application of differential calculus

Project Guidelines Math 165 Overview: One of the best ways to learn a subject deeply is to teach it. The goal of this project will be to explore some application of differential calculus in detail, and to write a worksheet that could teach this topic to your peers. You will work in your regularly assigned groups. Grades will be assigned individually. You will develop a 5-page worksheet on this topic, where the first 4 pages include necessary background information, and the fifth page is about more advanced material. Your topic must include both mathematically novel information, and a concrete application of calculus to other fields. Potential topics are included at the end of this document, but you may also pick another topic. Example: An inappropriate choice of project is to discuss the relationship between position/velocity/acceleration, because we have already talked about that in class. But, it would be appropriate to discuss a physics application we have not seen, like representing force with derivatives. Group component: As a group, you should submit a 4-page worksheet on your topic. This worksheet should contain all of the necessary background information and definitions necessary to understand your topic, spaced throughout the worksheet. But, the bulk of the worksheet should be devoted to problems about your topic, with space to write the solutions to these problems. Your worksheet should be written at a level where classmates not in your group can successfully work through your problems, and also would learn new information by completing the worksheet. The questions on your worksheet should explicitly involve the topic you picked, and should not be simple modifications of questions we have seen before. The questions and solutions should also not be directly copied from an outside source. Your worksheet must be typed, but the solutions may be handwritten. No space should be devoted to problems we have already seen this semester, or simple variations of them. Individual component: Each group member should also individually develop a fifth page to your worksheet which includes more challenging questions based on your topic. This last page to the worksheet should be a natural continuation of the rest of your worksheet. The goal of this page is to show your depth of understanding, and to challenge the reader’s mastery of the topic. You may discuss the fifth page with your group members, but no two submissions should contain simple variations of the same questions. Final Submission: As a group, you will submit the first four pages of your worksheet and a 4-page solution set, which may be typeset or handwritten. Individually, you will hand in a 5th page to your worksheet, solutions to that page, and a short write-up describing how each group member contributed to the project. Timeline: • Monday, April 19th: During lecture, your group will start working on this project together. Your group should decide on a project topic by the end of the day, along with at least two potential references. The same topic can be chosen by at most one group per recitation section. Topics can be claimed first-come-first-served by posting on your recitation channel in Teams. – Deliverable: Each group must also post in their group’s recitation channel on Teams to claim a topic by 11:59 p.m. CST on Monday. You cannot pick a topic another group in your recitation has already picked! • Tuesday, April 20th: In recitation, your group should create a solid foundation for the project. Find at least two sources for your project, neither of which can be a source we regularly use in class. Read through the sources you identified, and make sure you have at least two reliable sources. Come up with a list of key definitions and foundational material that will need to be presented in your worksheet. Develop at least two questions by the end of the day. – Deliverable: In Gradescope, upload the following: a brief summary of your topic, a list of the 2+ sources you found, and a commitment in writing to how each member of the group will contribute to the draft due next Tuesday. • Tuesday, April 27th: We will not work on the project during recitation. But, your group should have a draft of the first two pages of the worksheet by the end of the day, along with solutions. – Deliverable: On Gradescope, hand in the draft of the first two pages of your project worksheet and its solutions. • Thursday, April 29th: Clean up the first two pages of your draft based on feedback from your TAs, and work towards finishing the group portion of the project. Commit in writing to how each member of the group will contribute to the project by the due date. – Deliverable: On Gradescope, hand in written commitments to how each group member will contribute to the final project submission. • Friday, May 7th: All written components are due by the end of the day. – Deliverable: As a group, hand in the group portion of your worksheet and solutions through Gradescope. Individually, hand in your individual page, solutions, and write-up of your group members’ contributions. Also, upload a blank copy of your worksheet labelled with your team name to Teams, under “Final Project” → “Files”. • Monday, May 10th, 1-3pm: Final project activity. During our final exam period, we’ll share some of the worksheets you wrote with each other. We will merge groups together, and you will walk your classmates through your worksheet. 