If you took mathematics in high school or any other level, then most likely you have come across calculus. Well, do you remember its definition or where it is applied? Calculus is a branch of mathematics that deals with finding the properties of derivatives and integrals of functions. In simple terms, calculus involves the study of rates of change. There are two branches of calculus: differential calculus and integral calculus. Differential calculus deals with instantaneous rates of change and slopes of curves, while integral calculus focuses on the accumulation of quantities and areas under curves.
A Brief History of Calculus
There is a lot of history behind the development of calculus, like the ancient, medieval, and others, but we shall focus on two known fathers of calculus: Isaac Newton and Gottfried Wilhelm Leibniz. They both complained that the other stole the idea from them, which up until now has never been revealed who developed it first. All we know is that the two worked independently. Newton was the first to apply calculus to general physics, while Leibniz did a great deal of work developing consistent and useful notation and concepts. Both Newton and Leibniz deserved credit, as some aspects were developed and applied independently. Newton came to calculus as part of his investigations in physics and geometry. He viewed calculus as the scientific description of the generation of motion and magnitudes. Leibniz, on the other side, focused on the tangent problem and came to believe that calculus was a metaphysical explanation of change.
Other pioneers of calculus known today are ancient and medieval. The ancient period introduced some of the ideas that led to integral calculus, but did not develop these ideas rigorously and systematically. In medieval times, Arab mathematician Ibn al-Haytham (Alhazen) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration.
Applications of Calculus
Calculus today has been widely used in almost every field, like economics, statistics, and others which are the backbones of problem-solving.
Calculus in Economics
Calculus is an important tool in economics. It can be used to predict supply and demand and to calculate expected profits. Economists use calculus to determine price elasticities, i.e. the price elasticity of demand and price elasticity of supply. With these price elasticities, one can determine the right time to buy a commodity and the right time to sell. When performing these business calculations, one will have to incorporate several mathematical derivatives to determine specific points in the demand and supply curves.
Calculus in Modeling
Ever wonder how some huge shapes fit together perfectly? Geometry works in conjunction with calculus to evaluate these perfect shapes. In real life, there are a lot of irregular shapes that are gigantic and cannot be handled physically. With the concept of calculus, all can be solved in a paper form where it can be easily handled.
Calculus in Structural Engineering
The tallest building in the world, the Burj Khalifa has a gigantic mass and is supported and interconnected by concrete. It is difficult to determine its total weight, due to different shapes and densities of materials used, but vector calculus can help in determining the total weight. It is also situated in a desert where there is so much heat that it has to be designed in a way that there is a huge heat loss. To do this, a graph of heat loss vs time can be plotted, and then integral geometry can be used to determine the area under the graph.
Calculus in Finance
Using calculus in finance sounds like a lie to most people who have never developed interests in anything revolving around finance. In investment banking, calculus helps banks in determining portfolio risk, bank mandate, and many more. Meanwhile, the famous Black-Scholes equation uses calculus.
Calculus in Other Disciplines
Physics
Calculus is used in a lot of physics concepts. Among the physical concepts that use calculus include motion, electricity, heat, light, harmonics, acoustics, and dynamics. Advanced physics concepts, including electromagnetism and Einstein's theory of relativity, use calculus.
Biology
Biology is life, and that means calculus is also life since there are many uses of calculus in biology. Calculus is used to formulate critical rates, such as birth rates, death rates, epidemiology, bacterial growth, cardiac output, patient diagnosis, and many more.
Chemistry
In radioactivity, radioactive elements multiply at a proportional rate. Calculus helps in calculating the rate of decay of isotopes. Chemical reactions also use calculus concepts.
Research Analysis
Calculus is also an important tool in performing research programs. In manufacturing, it can be used to study different variables to determine operational efficiency.
Meteorology
Weather prediction also uses calculus. Meteorologists can calculate the effects of changing weather conditions in the atmosphere by applying different variables, such as temperature, pressure changes, and speed of the wind.
Solved Examples of Calculus
Example 1:
Differentiate y=2x^3-5x^2+10
Follow these steps to differentiate this equation
Multiply the coefficients with the powers.
Reduce the powers by one.
Eliminate the constant.
Answer: y=6x^2-10x
Example 2:
To find the slope of the curve y=6x^2+2x+5 at the point (-2, 0), we need to first differentiate the equation: y=12x+2
Slope at point (-2, 0) is m=12(-2)+2=-22
Answer: The slope of the curve is -22
Conclusion
Any event that has already happened, is happening now, or is about to happen has calculus in it. The list of real-life applications of calculus is unending.