Highlights of Calculus

- Functions

Calculus is about functions. In calculus, a function is a rule that transforms one real number into another real number. A graph is a geometric representation of that rule. 

Following are three examples:

• Linear function:  2x
• Squaring function: x²
• Exponential function: 2^x

The first point is that those are not the same! Their formulas involve 2 and x in very different ways. When we draw their graphs (this is a good way to understand functions) you see that all three are increasing when x is positive. The slopes are positive. 

When the input x increases (moving to the right), the output y also increases (the graph goes upward). The three functions increase at different rates. 

The first point is that those are not the same! Their formulas involve 2 and x in very different ways. When we draw their graphs (this is a good way to understand functions) you see that all three are increasing when x is positive. The slopes are positive. 

- Graphs

When the input x increases (moving to the right), the output y also increases (the graph goes upward). The three functions increase at different rates. 

A function has inputs x and outputs y(x). To each x it assigns one y. The graph of a function is the set of all points in the plane of the form (x, f(x)). It can also be defined as the graph of the equation y = f(x). 

To graph a function, you can evaluate the function at a few points. For example, to graph f(x) = 2x, you can plug in a few x-values to determine the corresponding y-values. 

- Functions and Graphs

Here are some other things to know about functions and graphs:

• Graphs are tools used in many areas, including science, engineering, technology, and finance.
• Graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis. 
• The most common graphs name the input value x and the output value y. 
• In modern foundations of mathematics and set theory, a function is equal to its graph. 

Please refer to the following for more details:

• Wikipedia: Function

• Wikipedia: Graph of a Function

- Special Functions

Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. 

Some of them are listed below: 

• f(x) = ln(x) 
• f(x) = x^2 
• f(x) = cos (x) 
• f(x) = 1 / (x^2 – 1) 
• f(x) = tan (x)

We encounter functions when we need to adjust an algorithm to minimize it. We will encounter different algorithms when practicing machine learning models, and it is crucial to understand how these functions react and transform it. 

[More to come ...]