Calculus through Data & Modeling: Limits & Derivatives

The Limit of a Function

One of the goals in studying Calculus is to analyze rates of change and movement In this module, we introduce the central ideas which will help us achieve this goal: the notions of the limit and the derivative Rather than evaluating a function at a single point, the limit allows for the study of the behavior of a function in an interval around that point In this module, you will find limits of functions by a variety of methods, both visually and algebraically Finally, we will apply limits to define the key idea of Differentiable Calculus, the derivative

The Limit Laws

Using calculators or graphs is an imprecise way to find the limit of a function In this module, we will state and use algebraic properties of limits, called the Limit Laws, to calculate the exact values of limits A solid understanding of these laws will allow us to derive theorems which in turn can be used to study the behavior of more advanced functions

Continuity

In the last module, there were several types of functions where the limit of a function as x approaches a number could be found by simply calculating the value of the function at the number Functions with this property will be called continuous and in this module, we use limits to define continuity We will see that the mathematical definition of continuity will correspond closely with the English meaning of the word continuity used in every day language

Limits at Infinity

In this module, we allow for x to become arbitrarily large in the positive or negative direction to understand the end-behaviors of functions This will allow for the formal definition of a horizontal asymptote and to provide classifications of end-behavior of certain types of functions

Derivatives

The problem of finding the slope of the tangent line to a curve and the problem of finding the instantaneous velocity of an object both involve finding the same type of limit This special type of limit is called the derivative and in this module, we will see that this notion of the derivative can be interpreted as a rate of change in any of the natural or social sciences or engineering

Final Project

In this final project, we will apply the tools and language of differentiable calculus to analyze trends in data This project will focus on modelling and analyzing gender ratios in educational attainment over time in several regions of the world