1. business and industrial
2. business operations
3. management
4. project management

AP CALCULUS AB FINAL PROJECT
Real Life Applications of Calculus
PROJECT DESCRIPTION
This project can be done as an individual or as a group. You will choose your own topic, and it should be
something from real world that requires technology to prepare and present the contents. I expect more
ambitious project with a bigger group of people. Some project ideas are provided as examples, but your groups
is more than welcome to come up with own topic. There are four requirements that you should think of when
designing your project.
1)
2)
3)
4)
Your project must use SOMETHING REAL: REAL DATA, or REAL VIDEO or a REAL IMAGES
You must use some sort of CALCULUS that you did not know before taking this course
You must use some sort of TECHNOLOGY (i.e. Desmos, Excel, Video Editing, Google Earth…)
You need to find a way to PRESENT your findings to both me and your classmates.
This project idea is heavily borrowed from Bowman Dickson of King’s Academy in Jordan. The link to the original
material is provided: http://bowmandickson.com/2012/05/13/and-the-calculus-final-projects-begin/
SCHEDULE
Due: May 13th, Wed (Section A) / May 14th, Thu (Section B)
Group Forming and Topic Description
Goal – Students will find their project group members and they will choose their topic.
Document – Name of members and detailed description of the topic. Description needs to explain the goal
of the project and how it will be achieved and presented.
Due: May 21st, Thu (Section A and B)
Outline of Tasks and Deciding Roles
Goal – Each group will decide role and responsibility of its members, and create list of tasks to complete
the project.
Document – Task list and each member’s task assignment.
Due: June 3rd, Wed (Section A) / June 4th, Thu (Section B)
Completing the Project
Goal – Three class periods will be provided as work days.
Document – Completed project.
Due: June 8th - 9th, Tue – Wed (Section A and B)
Presentations
Goal – Two class periods will be used as presentation days.
SUBMISSION
On top of documents laid out in the schedule, each group has to submit evidence of their in-class work by the
end of each class meeting starting May 11th, Monday. Evidence can be anything from a phone photo of your
team’s brainstorming work to short clips of audio or video. To submit your work, create a Google Drive folder for
your team and then share it with Mr. Pak’s school account ([email protected]). If none of your team
members know how to do this, make sure to learn from Mr. Pak.
GRADING RUBRIC
 Completeness (60%): Your group’s finished project has to clear demonstrate four elements: it’s something
real, using calculus, incorporates technology, and presented in organized manner.
 Timeliness (40%): Each document has to be submitted by its deadline. Each instance of delay will cause 5%
deduction PER DAY.
PROJECT IDEAS
Graphing Stories Video
Create videos for many different aspects of differential calculus (i.e. a constantly increasing position, a constant
position, a constantly increasing velocity etc) in a story. You may do other quantities besides position vs. time,
for instance, pain vs. time.
Google Earth Land Project
With the help of Google Earth, analyze maps of the world to answer an interesting question. For example:
 Find a few nice plots of land on Google Earth that you would like to own. Find their areas and think of
some other things you would like to know about them. For example, how much does the land cost per
acre, how many houses could you put on it, how much seed would you need to plant it?
 Compare the plots of land of various celebrities and find their area with various methods (Riemann Sums,
modeling functions etc). Compare your answer with stated land areas.
 Answer an interesting question like “How much concrete does the city of Cheongna use?”, “How many
miles of roads are there in Cheogna?” etc.
Related Rates with Filling a Container Up
(adapted from Sam Shah’s Math Blog - http://samjshah.com/2011/02/01/related-rates-see-em-in-action/)
If you pour a liquid into a straight-sided glass at a constant rate, the water level will rise at a constant height.
However, if you pour it into a container that does not have straight sides, the water level will rise at a varying
rate depending on the shape of the container. Collect a few different containers (interesting flower vases,
Gatorade bottles, various household glasses) and film yourself filling up the containers with a liquid. Then
analyze the height of the liquid at any time t, and come up with a way to talk about dh/dt.
