Calculus: Applications of Calculus in Everyday Life
Preamble:
Although it may not always be obvious, we actually use calculus quite often in our daily lives. Various fields such as engineering, medicine, biological research, economics, architecture, space science, electronics, statistics, and pharmacology all benefit from the use of calculus. Although the average person isn’t solving differential or integral calculus problems daily, we are using technology and equipment developed through the application of calculus in almost every aspect of our lives.
What is Calculus?
Calculus means ‘small pebble’ in Latin, and is named after the rocks used by the Romans for counting. Sir Isaac Newton, one of the greatest scientists and mathematicians in history, is credited as the founder of calculus. Over centuries, many mathematicians have contributed to the further development of calculus as a branch of mathematics and physics.
What does calculus actually do? Well, calculus measures the rate of change that occurs in almost every phenomenon in the universe. There are two branches of calculus, integrals and derivatives. The derivative component of a mathematical function defines the rate of change at a specific point, whereas integrals show the accumulation of discrete values of a function over a range.
Derivatives and integrals are fundamental concepts in calculus that have been studied and applied for centuries. These concepts were first formalized in the late 17th century, notably by mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz.
Derivatives:
The concept of derivatives can be traced back to the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. In his seminal work "Philosophiæ Naturalis Principia Mathematica" (1687), Newton introduced the idea of fluxions, which later evolved into the concept of derivatives. Similarly, Leibniz independently developed the concept of derivatives and introduced the notation dy/dx to represent the derivative of a function y with respect to x. This notation is still widely used today.
One of the earliest applications of derivatives can be seen in the study of motion. For instance, Newton used derivatives to describe the instantaneous velocity of an object moving along a curve. This led to the development of calculus-based kinematics, which forms the foundation of classical mechanics.
Integrals:
The concept of integrals emerged alongside derivatives in the late 17th century. Newton and Leibniz independently developed integral calculus as a means of determining the accumulation of quantities, such as area and volume.
In "Philosophiæ Naturalis Principia Mathematica," Newton introduced the method of exhaustion, which laid the groundwork for integral calculus. This method involved approximating the area under a curve by dividing it into smaller, manageable regions and summing their areas.
Fundamental theorem of Calculus
The fundamental theorem of calculus, which establishes a fundamental relationship between derivatives and integrals, was formulated by Isaac Newton and Gottfried Wilhelm Leibniz.
This theorem states that the integral of a function over an interval can be computed by finding an antiderivative of the function and evaluating it at the endpoints of the interval.
Applications:
The applications of derivatives and integrals are vast and diverse, ranging from physics and engineering to economics and biology. Some notable applications include:
1. Engineering
Real life applications of calculus is incredibly prevalent in the field of engineering. Let’s discuss a few of its applications below:
Structural Engineering:
One of the most critical applications of calculus in real life is in structural engineering. Calculus is used to calculate heat loss in buildings, forces in complex structural configurations, and structural analysis in seismic design requirements. Architects use calculus to determine the ever-important quantity of materials required for constructing support systems that can withstand stress over long periods of time. Even notable monuments such as the Eiffel Tower were constructed using calculus to predict the impact of wind resistance.
[Read: Applications of Trigonometry]
Civil Engineering:
When designing bridges, civil engineers need to consider three structural elements: beams, tension, and compression members. Using calculus, we can determine the amount of force acting on the beam including the weight of pedestrians, weight of vehicles, and the amount of traffic anticipated on that bridge. Based on these factors, the materials, size, and capacity can be computed.
Electrical Engineering:
Integral and differential calculus are crucial for calculating voltage or current through a capacitor. Integral calculus is also a main consideration in calculating the exact length of a power cable necessary for connecting substations that are miles apart from each other.
Mechanical Engineering:
Mechanical engineering is yet another great example. The pump used for filling an overhead tank, gardening tools, cars, motorcycles, robots, and many household appliances are designed using the principles of calculus.
Space Flight Engineering:
Before launching a rocket or exploratory probe, engineers must use calculus to figure out the gravitational pulls of the sun and moon in order to know how to launch a probe or hit the velocity needed to orbit the earth.
Derivatives are essential in optimization problems, where engineers seek to maximize or minimize certain quantities subject to constraints. Integrals are used in the analysis of continuous systems, such as fluid flow and heat transfer.
2. Medicine :
Calculus can also be applied in medicine. Applications of calculus in medical science include:
Bacterial growth:
Biologists use differential calculus to compute the exact bacterial growth rate in a culture by varying environmental factors such as temperature and food source.
Patient Diagnosis:
Calculus is crucial in medical tests used for measuring cardiac output, blood flow, and tumor growth. This data is needed to properly assess a patient’s health condition.
