Leah Parsons

The graph known as the Keeling Curve is the longest uninterrupted record of atmospheric carbon dioxide in the world, with data collection beginning in the 1950s. There are so many different angles from which to interpret and discuss this data, from a science lens to an ethics and values viewpoint to a sustainability interpretation. However, due to time and space limitations, as well as the purpose of writing on a blog called Got Calc?, the important information about the Keeling Curve that we will discuss here is this: It is a perfect real-life example from which we can explore different period functions, and examine the rate of change (i.e. derivatives) in each of the three models (linear, exponential, and polynomial) during a certain year.

Before we get into all of this though, here’s a little more background about the Keeling Curve:

Beginning in 1958, the curve displays data that describes seasonal and annual changes of CO₂ buildup in the middle layers of the troposphere. Climate scientist and namesake Dr. Charles David Keeling of the Scripps Institution of Oceanography first devised the graph at the Mauna Loa Research Observatory in Hawaii, where he managed air sampling efforts between 1958 and 1964. 

The y-axis unit used on the graph is parts per million (ppm), which represents the number of CO₂ molecules present per every million molecules of air. Overall, the Keeling Curve shows that average CO₂ concentrations in the air per year have increased substantially since 1959; average concentrations used to rise by about 1.3 to 1.4 ppm per year until the 1970s, but they began increasing by more than 2 ppm per year in the 2010s. The graph also shows seasonal trends; in general, CO₂ concentrations decrease during the Northern Hemisphere spring and summer months. This is due to an increase in photosynthesis as a result of the rapid vegetation growth during this season.

Now that we have some background, we can get into the calculus portion. How do we fit functions to the Keeling Curve data to actually model carbon dioxide concentrations in ppm?

Well, here are three functions that can model this general trend, with t representing number of years since 1950 for the sake of simplicity:

,  , 

The first function, , is a linear function model, which is the simplest and therefore best used for quicker approximations. This linear model shows us right away that there is a positive continuous growth trend in the data with respect to time, with a positive slope of 1.3 ppm/year; meaning that if the data were to fit a simple linear model starting with 303 ppm of CO₂ in 1950, the concentration should increase by 1.3 ppm each year. The linear model means that the rate of change of carbon dioxide is constant from year to year– if we are looking to predict the rate of change in a specific year, such as 2010 which is after exactly 60 years, it would still be 1.33 ppm/year; which also happens to be the derivative of the function after .

The next model, , is an exponential function model. This function is able to give a more detailed description of the variation in carbon dioxide concentrations since 1950, because as an exponential growth curve, it now tells us that the slope of the function becomes steeper as time moves forward. In other words: the rate of change that determines the amount of carbon dioxide concentration in a specific year gets larger as time goes on. This is a representation of how a portion of carbon dioxide becomes trapped in the atmosphere each year, compounding onto the portion that was already trapped. So, the rate of change is not constant. Therefore, in contrast to the linear model, if we wanted to predict what the rate of change of CO₂ would be in a specific year like 2010, we would need to find the slope of the line tangent to the curve at that specific data point in the year 2010. This means finding the derivative of the function, and then plugging in 60 years for (because ). To find the derivative, we use the exponential rule, which is a variation of the chain rule: the full function multiplied by the derivative of just the power, which is . Plugging 60 years for into the derivative, we get a predicted rate of change of 1.45 ppm/year of CO₂ levels in 2010; this is a larger predicted rate of change for 2010 than we calculated with the linear model.

The last model, , is a quadratic polynomial function model. This model insinuates that the graph is shaped like a parabola; however, we are only looking at one side of the parabola in terms of growth rate of CO₂ levels in relation to time, which can only be where . Therefore, this model shows that when , the graph crosses over the y-axis at 310.5 ppm of CO₂. Using this model to predict the rate of change for 2010, we would again start by finding the derivative of the function, which in this case would simply be . Plugging in 60 for and multiplying, we get: The rate of change in CO₂ concentration levels in 2010 based on this polynomial model is 2.133 ppm/year. This is the largest predicted rate of change for our example year of 2010 out of all the models shown here, and, in comparison to actual data from 2010, also the most accurate prediction.

Here are the three functions plotted together using the Desmos graphing calculator, placed directly above a graph of the Keeling Curve:

Sources used:

https://www.britannica.com/science/Keeling-Curve

https://keelingcurve.ucsd.edu/

https://www.amnh.org/explore/videos/earth-and-climate/keeling-s-curve-the-story-of-co2/dataset-information