Why?

Calculus was created to solve some problems that other branches of math were not adequate to treat:

• Determination of tangents to various curves (e.g., to determine the course of a light ray after it strikes the surface of a lens).
• Finding the minima/maxima (e.g., determination of the maximum range of a projectile, maximum/minimum distance of a planet that is moving about the sun).
• Length of curves, areas, and volumes of figures bounded by curves.

To solve these problems, the following concepts are needed:

• Limit (fundamental to formulate the derivative and the integral).
• Derivative.
• Integral.

The Concept of a Function

A function is the relation between variables (whose value can be expressed numerically). The most effective mathematical representation of a function is through a formula like the one below:

The above formula says that when $t=2$, then $s=16\cdot 2^2=64$ and is represented as $s_2$. For each value of $t$, there’s a corresponding value of $s$. In the above form, $t$ is the independent variable, and $s$ is the dependent variable. If we solve the equation above for $t$, we have:

Now $s$ is the independent variable, and $t$ is the dependent variable.

The notation $f(x)$ can also represent functions without extensive verbiage, e.g., $f(x)=x^2-9$. This notation also has the advantage of telling us which is the independent variable. If we want to calculate the value of the function, we can use something like $f(3)$, which is the value of $f(x)$ when $x=3$.

A formula can also be represented as a curve (this method of interpreting formulas geometrically is known as analytic geometry). Let’s represent the following function below using a curve:

The function above is simple in that for each value of $x$, there’s a corresponding value of $y$. However, the concept of a function does not require this. For example, the function:

does not have a valid value when $x=0$. This means that the function exists for each value of $x$ other than $0$.

The concept of a function, then, doesn’t require that there’s a $y$ for every $x$, but it does require a $y$-value for each value $x$ in some collection/set of $x$ values. The collection of $x$ values for which a $y$ value exists is called the domain, and the collection of the corresponding $y$ values is called the range.

Image taken from https://wordsmithofbengal.wordpress.com/2021/08/02/dr-philos-the-creative-fantasy-of-differential-and-integral-calculus/