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Introduction To Calculus

The word calculus comes from Latin, originally meaning “small pebble” and it refers to a branch of study in mathematics of continuous change. Throughout the years, mathematicians have divided calculus into two main branches: Differential and Integral Calculus. (If you’re looking for help to learn more about this subject click here LINK). 

Differential calculus studies the rates of change, curves, and slopes, and integral calculus focuses on the accumulation of quantities and the spaces under and between curves.

Therefore, these two branches can be considered as opposites but related by the fundamental theorem of calculus and are considered a high-level course. Definitely, something you will be required to know in university and something a tutor can help you with.

Isaac Newton and Gott-fried Wilhelm Leibniz are known as the fathers of modern calculus. The mathematical theories they each developed have surpassed the test of time and are now being used by different fields, some of which are Sciences, Economics, and Engineering, to name a few. This proves that calculus opens the way to other, highly advanced areas of mathematics.

Below we’ve listed individual principles of Calculus, dominating these subjects will score you a high grade in any calculus test you might have:

  • Limits and Infinitesimals
  • Differential Calculus
  • Leibniz Notation
  • Integral Calculus
  • Fundamental Theorem.

As mentioned above, the principles of calculus are applied to many subjects aside from math. The perfect example of this is Physics, which makes particular use of advanced level Calculus, but also these principles are applied in Computer Science, Statistics, Business, Economics, Engineering, and Medicine. Which is why these concepts are so broad and applicable to many areas of study, they are universal!

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Calculus: A Quick Introduction

Let's face the truth, Calculus is not a subject you can teach yourself overnight, hardly anything is. But you can change your point of view by getting a head start and approach the subject little by little and make learning Calculus a less daunting process. By taking small steps you are moving forward in your education, which will help you get into a good college and finish school with good grades.

Another alternative you might find useful is hiring a tutor to guide you through the lessons. Tutors are trained to make challenging subjects look easy, so private tutoring can always be a good idea that can get you ahead in a math class.

As a student, it can also help if you understand what Calculus is all about before you start learning about it in-depth, if you come prepared for the course, your teacher will appreciate it. If you have a better picture of what’s ahead, then the chances you are surprised during the course are slimmer and you can retain the information best. Acquiring good math skills is all about practice and you can perform best if you prepare yourself through private tutoring lessons.

Don’t forget that finding a private tutor to improve and understand better a subject is always a good option. They could guide you through tough subjects, teach you tricks to solve challenging equations, and ultimately help you score high scores in no time, definitely faster than trying to do it only through class.

This is because tutors, especially someone with experience, are prepared to attend to your needs and teach you how to properly go about solving equations or a math problem. And the best thing is that now, during the pandemic, you can still get your private class online.

You can also check out more on Calculus lessons in our blog. 

What Is Calculus?

A first step is to know the definitions of the field of study you’re about to start learning. Calculus, as we defined previously, is a term used to describe the study of continuous change – but you might be wondering what this means? Basically, Calculus helps you find, split, and rearrange patterns and shapes - skills that are in high demand for students to master in university.

For example, let’s take a look at a circle. Imagine that the circle is made up of many rings, then, if you rotate the 3D image, you can see how what once was a disc now has taken the shape of a dome. You can find free online tools that will show you this we just mentioned. 

You can search Superprof for an online math tutor and get more experience in the field.

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

If then, you unroll the rings that make a triangular shape and then sketch them on a graph, you will use a formula to measure the circle’s area (Area = ½ base x height - go check previous math lessons if your memory is a little foggy on the subject).

There are more ways to see how fascinating Calculus can be, aside from calculating the area of a circle (although they will for sure be teaching and testing you on this). Another way to think of Calculus is to look at it as a branch of math that studies huge and small numbers; what mathematicians refer to as ‘infinite and infinitesimal'.

Although Calculus studies not only these numbers it allows you to take the numbers and use them to describe an intangible event, like finding out how fast objects move in a certain direction. This is referred to as 'acceleration and velocity' by mathematicians and is something that could come up in a test.

If you take all of this into account, this subject uses a combination of Algebra and Trigonometry and afterward adds some extra rules and theories to take on board. Prepare yourself to hear your teacher use words and terms like parabolas, to have lessons about new symbols, and finally having courses about the famous Pi! As a student, a subject will be a little easier to prepare for if you read about the class before starting. 

What Is Interesting About Calculus?

No one has ever said something like “high-level Calculus is such an exciting lesson”! But teaching you about calculus is very important and there are many elements within the subject that are interesting. 

For example, your teacher will teach you about concepts that have a role in many facets of our day to day lives, like cars, computers, social media, and anything else you can think of. Calculus has been a tool for humanity and mathematics has resolved many problems we’ve encountered throughout history.

