Calculus Honors Quarter 1 Project
Background:
This project will be a large-scale project which will count as 15% of your quarter grade. This should be taken very seriously and academic honesty is of the upmost importance. Workdone should be your own. You are expected to work on these individually and follow all required guidelines. Academic Dishonesty of any kind is not tolerated and will be given a 0 for the grade. Due Beginning of Class Friday Oct 11. Late submissions will not be accepted.
Purpose:
This project is aimed to help you gain in-depth knowledge of the continuity and differentiability of functions. Limits and differentiation are two critical topics in Calculus which have numerous real world applications.
This project will require research, critical thinking, application of knowledge, calculation, creativity, and the use of technology. You will be asked to design a roller coaster using a piece- wise function with at least 6 functions that is everywhere continuous and differentiable, and create an A3 poster of your design.
Part One: Design a Roller Coaster
You willcreate a design for a rollercoaster. You must:
1. Use at least 6 piece-wise functions which includes at least 3 function types among the following: polynomials, root, rational, trigonometric (and obviously others as you wish!)
2. Show all calculations to prove
a. Functions are continuous
b. Functions are differentiable
3. Write the complete piece-wise function that represents the entire roller coaster path.
4. The roller coaster should start and end with a horizontal line which must be equal (loading area).
*Note: Loading area is not part of the piece-wise functions (in terms of differentiability).
5. Use desmos (or any equivalent graphing software/application as you like) to draw the 2D representation of the roller coaster.
See a picture of an example of the graph below.
Part Two: Create an A3 poster of the Roller Coaster
You willcreate an A3poster ofthe rollercoaster. You must include the following information:
1. Label the different pieces of the roller coaster.
2. For each piece, label the function that is selected.
3. Explain why those functions where chosen (as in how they play a role in your Roller Coaster).
4. Name the roller coaster!
5. Clearly show your calculations for functions being continuous and differentiable at the points where the pieces meet.
6. At the minimum your graph of the roller coaster should be colorful (each piece should be a
different color), and display all required information. Bonus points will be rewarded (see rubrics below) for visually pleasing and neat graphs, whether it’s hand-made or printed.
7. Flexibility in general is encouraged, if there is anything else that you deem fit to be added to the poster, feel free to do so.
Scoring Rubric
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Category
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Points Possible
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Points Earned
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Regular Scores
• Explain why the
differentiable functions [at least 6] used to create the piece-wise path were chosen (10 pts)
• Complete calculations to prove differentiability and continuity of functions (25 pts)
• Complete final functional expression for the
piecewise path (5 pts)
• Complete and well-
formatted and labelled roller coaster path graph (10 pts)
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50
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Bonus Scores
• Complexity in the functions chosen
• Exceptional design
• Neat, creative, visually stimulating
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Up to 5 (to be
rewarded through
other means if
exceeding 100% on
the project)
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