# Calculus 1

## Examine a new world called Calculus 1. With these articles and videos that explain everything step-by-step, you will master calculus.

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.

## Limits (Calculus Starts with Limits)

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

## Evaluating Limits Analytically (Using Limit Theorems) [Video]

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.

## Intuitive Introduction to Limits (The Behavior of a Function) [Video]

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.

## Related Rates (Applying Implicit Differentiation)

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.

## Numerical Integration (Trapezoidal and Simpson’s)

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.

## Integration by Substitution (by Example)

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.

## Fundamental Theorem of Calculus

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.

## Integral Definition (The Definite Integral)

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?

## Area and Limits of Riemann Sums

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.

## Indefinite Integrals (What is an antiderivative?)

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.

## L ‘Hopital’s Rule and Indeterminate Forms

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.

## Parametric Equations and Calculus (Finding Tangent Lines)

Parametric equations are incredibly intuitive, especially when the parameter represents time. In this article, I consider how to sketch parametric curves and find tangent lines to parametric curves using calculus. I also describe motion along a curve and parametric equations in polar form.

## Linearization and Differentials (by Example)

In this article, I discuss what differentials are and how to use them. I also explain what linearization is and demonstrate how to linearize a function at a point. So, you will understand the so-called tangent line approximation with these examples.

## Optimization Problems (Procedures and Examples)

I discuss several types of optimization problems, such as: optimizing with numbers, volume, geometry, area, angles, distance, time, and others. In the beginning, I present you with an optimization procedure and then take you through it with each of these examples.

## Inflection Points and Concavity

What is the difference between the functions: taking the square of a number or squaring a number. Both of these functions are increasing. But one of them is growing must faster than the other. Comparing the growth of functions is the idea behind concavity and is foundational in computer science.

## First Derivative Test (and Curve Sketching)

When finding the relative extrema of a function, if any, it is not enough to find a function’s critical numbers. In this article, I discuss the First Derivative Test and show how to perform curve sketching.

## Monotonic Functions (Increasing and Decreasing)

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.

## Mean Value Theorem (and Rolle’s 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.

## Extreme Value Theorem (Finding Extrema)

The Extreme Value Theorem is helpful. It says that if a function is continuous on a closed bounded interval, it must attain its maximum and minimum values. But what are critical numbers, relative extrema, and absolute extrema? We go through all of these concepts in detail.

## Derivatives of Inverse Functions

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.