If the mere sight of the word “calculus” scares you, you’re not alone. Just like algebra and trigonometry, this one’s a tough subject, and not everyone experiences the joy of advanced math “clicking” for them right away. You might never “love” calculus unlike some math-inclined folks out there; but at the very least, in the next five minutes you should be better able to understand calculus through some common real-world applications.
In tutoring thousands of students, we often hear students say that calculus feels like a far-off and abstract land, filled with chalkboards, mean teachers and confusing formulas. But actually, the subject of calculus is just another tool that you can wield as you please, like history or science or English. Calculus is used to describe and solve real world problems with extreme precision, so let’s look at a few common examples of how Calculus is far from being a purely theoretical discipline.
The Foundations of Calculus: More Than Just Numbers
Calculus is all about understanding how things evolve. So, imagine if you had to track the speed of a car, predict population growth, or analyze the trajectory of a rocket. Without calculus, the best analysis you could do is say, “it started going kind of fast, and then it really took off.” But with calculus, you can get down to hard numbers to paint a better picture. The first two branches of calculus, differential calculus and integral calculus, give us two different ways of looking at the world.
Differential Calculus: the Science of Instantaneous Change
Using differential calculus we are able to understand rates of change and slope at certain moments. Consider this example: Your speedometer while driving doesn’t just show the average speed during your entire trip, but it shows the exact speed at that exact instant. This is calculus in action, specifically, differential calculus. With something called “derivatives,” engineers can infer optimum machine performance, economists can anticipate future changes in the market, and scientists can model difficult natural phenomena.
Real-World Example: Engineering and Design
The use of differential calculus in automotive design can be seen in the way by which engineers analyze how a car’s aerodynamics vary at different speeds. Understanding instantaneous rates of change helps car manufacturing teams to do a better job of designing more fuel efficient and stable vehicles. With calculus principles, the curve of the body of a car, the angle of the spoiler, and even overall performance can all be refined, and the refinement can be calculated with pen and paper, just like textbook math problems.
Integral Calculus: the Science of Accumulation and Total Change
If differential calculus is all about measuring things that are changing instantaneously, the what about things that change at various speeds? That’s where integral calculus comes in. Integral calculus helps to understand measurement of effects that build up (or down). Can you calculate the total distance traveled by a car whose speed changes as it accelerates and slows down? If it were always going at a constant rate, without speeding up or down, you could just use differential calculus; but, most things in life vary their pace over time. So, by using integral calculus, we can quickly add up the numbers and get the whole picture.
Real-World Application: Environmental Science
Using integral calculus, data enthusiasts can calculate total carbon emissions, predict potential temperature changes, and study complex ecological systems. You don’t have to understand sustainability to understand how numbers work. And so, the experts must rely on math specialists to make sure their conclusions are inferred correctly. Environmental analysts are able to combine many data points together to make comprehensive models, which help not just sustainability experts, but also politicians, teachers, historians, business leaders, and even kids, to better understand longer-term environmental trends.
Calculus in Sports: Performance and Strategy
Mathematical modeling of professional sports is no longer a “specialty” job. With the help of software, calculus can help to manage anything from cameras in stadiums to advising coaches on player rotation and business leaders on hiring. Without knowing where the puck is going, hockey players wouldn’t know where they should be skating towards. As you’ll see in a moment, you probably already do calculus in your day-to-day without realizing it.
Real-World Example: The Perfect Basketball Shot
The shot arc of a basketball player can be modeled perfectly with calculus. Through a calculation of initial velocity, angle of release, and gravitational forces, basketball coaches can improve athletes’ shooting technique down to mathematical precision.
Financial Modeling and Economic Predictions
Calculus is the key to financial analysts being able to make predictions about market trends, estimating investment risks and creating complicated economic models. Principles of calculus are essential for making choices in derivatives trading, forecasting risk and predicting economic future.
Real-World Application: Your Piggy Bank
If you know that this week you can only put away two dollars, but next week you will be able to put away three dollars, and the week after that you will be able to put away six dollars… That’s calculus at work! Similarly, calculus is used by banks to determine mortgage rates, and by investors to compute potential returns when they try to model more complicated financial scenarios that can involve dozens or hundreds of knowns (and some unknowns) that may evolve over time.
Medical Research and Healthcare Innovations
Calculus is used by medical researchers to model how a disease spreads, to analyze how effective a drug study may turn out, and to understand highly complex biological systems. Calculus is important in healthcare and is not purely historical; it can be used by forward-thinking scientists to predict pandemic trajectories and to design more effective medical treatments.
Real-World Example: Epidemiological Modeling
Epidemiologists used calculus-based models to predict infection rate, understand transmission dynamics and develop containment and vaccination strategies during the COVID 19 pandemic. Every fall, a new strain of the flu seems to go around – without calculus, it would be impossible for doctors to prepare for treating patients who will have the upcoming season’s illness. Because there could be any one of the thousands of virus strains that becomes dominant, calculus has to be used in labs to understand the direction of where the next diseases might come from, so doctors can help the world to stay better prepared.
Final Thought… Do You Like Video Games?
Just as with “2+2=4,” calculus is all shown in the work: it is at the heart of the visual effects we see at music concerts, the 3D modeling for graphics, the animation in movies and the realistic simulations in video games we rely on today. Calculus doesn’t have to be intimidating to understand. Here are some strategies to make learning more accessible:
- See concepts through real world examples
- Use online interactive tool and simulation
- Learn to apply calculus to real world situations
- Apply mathematical principles to the things you like.
- Seek out proactive lessons from tutors with more examples.
Calculus is a Universal Language
As you’ve come to learn by now, calculus isn’t a math discipline, it’s a way of seeing the world, of imagining change, of visualizing all the amazing interconnectedness and complexity. Calculus gives you tools to analyze and predict sophisticated systems, whether you’re interested in science, technology, economics, healthcare, video games, or sports.
But it doesn’t come easy, and it is a process. If you trust the process, your reward will include a new way of seeing the world. When we learn to use calculus as a skill, we open up new ways of looking at the world. What makes calculus beautiful is that it is not all that complex once you get started, and it can help you to radically translate complex changes into actions, to give you new insights.
With practice, with patience, and with an ability to link mathematical principles to real world situations, anyone can come to love this fantastic mathematical discipline. OMC has tutors who can help you to improve your understanding of calculus and its applications. But if you feel you need a little extra help, our experienced tutors can offer you personalized guidance and help apply calculus concepts to the real world situations you care about. Learn more and get started today at our website.
