Our life is represented by the shadow being enlarged. As we’re getting older, our responsibilities are getting larger just like the man’s shadow as he keeps walking forward. In the beginning of the poem, it represents our childhood, past, where it felt like we had endless time, our life was like an open equation, where we didn’t have any responsibilities, doubts, or concerns. All we were interested in was having fun and taking numerous of opportunities to find our self identity. The mathematical components were subtly incorporated in the poem, such as the mentioning dx/dt which represents how fast the person is moving, representing our life moving forward, and  dy/dt representing the shadow’s length rate of change, which represents our responsibilities accumulating over time as we grow. As the man keeps walking, it goes through the cycle of life, where he has a job, responsibilities, a family to start to uphold as a citizen and parent. And finally, whenever he stops and gets to his destination, that also signifies when our responsibilities have stopped (whether that may be starting a family and children, passing the responsibilities onto our children, or etc).

The mathematical topic we were inspired to create the poem about was related rates, specifically the shadow problem, which we’ve all done before, it’s the lamp and the man walking, and as the man's shadow gets larger, we have to calculate the rate at which the shadow is changing.  The theme of the poem is about how rapidly our lives are moving.  Our life is represented by the shadow being enlarged. As we’re getting older, our responsibilities are getting larger just like the man’s shadow as he keeps walking forward. In the beginning of the poem, it represents our childhood, past, where it felt like we had endless time, our life was like an open equation, where we didn’t have any responsibilities, doubts, or concerns. We also mentioned some related rates concept to connect the illustration and poem such as deriving the base equation to get the derivative, so we're able to find the equation for dy/dt, the shadow's flow.

In one of the discussion board assignment, I learned about the fact that sunflower seeds are arranged in spirals that follow the Fibonacci sequence.

Often there are 34 spirals in one direction and 21 spirals in the other direction. Smaller sunflowers may be composed of 8 and 13 spirals. To optimize space and reduce shadowing among seeds so they get the most sunlight, the angle between successive seeds in a sunflower head approximates the golden angle (about 137.5 degrees), which is derived from the golden ratio - phi, the exact limit of a(n+1)/a(n) when n->infinity in Fibonacci Sequence, approximately 1.618. Why 137.5? Becuase (360-137.5)/137.5 = 1.618.

I was intrigued about the regularity, yet could only understand the phenomenon but not the proof or reason. So I built an interactive demo on Unity Engine to visually and intuitively demonstrate how the Golden Ratio of 137.5 degrees actually maximizes the area each single seed is distributed with. For the purpose of playability, I also made sound effects with synthetic instruments and gradiently-changing visual effects for aesthetics. I also put in some easter eggs (too stealthy for anyone but myself), just for kicks. Even though the whole project only consists of one single game object, I found it very difficult to make the animation effect smooth with the use of parameters. I had to go through a lot of tutorials to learn where exactly I should place a parameter and which parameter controls which effect, such as the angular speed and acceleration/decceleration, and how sensitive the pattern is to the mouse's movement. The audio part took much more time than I expected because I had a hard time finding the perfect synthetic effect and chord. For some this can be a very easy program, but I had to learn a lot of things from scratch, and it took me a lot of time to implement this seemingly easy effect.

Nevertheless, when playing with the model to tweak the parameters, I actually kind of have a better understanding on why 137.5 is so special. It is most irrational angle, causing the least overlapping of the seeds so each gets the most area. If 120, after 3 turns there is an overlap. But for 137.5, or more rigorously, 360*(1-phi), There will never be overlaps, and the seeds are not even showing any tendency of [approaching one another]. Though I still can't fully understand the mathematical proof, visually I can see why, which is the whole point of this project.

At first, it was challenging to bring a creative and artistic approach to a topic I usually only see in equations and graphs. I had to think about how to visualize calculus in a way that was both accurate and visually interesting. Once I decided to use a roller coaster as my inspiration, it became really fun. I realized that loops and smooth curves naturally connect to sine functions, and the space between the rails is exactly what the area between curves measures. This project pushed me to think more visually about calculus and helped me understand how integrals don’t just calculate numbers, they also measure space and shape. Now I see that art and math can work together to make abstract ideas more concrete and creative.

-Leslie Ortiz-Patino

This video displays how an Alternating Series can oscillate from positive numbers to negative numbers. It also displays how an Alternating Series can converge or diverge through the contraptions I built in the video.

In terms of calculations, I did not do many calculations. However, in terms of graphs and planning, I had to map out how many blocks I would need to make these contraptions to get the desired result. Also, as you can see, the contraptions represent graphs, which is also something I planned before creating.

Initially, it was somewhat challenging to represent an Alternating Series in Minecraft because I was not completely sure how I would represent it. So, I began thinking about how an Alternating Series behaves, and concluded that I could essentially build a contraption that represents a graph, that could display the idea of an Alternating Series, along with how it could converge or diverge.

