Calculus

Calculus deals with the properties of derivatives and integrals of quantities. This process is initially dependent on the summation of infinitesimal differences. It helps in determining the changes between values related to the functions. It is a tool for studying how quantities change over time in relation to other variables.

Ancient Times
In mathematics, the foundation of calculus dates back thousands of years. In the fifth century BC, Democritus worked with ideas based on infinitesimals during the ancient Greek period. If it is possible to support it with a geometric proof, the Greeks would consider a theorem accurate. The Greek philosophers also viewed ideas based on infinitesimals as paradoxes, as it is always possible to divide an amount again, no matter how small it becomes.
In the 3rd century BC, Archimedes developed the method of exhaustion, which is used for calculating the area of circles.



The Middle East and India
Like many other areas of science and math, the growth of calculus slowed down in the Western world during the Middle Ages.
At the same time, on the other side of the world, people were discovering and studying both integrals and derivatives. An Arab mathematician, Ibn al-Haytham, used formulas he created to find the volume of a paraboloid. A solid shape made by spinning part of a parabola (a curve) around a line.
During this time, ideas moved between the Middle East and India, since some of them were also found in the Kerala School of Astronomy and Mathematics. By the start of the fourteenth century, many pieces of calculus were already known, but differentiation and integration had not yet been joined together as one subject.



The Fourteenth Century
In the 14th century, a group of mathematical scholars revived the study of mathematics. The group was known as the Oxford Calculators. They explored the philosophical questions through the lens of math.



The Seventeenth Century
The early seventeenth century was the most critical time in the history of calculus. During this period, René Descartes created analytical geometry, and Pierre de Fermat studied the highest and lowest points of curves, as well as their tangents. Some of Fermat’s formulas are very similar to the ones we still use today, even after almost 400 years.



The Eighteenth Century and Beyond
The debate surrounding the invention of calculus became more and more heated as time wore on, with Newton’s supporters openly accusing Leibniz of copying. Britain’s firm claim that calculus was Newton’s discovery.
 

Based on the change and mathematical analysis, calculus can be divided into four parts. 

• Differential Calculus 
• Integral Calculus
• Multivariable Calculus
• Vector Calculus
   

Differential calculus is used to find the rate of change of a quantity with respect to another. The key concepts are limits, continuity, and slope of a curve.  

To find the quality when the rate is known, integral calculus is used. An integral is the inverse process of differentiation. We use it to calculate the area and volume. 
 

It is used when the function of two or more variables is calculated. It is an extension of differential and integral calculus. 
 

Vector calculus is about the use of differentiation and integration of vector fields in Euclidean space.  

The integral is the inverse process of the differential. The fundamental theorem of calculus links derivatives and integrals. There are two theorems that are the first and the second fundamental theorem of calculus. 

In calculus, for each function, we have different formulas. And students should remember and use the correct formula. For a better understanding of calculus, it's important to remember all the basic formulas. 

d / dx(xⁿ) = nx^n-1

d / dx(C) = 0  → C is the constant

d / dx(Cf) = C.d / dx(f)

d / dx(f±g) = d / dx(f) ± d / dx(g)

d / dx(fg) = d / dx(fg) + d / dx(gf)

d / dx(fg) = g(d / dx)(f) - fd / dx(g)

                            g²

d / dx(sin x) = cos x

d / dx(cos x) = -sin x

d / dx(tan x) = sec²x

d / dx(cot x) = -cosec²x

d / dx(secx) = sec x tan x

d / dx(cosec x) = -cosec x cot x

d / dx(a^x) = a^x In a

d / dx(e^x) = e^x

d / dx(In x) = 1 / x

d / dx(log_ax) = 1 

                   x In a

∫1dx = x + C

∫a dx = ax + C

∫(1 / x) dx = In |x| + C

∫1 / √ 1 - x²dx = sin^-1x + C

∫1 / √1 + x²dx = tan^-1x + C

∫1 / |x| √1 - x²dx = sec^-1x + C

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

∫sec x (tan x) dx = sec x + C

∫csc x (cot x) dx = -csc x + C

∫sec² x dx = tan x + C 

∫csc² x dx = -cot x + C 

∫e² dx = e^x + C

∫a² dx = a^x/ In a + C 

∫1x dx = In |x| + C

∫log_a x dx = x In x -x  + C

                        In a

We have talked a lot about calculus. We know that we use it in our daily life. Now let’s see the applications of calculus.

 

Physics:  In Physics, calculus is used to calculate motion, force, change, energy and many more. The speed of a vehicle, rocket, or even the light can be calculated. 

 

Engineering: From building bridges to developing circuits, we use calculus in different fields of engineering.   

 

Economics: Calculus is used to study the market and its behavior. To understand the minimum cost and maximum profit .

 

Medicine: To study the growth of bacteria and infectious diseases' calculus is used. This data is used to treat cancer and diagnose patients.
 

Calculus is a broad topic that includes limits, derivatives, integrals, and many more. Let’s learn some tips and tricks to make it easier for students to learn

 

Solve calculus problems differently 

Performance in math subject is based on solving problems. However, with calculus, it is difficult to understand the underlying principles in each problem solved.

 

Seek external visualizations

For better understanding of the concept, simple, compelling and effective visualizations from other sources like, YouTube is highly revered for the quality and clarity of the computer generated visuals. 

 

Keep a dictionary of calculus notations and terms

This is the best way to keep track of all the definitions and notations for studying is to keep a dictionary. 

 

Don’t hesitate to ask for help
When you run into trouble understanding part of a problem or concept, reach out to your teacher for help.
 

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