Calculus 1 is a high-school or undergraduate mathematics course. Newton and Leibniz invented calculus, and then it was significantly improved upon by Euler, Gauss, and many others. Topics center around the concepts of the derivative and the integral. However, to correctly understand calculus, an understanding of limits and continuity is essential. For this reason, students first study limits of functions, followed by the continuity of functions. Then the main topics of differentiation and integration fill the rest of the course. Towards the end, instructors surprise students with a magnificent theorem.
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.
This article covers different forms of numerical integration: the midpoint rules, trapezoidal rule, and Simpson’s rule. For each technique, I discuss the setup and work through examples. I also explain the error in the Simpson Rule.
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.
What are the most beautiful theorems of calculus 1? This article reviews the Intermediate Value Theorem, the Extreme Value Theorem, the Mean Value Theorem. I then explain the first and second fundamental theorems of calculus.
In this article, I discuss the definition of the definite integral and several properties of the definite integral. I cover many examples and also consider displacement and the area under a curve. So why is the definite integral so important?
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.
Who is L’Hospital, and why does he have a rule? What are indeterminate forms? I go over all these questions with many many examples. In fact, I give at least one instance of each of the seven types of indeterminate forms.