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 VisualHunt.com

For example, there are many elements involved while building an airplane that use Calculus, especially when it comes to building parts like the wings.

For this, Engineers have to study lift force and then inform about the design that should be adopted for the wings because they have to adjust the curve according to the purpose. Additionally, they must always consider the pressure as velocity increases.

This is why there are many calculations and equations to solve to guarantee that the airplanes are safe and fit for their purpose. This is why math class is compulsory for students studying engineering and teaching these subjects in school and university years is not just for nothing.

Talking about speed and velocity, here is a point that will help you understand why Calculus is so interesting to learn about.

If you don’t feel like becoming a mathematician is not in the cards for you, and that this subject is challenging for you, remember that you can always find support in tutoring. Tutors are here to help us navigate difficult topics, and if you choose to start private lessons you will be on your way to getting an A. 

Are you familiar with Algebra? If not, the first step you have to take is to revise previous courses. Then you will comprehend that if you have two apples and add three apples, you would have a total of five apples, are you able to visualize a formula? Most likely you can. 

On the other hand, Calculus is here to help us see things that are less obvious and that can’t be seen by the human eye. For example, with this subject, you can determine how fast one of those apples are going to hit the ground if it’s dropped from your hand. 

These kinds of calculations are very useful for people in fields like Chemistry, Technology, Engineering, and Physics because the answers and solutions will have a direct impact on the working processes they undertake.

If you are falling behind in school, remember that private tutors can guide you to understand best all of these ideas and concepts. And once you can see the concepts clearly, your point of view might change into one that likes math. 

Untangling Calculus 

Functions, Variables, And Limits

It can be challenging to understand it if you don’t first know about what makes up this branch of Mathematics.

For starters, we recommend you check previous math courses and refresh your memory a little. There is a basic element in Algebra called a 'function'. This refers to a group of letters that are separated by the '=' sign and is described as a 'rule'.

In that function, when there is a value added to the mix, it has to be reflected on the other side too, just as it happens in a standard algebraic formula, you know the step! You have to understand that in each function, there is an 'independent variable' (input) and a 'dependent variable' (output). These are concepts that they are for sure going to be teaching you in school.

But the thing to pay attention to the most is the 'limit': the value that the dependent variable approaches as the independent variable approach a given value.

Also, you’ll learn that a slope is used to describe the change in y divided by the change in x. For example, the slope between two points on a graph. Well, this calculation then will help you find the average rate of change between y and x too.

Therefore, derivatives use this slope to work out the gradient of it at a point on a curving line, which is where limits come into play. If this seems a little intimidating, you can always check with a tutor outside class time. Private tutoring can always come in handy and give you an advantage during a test and overall during the school year.  

The integral is a word used to describe the way of finding an area. You’ll get the lesson on integrals when you start learning to find areas of shapes like a square or a circle and any other kind of irregular shape. This is considered as the opposite process of a derivative, and it also works as a way to take even more data from a graph. 

Be sure to understand this since it will most likely appear in a test, but don’t worry you can check out our guide for differentiation and integration problems here (LINK).

Calculus enables us to find the area of circles, squares, and irregular shapes. Photo by pedrosimoes7 on VisualHunt.com

Guidance for what's to come

Throughout the course lessons, you will have to acquire skills on everything already mentioned, plus many others like Graphs, Tangents, and Areas.

Don’t feel intimidated, even if this looks like more than what you think you are capable of! One problem students have is doubting themselves while faced with a challenging subject during their education years. Well, rest assured, school is designed to provide us with the experience that will come in handy in the outside world.  

As a student, you need to recognize if you require guidance for a class or a lesson and consider that a tutor can be the best way to overcome an obstacle and better prepare you for university. Keep in mind that free tutoring lessons are often offered on campus, or, for a more personalized experience, you can find a tutor online. 

Math class is designed to give you an in-depth and well-rounded view of Calculus and its many themes. You’ll reach a point where you’ll know each of these themes inside out and you’ll be ready for college level mathematics.

And consider that not only will you comprehend how it works, but you will most likely be able to understand how each theme is linked to one another, something that will come in handy during your years at university.

Until that time, you need to focus on taking one step at a time and be patient with yourself. 

It might come to a point where you feel you would like extra support for a lesson or subject, whether it is with problem solving, homework, preparing for an exam or if the subjects are turning out to be a little overwhelming. Well, if this is the case, why not consider contacting a private mathematics tutor and get the one-to-one experience.

Don’t forget, there is no shame in hiring a tutor, after all, tutoring can help you navigate this very tricky subject, one that is also challenging for advanced mathematicians. 

For more on A level Maths problems, see our blogs on Solving Exponentials and Logs and Solving Mechanic Forces. And consider private tutoring as an option to improve your skills in math.

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Maria