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Our work is about the application of calculus in calculating the lift provided by an airplane wing, and the downforce provided by a car’s rear wing.

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

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For my project, I have done String Art Parabola and used derivatives to calculate each tangent line for 20 tangent points.

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i tracked my heartrate for two weeks and found how much my heart is expected to beat in one year according to a model I designed from the data I got.

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Derivatives in Motion: The Hidden Story Behind Acceleration

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Derivatives in Motion: The Hidden Story Behind Acceleration

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My project focuses on higher-order derivatives, particularly using the rules of differentiation to explore motion. This concept was often introduced through the position-velocity-acceleration model in calculus. I applied the rules of differentiation to a cubic function:

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.

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The topic i chose were parameters,and when you explained about paramters like a boat circle around a certain point. It made me think about how the earth circles the sun with its orbit.And its parameters can dertermine where the earth is currecntly located as well.

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The legend of Sisyphus is a Greek story about a man who is being punished for cheating death. Rather than giving a simple punishment such as eternal pain or torture, Zeus decides to give him the task of pushing a boulder up a mountain, and once that task is complete, he is free to go. The catch is that the mountain is impossible to get to the top of, no matter how close it may seem or how hard he tries.

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

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I have written a poem about my math journey at De Anza, starting from summer 2024, when I started with pre-cal 1, all the way until Math 1B. I tried to include every core concept that I learned along the way in this. Hope you guys like it! Thanks

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For this project, my classmate, Nisa, and I decided to make a beat because of our shared love for music. With no prior experience, we layered a variety of instruments, created an arpeggio, and extracted sound waves from the piece to find the derivative. The rate of change in a sound's amplitude or frequency gives insights into the quality and clarity of sound. Derivatives can be used in musical composition as they allow us to manipulate and synthesize sound more effectively.

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The chain rule is the great representation of how the sequence of the functions matters on the derivative. The same way that the sequence of pouring the shots and the milk into cup will affect on your coffee.

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.

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I chose this 3d printing project because I love parametric equations and the concept of limits. I think these two ideas together can help us see how the world slowly takes shape.

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.

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This is the PDF of my comic strip story called "Tommy's Integral Mountain Aventure." I attempted to create a simple concept that combines real-life examples of integration techniques with an educational approach similar to those found on PBS, which breaks down complex concepts in a way that everyone can understand.

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https://drive.google.com/file/d/1lHVfCvM5dhK-D8-ezVjHNC4vyfdq81hz/view?usp=drive_link

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This is my final project, it is a book with 3D graphics to be able to get more learning, and I could really enjoy doing it, because I was able to mix all the criteria that the teacher asked for in a single project, and I think it is quite useful to be able to remember formulas, examples and processes of how we find the answer of the Different functions. It took me about 12 hours to carry out this project, because I had to cut everything with precision and with an exact measure.