An Introduction To Calculus

An Introduction to Calculus

When a student hears the word “calculus”, it is often followed by ideas of dread and a general lack of enthusiasm. Why is this? Well, calculus is often the first idea in mathematics which is “out of the norm” for students, not following the general patterns that arithmetic or geometry do.

What this means, is that it seems very unusual and complex to the average student. In actuality, calculus is a very intuitive topic in maths with some amazing theories and ideas that can make even the most unenthusiastic mathematician get excited.

So what exactly is calculus? Simply put, calculus is the study of change. It studies infinity and how we can use it to create things.

Limits

Before we start, we need to look at what “infinity” actually is, because infinity as we know it (∞) is not actually a number, but rather an idea. To start, lets look at some notation:

\lim_{x \rightarrow \infty} f(x)

It is easiest to understand what this expression is actually saying by wording it out. $\lim_{x \rightarrow \infty} f(x)$ means “taking the limit as x tends towards infinity”. It is basically looking at how the function behaves as it approaches infinity. Note that I put “tends towards” and “approaches” in bold, as the function can never reach infinity.

Now it is time to look at an example: $\lim_{x \rightarrow \infty} \frac{1}{x}$ . To find this, we can examine what happens as we input very large values into $\frac{1}{x}$ : As the input increases the outputs denominator increases, meaning the output will decrease towards zero. So in this case: $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$

Note that the value that a function tends towards does not have to be infinity, and can be an arbitrary number, but this will just have the same affect as plugging the number in if the function behaves nicely at that point.

\lim_{x \rightarrow a} f(x) = f(a) \\  \\ \text{If $f(x)$ is continuous at $a$.}

The Derivative

Next up is the derivative, which basically is a way of looking at the rate of change or gradient of a function. There are many notations for a derivative, but I will be using the Lagrange notation for the most part. If $f(x)$ is a function, $f'(x)$ is the function’s derivative, which in itself is another function whose output for a certain value of x is the gradient of f(x) at that certain value. The formula for a derivative is as follows:

f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}

Breaking Down The Formula For The Derivative

Although it looks daunting, if you remember the formula for gradient, $\frac{\Delta y}{\Delta x}$ , then you will be fine as this is just a modified version of this formula. To find the derivative, we take a small nudge in the x value, $h$ , and we want to see by how much the y value changes, which in this case can be found using $f(x+h)-f(x)$ . Then using the gradient formula, we get the fraction in the formula for a derivative. Remember, at this point we can choose what value to take $h$ as, but we know that as $h$ gets smaller, the fraction gets a more accurate value for the gradient. What we then can do is use the limit notation to “make” $h$ tend towards zero. This gives us the formula!

\text{Some General Results:} \\  \\ \frac{d}{dx}\left(f(x)\right)=f'(x) \\  \\ \frac{d}{dx} (a)=0 \\  \\ \frac{d}{dx}\left(x^n\right)=nx^{n-1}

The Integral

On our final list of notations to cover, we have the integral, which is one of the more complex ideas in introductory calculus, an integral is used when you want to find the area “under” a curve of a function (although there are many more applications than just this), the following notation for an integral is used:

\int_a^b f(x) dx \ \ \ \text{means the area under the graph from x=a to x=b}

As the intuition and formulae for integrals are far more complex, I will not be covering them in as much detail. But there are some amazing resources which I have linked at the end of this article.

The Fundamental Theorem Of Calculus

This theorem states that there is a link between integration and differentiation, and that this link is that they are the opposite of each other this theorem is shown with the following formula on the right. Note that the integral has no limits, as this is an indefinite integral which is a function rather than a definite integral which spits out a number.

\frac{d}{dx} \left( \int f(x) dx \right)=f(x)

Other Resources

“The essence of Calculus” is a youtube series created by 3Blue1Brown, which is an amazing way to get new perspectives on the topic, and I would highly recommend it as well as all of 3Blue1Brown’s other videos.