Epidemiology:
Epidemiologists who conduct research on infectious diseases use calculus to discover the spread rate and area where an infection will likely occur. Calculus can also help create a containment plan and investigate the source of an infection. 
Cancer:
Calculus is used to determine the growth or shrinkage and number of cells of a cancerous tumor. Using an exponential function, oncologists analyze the progression or regression of a disease.
Surgical Control of Red Blood Cells:
The blood in the human body is made up of red blood cells. While undergoing surgery, a patient’s blood volume has to be maintained by injecting a saline solution that mixes quickly with the blood and dilutes as time passes. Calculus is used for computing the volume of red blood cells so the proper amount of saline solution can be given to the patient during surgery.
[Read: Probability in Real Life]
Cardiology:
Cardiologists use differential calculus to understand the blood flow dynamics needed for building an artificial aorta model in order to make sure it is placed correctly during transplant.
Pharmacology:
Understanding drug sensitivity is crucial in finding the proper dosage for maximum output of drug integration. The side effects of a drug, brought about by factors like body temperature change, can be computed through integral calculus. Pharmacologists can make a kidney a changing function in the calculus-based Cockcroft-Gault equation and determine the creatinine levels in the blood to assess the kidney’s functionality. Using these results, the dosage of drugs for treatment can be finalized.
Neurology:
Neurology is the study of the human nervous system, which is a very complex network of neurons that pass the brain’s signals to the entire body and control all physical activities. The system consists of the central nervous system, the brain, and the spinal cord. Integral calculus is used to compute the voltage of a neuron at a certain point. In contrast, differential calculus is used for calculating the change of voltage in a neuron with respect to time.
Entomology:
Entomology is the study of insects. An entomologist conducts research, classifies insects based on species, and finds safe solutions to control insect infestations in indoor and outdoor environments. Calculus is used for modeling and generating insect proficiency through partial derivatives.
3. Business
Modern business also benefits from the application of calculus. Some of its uses include:
Credit card payments:
Calculus is used for setting payment structures and the minimum due amount by the credit card company by considering variables such as interest rates and fluctuating balance.
Economics:
The price elasticity of supply and demand is determined using calculus. Economists use variables such as the supply-demand curve to find the exact measure of elasticity at a particular point according to changes in price.
4. Music
You may be surprised to know that the use of calculus is not restricted to engineering or medical science, but can also be applied to music. Some applications include:
Harmonics:
An oscillation created by a damped harmonic is not infinite, as friction and air resistance will dissipate the energy. Calculus is used to anticipate these motions to make the proper adjustments and provide the best musical experience to the listeners.
[Read: Applications of Algebra]
Acoustics:
Resonance and forced oscillation can be computed using calculus. Air resistance varies at different frequencies and resonates throughout an enclosed space whenever a musical instrument is played. Through calculus, we can make improvements on acoustics and improve the listener’s experience.
5. Research Analytics
The application of calculus in research and development has paved the way for manufacturing, data management, gaming, and other service industries to grow exponentially. These applications include:
Variables in manufacturing:
Research analysts use calculus while observing different processes. Based on collected data, companies can optimize their output, productivity, and efficiency, which improves the industry’s quality and revenue.
Evaluate survey data:
Statisticians make use of survey data to find a range of answers for different questions. Using calculus allows them to make accurate predictions, which helps their decision making process.
Computing birth and death rates:
Ecologists use calculus to make dynamic population models that showcase growth without any environmental constraints. Through calculus, accurate predictions on population changes can be made, taking birth and death rates into account.
Simulation for video games:
Graphic designers use calculus to understand 3D models created through changing conditions. Based on the results derived from calculus, video games and animated movies are made with a real world perspective.
6. Meteorology
Understanding weather patterns is crucial for agriculture, weather warnings, and for preparation in case of extreme conditions.
Computing Weather Patterns:
Meteorologists use differential calculus equations to predict the effects of varying weather conditions on the atmosphere with respect to temperature, humidity, and pressure changes.
[Read: Father of Mathematics]
Conclusion:
In conclusion, derivatives and integrals are fundamental concepts in calculus that have found widespread applications across various fields since their formalization in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.
References:
- Newton, Isaac. "Philosophiæ Naturalis Principia Mathematica." 1687.
- Leibniz, Gottfried Wilhelm. "Nova methodus pro maximis et minimis." 1684.
- Struik, Dirk Jan. "A concise history of mathematics." Dover Publications, 1987.
- Courant, Richard, and John Fritz. "Introduction to calculus and analysis." Springer Science & Business Media, 20135.
- Lester R. Ford, Sr. & Jr. (1963) Calculus, McGraw-Hill via HathiTrust
- W.Mellen W. Haskell (1895) On the introduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1(6):155–9.