-Justin Ho

In this project, we explored how to find the volume of a 3D shape using calculus. We used the function f(x)=square root of x and rotated it around the x-axis to form a solid. Then we built a physical model out of paper disks to represent slices of the shape. This helped us understand how integration works by adding up all the tiny pieces to make the whole volume.

Link to code and files: https://drive.google.com/drive/folders/1ablwouI8HnjPayko3lKyLbXTQLZuJenh?usp=sharing

The mathematical topics and key ideas used for this project are 2d vectors, Conservation of Linear Momentum and Kinetic Energy and the concept of elastic collisions (from my calculus based physics class). Most concepts in physics including these are derived from calculus, in fact, Isaac Newton the inventor of calculus and the inventor of the derivative: first defined his 2nd law of motion as F_net= dP/dt = ma  ( dP/dt is the derivative of momentum with respect to time). My car crash simulator represents the idea of Conservation of Linear Momentum which is derived from newton’s 2nd law, because each car only has an initial velocity but no net forces acting on them:

F_net=dP/dt → 0 = (P_f-P_i)/(t_f-t_i)  → 0 = P_f-P_i → P_f= P_i (line 116 of my code)

Momentum is defined as: P= mv, 

since there are two masses (the car and the bus):

m₁v_1i+m₂v_2i = m₁v_1f+m₂v_2f  (lines 81-84 of my code)

Elastic collision: Since this is an ideal simulation, both Kinetic Energy and Momentum are conserved

Kinetic energy is defined as KE = 1/2mv^2 so therefore (using the same principle as for conservation of linear momentum, W-KE theorem) I can write:
1/2m₁v_1i^2 + ½ m₂v_2i^2 = 1/2m₁v_1f^2 + ½ m₂v_2f^2

After solving both conservation equations for v₁ and v₂ (v_1f and v_2f) I get these two equations:

v₁ = ( (m₁-m₂)v_1i + 2m₂v₂ ) / (m₁ + m₂) and v₂ = ( 2m₁v₁ + (m₂-m₁)v_2i ) / (m₁ + m₂)  (lines 81-84 in the code)

After the collision I calculate the reaction force of the ejected driver on vehicle they collided with (again using newton’s 2nd law): F = dP/dt → F = (P_f-P_i)/0.1 (the 0.1 is just a constant that I think is pretty realistic for a real collision) (line 180 and 190)

And then if F exceeds the Lethal Force the driver is pronounced dead.

The most challenging part of coding this simulation was definitely ejecting the the driver properly, other than being annoying to code, at first the driver would just collide with the other driver causing death almost always, so I opted to let the drivers pass through each other. Also synchronizing the vectors was slightly tedious. The derivation was not that difficult but it also took a bit to put my idea into words and narrow down the scope of my project. 

This is a link to our google slide presentation:

https://docs.google.com/presentation/d/1tyDJ8p2ntvqax7e7p2wqQDTIZ82rq8DTla-IeOYz9vs/edit?slide=id.g36917592e0b_7_1#slide=id.g36917592e0b_7_1

In Calculus, a derivative measures how one quantity changes with respect to another. In game design, especially when simulating car mechanics, derivatives translate directly into the core physics that make driving feel authentic. By treating a car’s position along the track as a function of time, the first derivative gives its velocity, essential for rendering speedometers, motion blur, and engine sound pitch. The second derivative provides acceleration, which designers use to balance throttle response and collision forces. Understanding these derivatives lets developers fine-tune everything from drift behavior to braking distance, turning raw math into a realistic driving experience.

Let’s imagine a racing car in game speeding up a straight track. Its position over time is given by a function and from that, we can calculate its velocity and acceleration. We use the idea of a racing car to explore how derivatives describe motion, specifically position or  distance, velocity, and acceleration. In calculus, we start with a function that describes position over time,  like s(t)=2*t^2. So in the  video at the top right of the poster, we used the function s(t) = 2*t^2 to represent the position of a racing car over time. The dot which is Bebe on the graph is not just a point, it actually represents the real racing car which  is Olivier’s racing car in motion. As time goes on, both Bebe and Olivier  move together, and we move faster and faster. This shows how the car's  position changes with time, and how the math behind the curve directly  reflects real world motion.  

The first derivative (The left graph), s′(t) gives us velocity, how fast the car is  moving. For our function, s’(t)=4t, which means the car’s speed increases as time passes. The first derivative represents, the steeper the slope, the faster the car is moving at that moment, if the slope is zero, it means the car is at rest at that moment (which means it’s not moving), and if the slope is negative, it means the car is moving backward  (The 2nd derivative graph) Then the second derivative (the right graph) , s”(t) gives us acceleration, which means how fast the speed itself is changing. In our case, the acceleration is constant: s”(t)=4. That means the car is getting faster at a steady rate. The second derivative represents if the second derivative is positive, the car is speeding up, if it’s zero, the car is moving at a constant speed (which means no  acceleration), and if the second derivative is negative, the car is slowing down (which  is decelerating).