2 1. Base group score: 60%. (These scores may be raised or lowered for each group member based on group member feedback and also contributions in recitations and lectures.) • 15%: Your worksheet was problem-based, and developed mathematically sound, calculus-based questions that were not mathematically identical to questions from class. • 15%: The problems on your worksheet integrated a new topic or an application of calculus that we have not seen before. • 10%: The solutions to the group problems (pages 1-4) are correct, clearly written, and easily readable. • 10%: Your worksheet gave the necessary background information for a classmate to follow your problems. • 5%: Your worksheet was easy to read, neatly organized, and free of grammatical errors • 5%: You used outside sources and cited them. 2. Individual score: 40% • 10%: Your individual worksheet page was an extension of your worksheet that displayed your mastery of the topic. • 10%: The solutions to your individual problems (page 5) were correct, clearly written, and easily readable. • 5%: You assessed your group members’ contributions to the project. • 5%: You correctly uploaded your project to Gradescope and met all intermediate deadlines. • 5%: You were able to guide your classmates through your worksheet during recitation. • 5%: You actively participated in other group worksheets during recitation. 3 Potential Topics: Some topics have sample questions to give you ideas of where to get started. If you choose a topic that has sample questions, you do not need to use them to develop your worksheet. 1. Chemical reaction rates 2. Applications of Newton’s Method for finding zeroes • What is the method? What are examples of the method? Where is it used in reality, and why? 3. Physics. Do NOT talk about position/velocity/acceleration. We have already seen that in class. Examples could include: • Representing force using derivatives • Differential equations for waves, heat, and diffusion • Maxwell’s equations (more challenging) • Radioactive decay, Newton’s Law of Cooling, with derivations 4. Biology and modeling populations. Note: we have already seen some population modelling, so your worksheet would need to go beyond these examples. 5. Economics and business • Marginal cost, minimizing costs • Profit and revenue • Compound interest 6. Space math. Several ideas for where to start at NASA’s website. It is not OK to copy these questions, and your worksheet needs to contain new questions. • • • • Changing comet/asteroid sizes Carbon dioxide in the atmosphere Cell sizes Why are hot things red? 7. Probability • By using Riemann sums, how do we derive the formula, ˆ b P(a ≤ X ≤ b) = p(x) dx? a • What is the normal distribution? Why is numerical integration required to compute probabilities with it? What example problems involve the normal distribution? 8. Beam stress and deflection, the Euler-Bernoulli equation 9. The ideal rocket equation from aerospace engineering (more challenging) 10. Maximum likelihood estimators in statistics. 11. Numerical solving of differential equations (with applications). 12. Calculus and music, like here or here. 13. Calculus and sports, like here. 14. One of the many applications discussed in Calculus and its Applications. 15. One of the many applications discussed in Calculus in Context, like models for disease spread. 5 Project Guidelines Math 165 Overview: One of the best ways to learn a subject deeply is to teach it. The goal of this project will be to explore some application of differential calculus in detail, and to write a worksheet that could teach this topic to your peers. You will work in your regularly assigned groups. Grades will be assigned individually. You will develop a 5-page worksheet on this topic, where the first 4 pages include necessary background information, and the fifth page is about more advanced material. Your topic must include both mathematically novel information, and a concrete application of calculus to other fields. Potential topics are included at the end of this document, but you may also pick another topic. Example: An inappropriate choice of project is to discuss the relationship between position/velocity/acceleration, because we have already talked about that in class. But, it would be appropriate to discuss a physics application we have not seen, like representing force with derivatives. Group component: As a group, you should submit a 4-page worksheet on your topic. This worksheet should contain all of the necessary background information and definitions necessary to understand your topic, spaced throughout the worksheet. But, the bulk of the worksheet should be devoted to problems about your topic, with space to write the solutions to these problems. Your worksheet should be written at a level where classmates not in your group can successfully work through your problems, and also would learn new information by completing the worksheet. The questions on your worksheet should explicitly involve the topic you picked, and should not be simple modifications of questions we have seen before. The questions and solutions should also not be directly copied from an outside source. Your worksheet must be typed, but the solutions may be handwritten. No space should be devoted to problems we have already seen this semester, or simple variations of them. Individual component: Each group member should also individually develop a fifth page to your worksheet which includes more challenging questions based on your topic. This last page to the worksheet should be a natural continuation of the rest of your worksheet. The goal of this page is to show your depth of understanding, and to challenge the reader’s mastery of the topic. You may discuss the fifth page with your group members, but no two submissions should contain simple variations of the same questions. Final Submission: As a group, you will submit the first four pages of your worksheet and a 4-page solution set, which may be typeset or handwritten. Individually, you will hand in a 5th page to your worksheet, solutions to that page, and a short write-up describing how each group member contributed to the project. Timeline: • Monday, April 19th: During lecture, your group will start working on this project together. Your group should decide on