an uncanny combination, the use of calculus can actually provide a substantial advantage when it comes to competitive sports. Using calculus, the movement and plays of players can be detected to refine technique, identify pattern of movement, and adapt accordingly (e.g. maximizing weight loss, maintaining streamlined form, developing counter strategies to competitor tendencies).
Architecture
If you love design, you will have to learn how to love calculus – Calculus is an integral part of architecture when it comes to the design of dome-shapes and the infrastructure with curves and bends (e.g. bridges). Using integration an architect can calculate the amount of material required for construction and how to design the support systems of these non-linear structures. With this ability, architects can play around with a variety of designs and push the boundaries of innovation to stand out and by creating unique structures that have the utmost structural integrity. Other general uses of calculus that architects find incredibly useful involve differentiation to determine how to maximize the use of a space. This can inform the interior designers and architects about the limits and sizes of workstations, the placement of windows, open spaces, and so forth.
Biology and Medicine
Having the skills and ability to use mathematics in Biology is becoming more and more desirable. Such is particularly paramount when it is related to the study of growth rates, birth rates, and death rates. Since growth rates are applicable for a multitude of organisms ranging from bacteria to birds, calculus enables the biologists to venture freely into different fields to ask questions and test hypotheses that help us understand the phenomena we observe today. It can also educate us on the potential threats that we can expect to face – for example, what conditions allow bacteria to exponentially grow and how can that be stunted. Another example calculus can inform Biologists is related to conservation, particularly improving growth rates of endangered animal species that would not otherwise survive or successfully compete.
Saving lives requires a strong command of calculus – In medicine, using calculus enables one to determine the concentration of drugs in living organisms, theoretically test out new drugs and remedies with high accuracy while avoiding intrusive and expensive trial and error experiments. These rapid developments and highly accurate math aids help epidemiologists and doctors to constantly be ready for the next outbreak or the next overdosed patient.
Economics
Businesses rely on calculus to survive! With calculus businesses can determine what marginal expenditures and marginal profits are needed to break even. On the other hand, calculus can inform businesses how to generate maximum profits and what strategies to adopt when faced with specific market environments. For those in the business of buying and selling stock shares, the black scholes equation utilizes calculus to help predict stock prices so that companies can determine the right time to either buy or sell shares.
Graphic Artists
Calculus helps create some of the most complex and realistic 3D visual effects seen in todays gaming and movie industry. Calculus helps model and adapt 3D models to be able to function in unprecedented environments and bring to life sketches and graphic renders. Without calculus we would still be trapped in a 8-bit world!
Chemists find calculus particularly important in determining the rate of reaction. Knowing the rate of reaction is important for the development of nuclear power and assessing the lifetime of wear and tear of products (e.g. car, electronics, building materials). With calculus, warranty assessments can be accurately deduced and the efficiency of nuclear power increased.
Meteorology
Ever wonder why Siri, Alexa and Google Home are almost always right with the weather predictions for tomorrow? They use calculus! While the bulk of the work is carried out by complex algorithms and modelling, calculus equations form the backbone of assessing all the variables that go into predicting rain, shine, overcast. Variables such as wind speed, humidity, atmospheric pressure, wind direction, are all examined and processed through calculus computing.
Loving thy enemy
So what are some strategies we can adopt to help us appreciate calculus a little bit more?
1. Start em young: This isn’t a call to math-bash you, or your child with volumes of calculus (that is likely to lead to calculus aversion), rather we recommend early exposure to the ideas of calculus, practicing introductory problems and concepts, and identifying the basic utility of calculus. A basic foundation of calculus is as important as a basic grasp of trig, and algebra. Familiarity aids not only the understanding but appreciation. With all that being said, know that its never too late to dive into Calculus – a solid precalculus course will prepare you for a more pleasant experience in Calculus after gaining a strong holistic understanding of all the skills you will require.
- Help students see connections: We’ve seen enough of examples where balls get thrown into the air, cars moving from point A to B – conjure excitement by being realistic in showing how calculus is engrained in a number of careers and relevant even to the mathematically challenged. It’s also important to shed light on the mathematical links by helping students identify that they are not going into calculus ill-equipped and former knowledge from other math courses continue to be enhanced and developed in calculus.
In hindsight, we encourage you to consider again whether calculus is a friend or foe. With a little more preparation, practice and practicality we hope you can err to find reasons to keep calculus in your life.