Limits are used to study the behavior of quantities under a process of change. For example, limits can be used to describe the behavior of a function on its domain. Here we study one-sided limits and two-sided limits with emphasis on graphs. We discuss unbounded behavior and oscillating behavior with many examples given. An Intuitive … Read more
This video explains the limit theorems, including the Product Rule, Quotient Rule, and the Polynomial Functions Rule, and the Rational Functions Rules. I then work through many examples, including factoring, rationalization, and some limits involving trigonometric functions. In the end, I illustrate the Squeeze Theorem and how to find inequalities when working with the Squeeze Theorem.
Using tables of values and graphs of functions, I explain what limits are and how to use them. Then I discuss one-sided limits, two-sided limits, and when limits do not exist. I use several illustrations to develop intuition with limits. Because these methods of estimating limits are not rigorous, we explain how estimating can lead to incorrect results.
Okay, suppose that you know two rates of change of two corresponding quantities, and you know of a relationship between three quantities. Your goal is to learn the third quantity’s rate of change but have no physical way to determine it. Enter related rates problems. You can solve them using implicit differentiation.
Okay, so you know of some basic rules of integration. For example, the integral of sine is minus cosine plus constant. But what about the integral of the double angle of sine? For this, we can use the “reverse of the chain rule,” but that I mean, make a substitution and use the composition of functions. The process is easy after some practice.
In this article, I begin with sigma notation and working with the properties of finite sums. Once the reader is familiar with this, I motivate the definition of a Riemann sum. Then, estimating the area under a curve using Riemann sums is discussed. Finally, finding area using the limits of Riemann sum is detailed.
What is an antiderivative? What is an indefinite integral? Maybe you have heard of them? Their reputation does often proceed them. Well, can you answer this question? What is the function, or what is a function that I need to have a given derivative? Read on to see how this idea leads to finding an area.
Monotonic simply means either increasing or decreasing. A function is monotonic if one or the other holds. In this article, I discuss at length finding extrema of monotonic functions. An essential tool in this regard is the First Derivative Test. Finding extrema becomes much easier when you have mastered this theorem.
The Mean Value Theorem, and its special case, Rolle’s Theorem, are crucial theorems in the Calculus. They provide a means, as an existence statement, to prove many other celebrated theorems. Understanding these theorems is the topic of this article. I go into great detail with the use of many examples.
Ok, so you studied inverse functions in precalculus. You know these types of functions are useful but can be abstract. You also know implicit differentiation by now. This article demonstrates a fantastic relationship between the derivative of an inverse of a function and its derivative. To understand what I just said, read on.