When going over the possible calculus topics I could choose for this project, I settled for limits because it is something that I am familiar with and I feel that the rest of the class is too. When brainstorming what possible topics relate to limits, I thought of Sisyphus. I felt that it was a perfect representation of limits going to infinity because in the tale, he will never reach the top, or in my depiction, x=a. He will continue to push the rock up the slope of Lim x->a f(x) = Infinity and getting infinitely close to that limit.

Some challenges I had while translating the idea into reality is not really anything math related, but more of just drawing Sisyphus. I ended up doing 3 iterations of this just to try to perfect it.

Overall, this project enhanced my understanding of limits even more and I also enjoyed my time using some creativity to depict calculus.

-Douglas Jones

s(t) = 2t³ − 15t² + 24t

From this, I computed the velocity v(t) = s’(t) and acceleration a(t) = s”(t), eventually solving for a (5).

The first illustration shows a student solving for acceleration on a chalkboard, connecting classroom learning to the real world. 

In the second digital illustration, we see the derivatives broken up in terms of what occurs during real-time motion: the car moves (position), then accelerates (velocity), then jumps into a second (acceleration). This illustration, in a sense, takes the viewer through a visual triptych concerning how calculus comes to life and how each derivative layer alters the car's behavior.

The calculations is shown below.

s(t) = 2t³ − 15t² + 24t

v(t) = s′(t) = 6t² − 30t + 24

a(t) = s″(t) = 12t − 30

Evaluating at t = 5:

a(5) = 30

This was the important value I used to represent the idea of acceleration after 5 seconds.

Translating abstract math into relatable visuals was somewhat complicated. I was thinking the whole time about how I was going to relate derivatives to the car and make it feel real, not just symbolic. It was a challenge to design motion using still images, and at times I felt a bit distressed or stuck trying to bring the concept to life in a creative way. At the same time, however, I found the process of negotiating this, as being fun and interesting, and it forced me to think about calculus in a fashion that was more creative than I am used to. It was fun figuring out a story in relation to math, and I had not expected that to actually be a somewhat fun aspect of the project overall.

I gained a deeper appreciation for how higher-order derivatives are more than just symbolic — they describe the world. Acceleration is not just math — it is something we feel when we are in cars, on roller coasters, and in life. Making this helped me picture how correlated calculus is to physical experience.

The aerodynamic principle is that the upper surface of an airplane wing is more curved than the lower surface. The airfoil with this kind of shape has a name called “asymmetric airfoil”, which causes the airflow over the upper surface to move more faster than the airflow beneath. According to Bernoulli’s principle, this results in lower pressure on top and higher pressure underneath. The pressure difference between two surfaces generates an upward force called lift, allowing the airplane to rise into the air.

To model this, I drew a simplified airfoil cross-section and graphed two functions. In this graph, the variable x represents the position along the wing’s cross-section, from the leading edge to the trailing edge. This is also called the chord length. The vertical axis shows P, which is the pressure at each point along that surface. And, the red curve for the pressure on the top surface,while the blue curve for the bottom. Using calculus, we can calculate the lift by finding the area between these two curves.

This integral gives us the total lift force per unit width of the wing.

And speak to the downforce on the car’s rear wing. Interestingly, the rear wing of a car uses the same aerodynamic principle, but flipped upside down. The rear wing is designed so that the air over the top surface moves slower than the air underneath. This creates higher pressure on top, and lower pressure on the bottom, resulting in a downward force called downforce. The downforce generated by the rear wing helps keep the racecar stable at high speeds and improves its handling performance.

Even though the direction is different, the math behind them is exactly the same.

One of the challenges we faced during this project was understanding the concepts. Because it involved Bernoulli’s principle, we had to deal with several physical quantities and different units, which is not easy, and we also need to combine them with calculus.Drawing the graph was also a bit difficult, since the structure is quite detailed, it took us some time and effort to draw the graphs accurately.

This project helps us understand the importance of calculus in engineering. It gives engineers a tool to analyze data, enabling them to figure out which designs are more efficient and effective. While working on this project, we also gained a deeper understanding of how to calculate the area between curves. Drawing these graphs helped us visualize the concept more clearly.

By Xinyang Liu

Haoran Zhang

Derivative of F(g(x)) and derivative of G(f(x)) would differ depending on the actual functions you plug in.

In comparison, the result of switching milk and shots would result in different coffees you'll get.

The butterfly curve I used is defined by this parametric equation: x(t)=sin⁡(t)(e^cos⁡(t)−2cos⁡(4t)−sin⁡^5(t/12)), y(t)=cos⁡(t)(e^cos⁡(t)−2cos⁡(4t)−sin^⁡5(t12)). I wanted people not only to see how this beautiful pattern grows with time (t), but also to touch the path of its transformation. I printed the butterfly at different values of t: 4π, 8π, 16π, 48π, and 96π. One of my biggest challenges was turning the 2D curve into a 3D printable model. I failed several times, but through these failures I learned how to fix problems and understand the curve more deeply. This was my first time creating a 3D print from scratch. I discovered that math is not only logical, it can also be artistic, physical, and even emotional. Math not just as equations, but as a way to build and feel the beauty of the world. We can never truly reach limits, but as time goes on, it keeps growing and becoming more complete. Just like the butterfly curve, we are all shaped by the journey.