### 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:

\[ s = 16 t^2 \]

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:

\[ t = \pm \sqrt{\frac{s}{16}} \]

Now \(s\) is the *independent* variable and \(t\) is the dependent variable

The notation \(f(x)\) can also represent functions without an 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 the notation 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:

\[ y = x^2 \]

The function above is simple in that to each value of \(x\) there's a corresponding value of \(y\), whoever the concept of a function does not require this, for example the function:

\[ y = \frac{1}{x} \]

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 exist is called *domain* and the collection of the corresponding \(y\) values is called *range*.