Introduction to Calculus | ChemTalk

Core Concepts

In this article, we will introduce the concept of calculus. We will illustrate its origin, founding principles, and potential applications.

What is calculus?

Calculus is the study of change! More specifically, it’s the process of using math to model changes.

Credit is given to both Issac Newton and Gottfried Leibniz who, independently of each other, had both been developing calculus in the 17th century. At the time a key mathematical discovery, calculus is now considered a central piece of knowledge to engage in other areas. For example, physicists use calculus to model weather dynamics, explaining the dynamic and thermodynamic systems at play. In chemistry, calculus makes up a massive portion of our understanding of the way electrons move via the particle in a box model. Pharmacology uses calculus to study metabolism and improve drug absorption. There are so many more ways that calculus intersects with a myriad of other disciplines! First, let’s see an introduction of how calculus became the powerhouse it is today.

Principles of Calculus

Calculus is a vast subject that spans far beyond the reaches of only mathematics. It is built around four key concepts. Below are explanations of these concepts and how to apply them.

First, let’s review the meaning of a function and function notation. An expression that defines the relationship between one variable (the independent variable) and another variable (the dependent variable) is called a function. Functions are often denoted with the notation “f(x),” which translates to “y as a function of x.” The variable y is frequently represented as f(x), because it quickly explains the relationship between x and y.

Limits and Continuity

The first concept that we’ll explore in detail is the role of limits and continuity in calculus. A limit is the value that a function approaches, given an argument or a value that is chosen. Although limits are the basis of all calculus, their purpose is often overlooked.

Limits take into account the behavior of a function, and serve as a way of predicting how a function will proceed. As an example, look at this table, which shows values reprsenting the function: f(x) = x – 8

Note that, as we move x progressively closer to 9, f(x) moves closer to 1! We can describe this phenomenon as follows: “the limit of f(x) as x approaches 9.” This is a mouthful and not simple to write out, so limit notation was created to simplify things:

Limit notation is an easier way to express the information needed to identify a limit. The special thing about limits is that, whether you approach the x value from the left or the right, the resulting limit is the same value! See the graph below for an example:

Using the red arrow as a reference to follow the line from the left side towards positive 20, we can see that, as x approaches 15, the y value approaches approximately 17. Simultaneously, if we use the green arrow to follow the line from the right side towards 0, notice that the y value is still approaching 17! As a result of this line’s continuous nature, it will always have the same limit, irrespective of the direction you approach.

However, in cases where a line lacks continuity, limits take on new forms. Continuity describes the ability to draw a graph or line without lifting up one’s pen. In terms of limits, though, it more so refers to a situation where the limit as approached from the right is different from the limit as approached from the left. A graph may lack continuity if a point on the line is undefined, as shown in the following example:

In this example, the limit would be approximately 80. This is because, although the x value we are approaching is not defined, the limit from both the left and the right are the same!

However, not all graphs are continuous. Discontinuous graphs need to be addressed differently. The issue with discontinuous graphs is that the limit as you approach the left and right can be different, like in a piecewise function:

The graph above shows a discontinuous, piecewise graph where, as x approaches -1 from the left or right, the y value is not the same. As x approaches from the right, following the green arrow, y approaches positive 1. But when x approaches from the left, following the red arrow, it approaches -1. In cases like these, the limit does not exist (DNE for short).

DNE not only describes this scenario, but also cases where the graph increases or decreases infinitely as you approach the x value of interest. Another way to conceptualize a DNE limit is to think of x approaching an imaginary line at some value, as the curve proceeds into infinity while never actually reaching the imaginary line. This is also known as an asymptote. The limit does not exist as, with the asymptote present, there is no way to approach an x value from the left nor from the right while getting the same y value.

We’ve learned a lot so far! Let’s try some practice problems to put these concepts together:

Practice Problem #1: Limits and Continuity

Derivatives

Building upon limits, let’s introduce the concept of derivatives in calculus! Derivatives measure instantaneous rates of change at a point. Like limits, derivatives have a unique notation used when discussing them. There are three main notations, the most common of which is the format credited to Leibniz.

The derivative is similar to a slope in many ways. Mathematicians noticed this connection and incorporated it into the formulation of the derivative formula:

In order to use this derivative formula, we need to plug in our function, f(x), into it. For example:

Step one of solving for any derivative is to plug in all known information into the formula. Next, we distribute out the terms in the numerator!

With everything distributed out, it’s clearer to see that there are some like terms. Using subtraction, cancel out the like terms!

