PDF for mathematics from the official Punjab Board website. Access chapter-wise details for mathematics with the link to ...

In today’s data-driven world, graphs are everywhere—from COVID-19 curves and economic forecasts to climate models and ...

The truth is painfully simple: consistency isn’t just important; it’s everything when it comes to achieving extraordinary ...

Complete Explanation A logarithm is the power which a certain number is raised to get another number. Before calculators and various types of complex computers were invented it was difficult for ...

Transformations of graphs of exponential and logarithmic functions. Transformations of exponential and logarithmic function graphs involve dilating (also known as stretching or compressing), ...

In other words, Inverse functions are rules that relate each \(y\) value to only one \(x\) value. These kinds of functions are named one-to-one functions. However, not every function has an inverse.

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. The formula for exponential growth is V = S x (1+R) T, where ...

How India's luxury market is becoming a stellar example of exponential growth Ever since the pandemic subsided, Indians have come out in droves, with revenge shopping initially, and then backing ...

A new Lyapunov function is proposed to simplify the proof, which does not require the traditional mathematical induction method for intermittent control. Based on the Lyapunov method and inequality ...