One of the challenges I faced during the process is connecting art and mathematics because for me it seemed like two different worlds. However, I discovered that math can be anywhere, in any form which is beautiful.

At first, it was challenging to bring a creative and artistic approach to a topic I usually only see in equations and graphs. I had to think about how to visualize calculus in a way that was both accurate and visually interesting. Once I decided to use a roller coaster as my inspiration, it became really fun. I realized that loops and smooth curves naturally connect to sine functions, and the space between the rails is exactly what the area between curves measures. This project pushed me to think more visually about calculus and helped me understand how integrals don’t just calculate numbers, they also measure space and shape. Now I see that art and math can work together to make abstract ideas more concrete and creative.

At first, it was challenging to bring a creative and artistic approach to a topic I usually only see in equations and graphs. I had to think about how to visualize calculus in a way that was both accurate and visually interesting. Once I decided to use a roller coaster as my inspiration, it became really fun. I realized that loops and smooth curves naturally connect to sine functions, and the space between the rails is exactly what the area between curves measures. This project pushed me to think more visually about calculus and helped me understand how integrals don’t just calculate numbers, they also measure space and shape. Now I see that art and math can work together to make abstract ideas more concrete and creative.

We use the sine wave formula from physics:
f(t) = A·sin(2πft + ϕ)

Since we’re working with pure tones, we set the amplitude A = 1, and because there’s no phase shift or volume adjustment, we set ϕ = 0. This simplifies the equation to:
f(t) = sin(2πft)

For frequency, we have a standard frequency for each note. For example, G4 has a frequency of 392 Hz (the frequencies of the notes used in our song are included in the slide). In our composition, we use the chords Gmaj7 and Am7.

Gmaj7 has the notes G, B, D, and F#, representing a combination of sine functions at their respective frequencies (e.g., sin(2π·392t), sin(2π·494t), etc.).

Am7 includes the notes A, C, E, and G and is similarly modelled using the sine functions for those notes.

Our song is set at a tempo of 130 BPM (beats per minute).

To calculate timing:

One beat = 60 / 130 seconds

One bar = 4 beats → time per bar = (60 / 130) × 4

Then we got the time duration for each chord in our song. The final chord sheet is included in our presentation slide.

Additionally, we analyzed the chord pattern as a sequence. Our song repeats Am7 and Gmaj7 in a loop, so it can be called a periodic sequence. But some songs follow a non-periodic pattern. For example, "Shape of You" by Ed Sheeran includes different chords and note changes in some parts, and it can be called a non-periodic sequence.

The challenge we faced

It was a bit challenging for us to create the function sheet because we had never imagined the chord sheet we usually use to play instruments as mathematical functions. We tried playing the chords on a guitar using only the time durations provided by the functions, but we realized that in real life, it’s difficult to play music accurately without knowing the number of beats and bars. The function gives very specific time values, like 0.4615 seconds, which are hard to follow.

The presentation slides

https://www.canva.com/design/DAGqomIGJZQ/ZBo07FxlSneCPaRzY4NLGw/edit?utm_content=DAGqomIGJZQ&utm_campaign=designshare&utm_medium=link2&utm_source=sharebutton

The first stanza sets the tone by introducing the idea that curves (functions) have hidden meaning, and that taking the derivative is like uncovering a secret, finding the slope at any point. Each of the following stanzas then focuses on one of the major rules, giving it a unique personality. For example, the power rule is described as strong and simple, while the chain rule is shown as something magical that connects layers together. I used metaphors like dancing, singing, and weaving to give these rules more life and emotion.

Writing this poem helped me understand the rules better because I had to think about what each one really does, and how they are different from each other. Instead of memorizing steps, I thought about the meaning behind the math. I also wanted to make the poem feel peaceful and elegant, to show that calculus can be beautiful in its own way. Overall, this poem is meant to help others (and myself) remember that there’s a kind of creativity and depth in math, even in something as technical as derivatives.

-May San Htar Phyu

We will integrate from y=0 to y=r(radius)

Area of disc : A(y)=π(r^2 -y^2)

The volume of each disc : A(y)dy=π(r^2−y^2)dy

Now we integrate from 0-r.

V=π[r^2 y - (y^3)/3]

Plug in 3, and we will find that V=18π in^3

For this project, I used cardboard and acrylic paint, and paper and markers. I really enjoyed making this project-it was super relaxing in contract to the studying for my other finals. The white paint I used was the only issue I had, as the cardboard I used was black and the paint had to be layered multiple times to reach the opacity I wanted. I found it really wholesome to connect art to calculus, and especially in such a straightforward way. This project made me want to make 3D models for more irregular shapes, such as the buildings in modern architecture, and have a more genuine appreciation for calculus.