Solids of Revolution
Find the volume of interesting solids. Examples include Coke bottle, instruments like a tuba, or that sofa in the
high school corridor.
Real Data Function Modeling
Using real data, analyze some sort of phenomenon in the same sort of way that we investigated the population
growth and radioactive decay. Tell me something interesting about these phenomena in terms of functions,
their derivatives and their integrals. Fit function to various sections of the curves that you create and discuss
why different functions fit in different places. Some examples:
 Examine the spread of a real disease (malaria, AIDS etc) in a limited community and see how closely the
spread of the disease fits with various models for something like this,
 Graph the record for a specific event in the sporting world (the 100 m dash in track, the 200 m freestyle
in swimming, the Boston Marathon record) and fit functions to the data to try to determine a
theoretical limit for the record (in addition to answering some of the questions above),
 Investigate the rate of oil consumption or production or anything else having to do with the looming
Global Energy crisis,
 Investigate the life expectancies of a few different countries and try to model their life expectancy with
a function, or
 Examine the earnings of certain movies and use your investigation to predict how much a movie that is
currently in theaters will earn.
Write and Answer a Calculus Letter
(adapted from “Writing in Mathematics” by Dr. Annalisa Crannell)
Write and answer a fake letter from a businessman, politician or scientist that poses a simple problem for which
you need the use of Calculus to solve. The problem should be detailed and involved, and the answer should
involve data (even if it is made up), graphs and functions. Here are examples of what I am talking about:
https://edisk.fandm.edu/annalisa.crannell/writing_in_math/Myron%28calc-I%29/myron2.htm
https://edisk.fandm.edu/annalisa.crannell/writing_in_math/Myron%28calc-I%29/myron1.htm
https://edisk.fandm.edu/annalisa.crannell/writing_in_math/GOOF/goof1.htm
Write a Calculus Song
Write and record a song about a concept (or many concepts) in Calculus. It could be a song that explains
something, a song that will help students in the future remember something, or anything else.
Optimization Exploration
Collect as many commercial containers of a specific type and analyze how efficiently they use materials
compared to the optimal dimensions of an object with that shape. Rank your containers from most efficient to
least efficient. In your investigation, include possible reasons as to why companies would design containers the
way they do if they are not the most efficient shape.
History of Calculus: The Calculus Wars, Leibniz vs. Newton
Research the founding of modern day Calculus, or rather how two mathematicians independently came up with
the Fundamental Theorem of Calculus connecting integral and differential calculus. Instead of writing a paper
(that has been done thousands of times), come up with a creative way to present the information – a play, a
dialogue, a debate, a court case. Visuals including the notation that they used at the time would be wonderful.
Zeno’s Paradox
An ancient Greek philosopher posed a bunch of really interesting questions that get at the heart of Calculus. One
of them goes something like this: Everything that moves, must always go halfway to its destination before it gets
there. Then it must go halfway again, and again and again. How does it ever make it? Another goes something
like this: At any given moment in time, an object is still, so how does something move from one place to another?
Complete some sort of demonstration of Zeno’s paradox to explore this deep question of philosophy in a
creative way. For example, a student in the past told the story of someone trying to get out of a ticket for
running a stop sign because they showed a picture that seemed to indicate that they were still at a moment
when they were behind the stop sign. The cop and biker argued about this idea using the fundamentals of
Calculus to see who was right.
Design a Building
Design a building using Calculus to help you figure out something about the building, whether that be the area
of a curvy part, or calculating the optimal shape and size given certain constraints (like a greenhouse or
something with solar panels). You can make a model of your building, construct a model on the computer, or
draw up diagrams to display plans for your building.
Make a Calculus Game
Design and construct a game to help students either learn or practice a specific concept in Calculus. It should be
interesting, creative, and durable enough to be used in future years. Possibilities include both physical game
boards and some sort of computer or calculator based-game.