Calculus can be both immensely rewarding and truly difficult to master. Calculus is a branch of mathematics that examines numbers and lines in the real world and maps out how they change. Calculus is one of the most commonly used fields of mathematics in the world, despite the fact that it may not appear to be practical at first. Imagine having the ability to analyse how quickly your company is developing at any given time or plotting the trajectory of a spaceship and the rate at which it is using fuel. Calculus is a useful tool in engineering, economics, statistics, chemistry, and physics, and it has aided in the development of numerous real-world inventions and discoveries.
Calculus is a branch of mathematics that deals with limits, derivatives, integrals and functions and is frequently applied in engineering and the higher sciences. This subject is a major component of maths, and it’s often the foundation for a lot of the equations in physics and mechanics. While studying calculus can seem overwhelming, there are a few tried and tested ways that you can improve your calculus study to make things easier for you.
- Start with Basic Concepts
Starting with calculus will feel like a huge leap in your mathematical understanding. You might remember leaping from arithmetic to algebra and wondering what all these letters were doing in your maths book. Calculus is the next logical step forward from algebra, and just like algebra, relies on the same basic concepts.
Your basic arithmetic functions and rules of algebra will still apply. Make sure you’re across your times tables, order of operations, and balancing and cancelling of algebraic equations so you can fully come to grips with calculus. Sometimes when you’re missing something in an advanced equation, it was a basic mistake. Studies show that a solid foundational understanding is key to deep advanced working knowledge.
- Understand Calculus Concepts
Generally speaking, you can divide calculus into differential calculus and integral calculus. Calculus aims to address how equations can grow, shrink, and change over time. Accordingly, when doing calculus, we’re often looking to find the equation’s lowest and higher points. Just like with algebra, we have variables, typically represented by pronumerals.
You should know what concept you are engaging with when studying calculus. Understand the difference between these two branches of calculus and how they engage with other types of mathematics. If you are focused on rates of change like slopes of tangent lines and velocities, that’s probably differential calculus. Integral calculus, on the other hand, deals with total size or value, such as lengths, areas, and volumes.
- Understand Calculus Limits
Limits are extremely useful in calculus. Limiting an equation means finding its highest and lowest points, which means the solution exists between these two points. When you’re grappling with a complex function, finding its limits can help break it down into smaller, more manageable chunks.
Limiting in calculus is readily applied to the real world, too. It’s often used to find the strength of magnetic and electric fields, for example.
- Familiarity with Fundamental Theorem
The fundamental theorem of calculus is that integration and differentiation are inverse to one another. As you reach higher levels of mathematics (including calculus), these relationships and phenomena take on an elegant beauty, and this is a great example of it.
You’ll need to understand and familiarise yourself with the fundamental theorem if you want to get anywhere with calculus.
- Practice Problems
As the saying goes, practice makes perfect. More accurate, however, would be to say that persistent practice makes you better. This applies to pretty much any skill you might be pursuing but is particularly applicable to mathematics.
The 10,000 hour rule is the notion that ten thousand hours of active practice is what you need to master any given skill. The Beatles played three times a day for years in Hamburg to hone their craft and went on to become one of the most successful pop acts of all time. You don’t need to put ten thousand hours into your calculus practice, but we recommend practising new problems frequently to sharpen your calculus skills. You’ll also want to check not only your answers but your work, to ensure that you are approaching each new problem correctly.
- Use All Resources Available To You
These days, there are plenty of resources available for those looking for help with their maths homework. You don’t need to go through your calculus study in isolation. It’s always a good idea to engage with your peers and work on difficult problems together to see if they have a different approach. Ask questions in class, use online homework resources, and be sure to keep your mind open as you engage with various learning tools and resources available to you.
- Integral Calculus
Integral calculus involves the study of integrals and their properties. As the inverse process of differentiation, it is a critical component of calculus that you need to understand. Effectively, assume that if you know the f of a function that is differentiable in its domain, you can calculate f.
There are two types of integral calculus. There are definite integrals and indefinite integrals. Definite integrals are defined using upper and lower limits, while indefinite integrals are not.
- Differential Calculus
Differential calculus, the inverse process of integration, seeks to find the exact derivative directly from the formula of the function. In other words, you’ll be seeking to find the rate at which quantities change. This could be, for example, velocity, which can describe how distance changes with time.
Differential calculus is also readily applied to the real world. For example, the rate of change of global temperature is useful when predicting climate patterns. Similarly, differential calculus, particularly with graphs, can be used in business with respect to profit and loss.
- Double Check Your Understanding
Calculus can be tricky stuff! While you are learning calculus, it’s important to consistently check in with yourself. The question you must ask yourself is “do I understand this, or am I simply following a recipe?”
There is a tangible difference between understanding and memorising. For example, you might be able to memorise the steps to integrate an equation, but do you understand why you’re performing each step? Do you understand how each part of the process relates to the other parts? You’ll want to make sure you know why you’re doing things as opposed to simply knowing what you’re doing. Double check your understanding by engaging with fellow students, your teacher, and even your textbook to ensure that you know what’s going on.
- Adopt an Effective Study Routine
To get better at calculus, you’ll want to make sure you are habitually practicing. Adopt an effective study routine that allows you to regularly test your calculus ability. Studies show that students who adopt and follow an effective study plan tend to perform better than those who don’t.
The same is true for calculus. Ensure that you have a dedicated study space with plenty of natural light. Take breaks, avoid cramming, and regularly go over any notes. Calculus is advanced, but not insurmountable. Incorporate effective study skills from other areas into calculus and you’ll quickly improve your understanding.
As with any area of study, when it comes to calculus, you’ll want to do well. Ensuring that you have an effective working understanding of mathematical basics, like arithmetic, trigonometry, geometry, and algebra, is a solid starting point. From there, make sure you understand the fundamental theorem and the relationship between integral and differential equations.
Adopt an effective study routine that works for you, and stick to it so when you’re practising new equations ahead of your calculus test, you’re feeling confident and ready to take on the next challenge. Calculus can be a deeply useful and advantageous form of maths to learn, but a deeper working knowledge of mathematics and the world around us will be truly rewarding.