a project topic by the end of the day, along with at least two potential references. The same topic can be chosen by at most one group per recitation section. Topics can be claimed first-come-first-served by posting on your recitation channel in Teams. – Deliverable: Each group must also post in their group’s recitation channel on Teams to claim a topic by 11:59 p.m. CST on Monday. You cannot pick a topic another group in your recitation has already picked! • Tuesday, April 20th: In recitation, your group should create a solid foundation for the project. Find at least two sources for your project, neither of which can be a source we regularly use in class. Read through the sources you identified, and make sure you have at least two reliable sources. Come up with a list of key definitions and foundational material that will need to be presented in your worksheet. Develop at least two questions by the end of the day. – Deliverable: In Gradescope, upload the following: a brief summary of your topic, a list of the 2+ sources you found, and a commitment in writing to how each member of the group will contribute to the draft due next Tuesday. • Tuesday, April 27th: We will not work on the project during recitation. But, your group should have a draft of the first two pages of the worksheet by the end of the day, along with solutions. – Deliverable: On Gradescope, hand in the draft of the first two pages of your project worksheet and its solutions. • Thursday, April 29th: Clean up the first two pages of your draft based on feedback from your TAs, and work towards finishing the group portion of the project. Commit in writing to how each member of the group will contribute to the project by the due date. – Deliverable: On Gradescope, hand in written commitments to how each group member will contribute to the final project submission. • Friday, May 7th: All written components are due by the end of the day. – Deliverable: As a group, hand in the group portion of your worksheet and solutions through Gradescope. Individually, hand in your individual page, solutions, and write-up of your group members’ contributions. Also, upload a blank copy of your worksheet labelled with your team name to Teams, under “Final Project” → “Files”. • Monday, May 10th, 1-3pm: Final project activity. During our final exam period, we’ll share some of the worksheets you wrote with each other. We will merge groups together, and you will walk your classmates through your worksheet. 2 1. Base group score: 60%. (These scores may be raised or lowered for each group member based on group member feedback and also contributions in recitations and lectures.) • 15%: Your worksheet was problem-based, and developed mathematically sound, calculus-based questions that were not mathematically identical to questions from class. • 15%: The problems on your worksheet integrated a new topic or an application of calculus that we have not seen before. • 10%: The solutions to the group problems (pages 1-4) are correct, clearly written, and easily readable. • 10%: Your worksheet gave the necessary background information for a classmate to follow your problems. • 5%: Your worksheet was easy to read, neatly organized, and free of grammatical errors • 5%: You used outside sources and cited them. 2. Individual score: 40% • 10%: Your individual worksheet page was an extension of your worksheet that displayed your mastery of the topic. • 10%: The solutions to your individual problems (page 5) were correct, clearly written, and easily readable. • 5%: You assessed your group members’ contributions to the project. • 5%: You correctly uploaded your project to Gradescope and met all intermediate deadlines. • 5%: You were able to guide your classmates through your worksheet during recitation. • 5%: You actively participated in other group worksheets during recitation. 3 Potential Topics: Some topics have sample questions to give you ideas of where to get started. If you choose a topic that has sample questions, you do not need to use them to develop your worksheet. 1. Chemical reaction rates 2. Applications of Newton’s Method for finding zeroes • What is the method? What are examples of the method? Where is it used in reality, and why? 3. Physics. Do NOT talk about position/velocity/acceleration. We have already seen that in class. Examples could include: • Representing force using derivatives • Differential equations for waves, heat, and diffusion • Maxwell’s equations (more challenging) • Radioactive decay, Newton’s Law of Cooling, with derivations 4. Biology and modeling populations. Note: we have already seen some population modelling, so your worksheet would need to go beyond these examples. 5. Economics and business • Marginal cost, minimizing costs • Profit and revenue • Compound interest 6. Space math. Several ideas for where to start at NASA’s website. It is not OK to copy these questions, and your worksheet needs to contain new questions. • • • • Changing comet/asteroid sizes Carbon dioxide in the atmosphere Cell sizes Why are hot things red? 7. Probability • By using Riemann sums, how do we derive the formula, ˆ b P(a ≤ X ≤ b) = p(x) dx? a • What is the normal distribution? Why is numerical integration required to compute probabilities with it? What example problems involve the normal distribution? 8. Beam stress and deflection, the Euler-Bernoulli equation 9. The ideal rocket equation from aerospace engineering (more challenging) 10. Maximum likelihood estimators in statistics. 11. Numerical solving of differential equations (with applications). 12. Calculus and music, like here or here. 13. Calculus and sports, like here. 14. One of the many applications discussed in Calculus and its Applications. 15. One of the many applications discussed in Calculus in Context, like models for disease spread. 5
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