An Introduction To Calculus

Calculus, which takes its name from the Latin word for ‘small pebble’, is a Maths term that describes the study of continuous change.

There are two principal branches of Calculus: Differential Calculus and Integral Calculus. (Learn more about Differentiation/integration problems here).

While the first is concerned with rates of change and curves and slopes, the latter is focused on the accumulation of quantities and the spaces under and between curves.

As such, they can be as somewhat opposites of one another yet the two are directly related by the fundamental theorem of calculus.

Modern Calculus, was thought to have been developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. The mathematical theory has since been adopted by various subject fields, including the Sciences, Engineering, and Economics. Calculus itself acts as a gateway to other, more advanced areas of mathematics.

The individual principles of Calculus include Limits and Infinitesimals, Differential Calculus, Leibniz Notation, Integral Calculus and Fundamental Theorum. As previously mentioned, Physics makes particular use of Calculus and its principles, however it is also applied to Computer Science, Statistics, Business, Economics, Engineering and Medicine. This shows just how broad the concepts and theories reach.

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Calculus: A Speedy Introduction

Let’s face it, Calculus is not something that you can learn overnight. However, to make the principles of Calculus less daunting, why not approach the subject in bitesize chunks?

Also important is to understand what Calculus is all about before you set off on studying it in depth. By knowing a bit more about what to expect, there will (hopefully!) be no surprises during the course and you will be able to absorb the information given to you with more of a level head.

What Is Calculus?

Calculus, as explained above, is a term used to describe the study of continuous change – but what does that even mean? In simpler terms, Calculus is about finding, splitting and rearranging patterns and shapes. Let’s take a circle, for example.

If you imagine that circle is made up of a series of rings, then rotate the 3D image, you will notice that the once disc now takes a dome shape.

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Rotated onto its side, a simple circle can be seen as like a dome made up of many layers.
If you take a circle and flip it on its side to imagine a dome shape, this is just one of the ways that Calculus manipulates shapes to work out areas and other information related to lines, curves and spaces inbetween them. Photo by ANBerlin on VisualHunt

By unrolling these rings which make up the triangular shape and plotting them on a graph, we can come up with a simple formula to measure the circle’s area. (Area = ½ base x height)

Even more mindblowing than obtaining the area of a circle, another way to look at Calculus is to think that of it as a branch of Maths that covers tiny to huge numbers, called (in Maths terms) ‘infinite and infinitesimal'.

Not only does it study these numbers, it takes them and uses them to describe an intangible event, like figuring out how fast objects move in a certain direction. Mathematicians refer to this as 'acceleration and velocity'.

So, taking all of this into account, Calculus combines lessons in Algebra and Trigonometry and then adds a few extra rules and theories to take take on board. Be prepared to hear about parabolas, see a bunch of new symbols and to use the famous Pi!

What Is Interesting About Calculus?

“Calculus is really exciting”, said nobody ever! However, elements of Calculus are very interesting. For example, the concepts play a role in various aspects of our every day lives from cars to aeroplanes to mobile phones.

Calculus plays a role in many aspects of our everyday lives.
Calculus concepts are present in our everyday lives, from how vehicles and planes work to how we are able to use our mobile phones. Photo on

For instance, a number of elements linked to Calculus and its theories are examined in the build of an aircraft, specifically when it comes to parts like the wings.

Engineers must study lift force to inform them of the design that they must adopt for the wings, adjusting the curve according to purpose. In addition, they must consider the pressure on the part as velocity increases.

For this reason, many equations and calculations are required to ensure that the aircrafts are safe and fit for purpose.

On the subject of speed and velocity, take a look at our next point which expands further on why Calculus is so interesting to learn about.

If you are already familiar with Algebra, then you will understand that if we have two bananas and add three bananas, we will have five bananas in total, and be able to visualise a formula.

Calculus, on the other hand, highlights things that are less obvious and that can’t be seen with the eye, like how fast one of those bananas hits the ground when dropped from the picnic basket. This process is really useful for people studying the fields of Technology, Physics, Chemistry and Engineering as the results or solutions found will directly impact on the working processes they undertake.

Breaking Down Calculus

Functions, Variables And Limits

You can’t expect to understand Calculus without first of all learning about what makes up this branch of Maths.

First of all, just like in Algebra, there is a basic element called a 'function'. This is normally a group of letters separated by an '=' sign and is described as a 'rule'.

When a value is added to the mix, what goes in one side will be reflected on the other side, just like a standard algebraic formula. Within each function, there is also an 'independent variable' (input) and a 'dependent variable' (output).

The biggest thing to consider, however, is the 'limit': the value that the dependent variable approaches as the independent variable approaches a given value.

Derivatives And Integrals

Once again, you will have heard of slopes being mentioned in Algebra lessons. A slope is used to describe the change in y divided by the change in x, i.e. the slope between two points on a graph. This calculation then serves to determine the average rate of change between y and x too.

Derivatives take this slope and then work out the gradient of the slope at a point on a curving line, which is where limits come into play.

An integral is a term used to describe a way of finding an area. Integrals can be used to find the area of a circle, a square, or any other irregular shape. It’s basically the opposite process of a derivative, and it’s yet another way to take away even more data from a graph.

Calculus focuses on the area within shapes and curves as well as the gradient of lines and curves.
Calculus enables us to find the area of circles, squares and irregular shapes. Photo by pedrosimoes7 on

Additional Calculus Themes

Over the course of your study programme, you will be faced with the above themes, along with many more such as Graphs, Tangents and Areas.

If all of this sounds like too much to take, do not despair. Your course will be designed to give you a fully comprehensive and well-rounded view of Calculus and its many themes, to the point where you'll know each of them inside out.

Not only will you understand how each of them works, you will also be able to make links between the themes and how they all relate to one another to make up this complex mathematical subject.

Until then, focus on taking each step one at a time and to give yourself plenty of time to absorb the different themes covered before jumping ahead to the next.

Remember also that there is no shame in feeling unsure of yourself - this is after all a very tricky subject to tackle, even for advanced mathematicians.

If you feel you might need a little extra support in the way of homework assistance, revision maths help or you just want someone to go over the content at a slower pace with you, then why not consider contacting  private maths tutors to offer you that tailored one-to-one experience.

For more on A level Maths problems, see our blogs on Solving Exponentials and Logs and Solving Mechanic Forces.

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