Once again there are like terms, but now they are split by the fraction bar. Cancel these terms via division, and the resulting number is the derivative!

Solving for the derivative of 8x, the result is 8! This is an overview of learning how to solve for a derivative!

Integrals

Derivatives and limits are closely related, but let’s briefly take a step away from limits. On the other side of a derivative is an integral, the area underneath a curve. Also referred to an antiderivative, the integral is most commonly represented by the following notation:

Solving an integral through a Riemann sum is a common methodology. Named after Bernard Riemann for his work in formulating integrals as we often see them today, Riemann’s method allows for the approximation of the area under a curve using rectangles! To use Riemann sums, you must be given a derivative, as well as the range of x values you are evaluating. The resulting solution will explain how the function changes as x changes.

There are two ways to approach: from the left or from the right. The left-sided approach means that the top-left corner of the rectangle will touch the curve. The right-sided approach means that the top-right corner of the rectangles will touch the curve. Below are the formulas for both right- and left-sided Riemann Sums and how to use them

It’s important to note that both methods yield approximations, not perfect answers. Depending on the function, this method can overestimate or underestimate the true area under the curves. Nevertheless, it is a useful method for integration, and is one of the fundamental steps in the integration process. To illustrate this, let’s do an example of both right and left Riemann sums using the same equation! We’ll start with right Riemann:

Step one is always solving for Δx ( “delta-x“), or the width of the rectangles. First, plug in all the known values into the formula. Given this example, the upper bound (b) is 5 and the lower bound (a) is 0. Following the equation, subtract b and a, then divide by the number of rectangles (in this case, 5).

This results in the calculated width of each rectangle being 1, so the next step is the solve for x_i, which represents the height of each rectangle. Here is how we calculate x_i for right Riemann sums:

As stated previously, first we’ll plug in all the information we have! We just calculated Δx in the step above as 1. i represents the count of the rectangles. Since this is the first rectangle we are calculating for, i = 1.

Now we can evaluate the equation for the height of rectangle number 1. It is imperative to remember to adjust i per rectangle when solving for x_i.

Finally, to officially calculate the area under the curve, we use the area formula! The first step is to plug in the values into the summation. f(x_i) indicates that for every x_i value, plug the result into f(x) and solve. Then, add each f(x_i) together and multiply by Δx. The resulting integral is the approximate area under the curve!

In this case, the area under the given curve for right Riemann is approximately -40.

Subsequently, for the left Riemann sum, the process is very similar, up until the calculation of x_i. Notably, there is a slight formulaic change between the two methods, which yields different rectangular heights. Recall that Riemann sums yield approximations, so the rectangular heights will be slightly different.

In the left Riemann, x_i is 0 for the first rectangle, rather than 1 like in the right Riemann. This indicates that, in this scenario, the left Riemann will be a smaller value than the right Riemann sum. In calculating for the final integral using the area formula, the method is exactly the same as before!

In the case of the left Riemann, the area under the curve is approximately -40! As you can see, the left and right Riemann sums approximate very similar areas. Either one can give you a solid estimation of what the area under the curve looks like. For visual learners, here are the graphical representations of left and right Riemann sums:

For the problem above, the “true” area is actually -35.83, which falls in between the left and right Riemann sum results! As you advance in calculus, you will learn more about how to calculate a more specific and accurate area. For now, here are some practice problems illustrating the lessons above:

Practice Problem #2: Integrals

Conclusion

The discovery of calculus has allowed for the study of things that change over time! For instance, the study of limits empowers us to observe how a curve changes over time. A derivative represents the state of a curve at a specific moment in time. Contrarily, an integral shows the space under a curve over an amount of time. These concepts allowed mathematicians and scientists around the world to answer questions that were too complicated to be addressed by linear equations. Calculus continues to grow and change, answering newer, more complex, and more exciting questions about the world around us.

Mini-Quiz

Take this quiz to explore your understanding of the fundamentals of calculus. These problems are slightly more difficult than our previous practice problems, but the goal of this quiz is to test your learning, not to judge your perfection. Attempt each problem on your own first before referencing notes. Answers are provided below to help you check your work.

Question 1: What is the limit for f(x)= x² as x approaches 2?
Question 2: What are the derivatives of f(x)=x³, 5x, and x² + 4x – 4?
Question 3: Given f(x) = x³ + 2x² + 5x – 10, what are the right and left Riemann sums? Use 3 rectangles, given an upper bound of 10 and a lower bound of 0.

Answers

Please make attempts at all questions before viewing the answers!