MAT 223 Final Project Guidelines and Rubric Overview At its essence, calculus is the study of how things change. In the field of information technology, the practical applications of calculus span a wide variety of industries and other areas, from data analysis and predictive analytics to image, video, and audio processing; from physics engines for video games to modeling software for biological, meteorological, and climatological models; and from machine learning and artificial intelligence to measuring the rate of change in interest-accruing accounts or tumors. What all these applications have in common is understanding how objects change with respect to time. The derivative function represents a rate of change. We can take the derivative of a function by using either the limit definition of a derivative or the different differentiation rules. What do we do when we don’t have a given function, but only a set of data points? There are two possible scenarios for the final project in this course. You must choose only one of the following options, which are outlined in the Final Project Scenarios document: 1. Motion Problem 2. Decay Problem You will create a report that illustrates your final answer, process, explanations, and detailed solutions. You will defend the validity of your solutions and demonstrate your ability to effectively communicate using calculus notations, conventions, and terminology. The project includes one milestone, which is an important opportunity to submit a draft of Part II and ensure the accuracy of your calculations. This milestone will be submitted in Module Five. The final product will be submitted in Module Seven. In this assignment, you will demonstrate your mastery of the following course outcomes:     Interpret real-world problems by selecting mathematical theorems that appropriately address the problem Utilize appropriate calculus techniques for solving real-world problems Determine the behavior of functions by analyzing a real-world model through appropriate calculus techniques Defend mathematical processes and solutions using appropriate calculus terminology Prompt Specifically, the following critical elements must be addressed: I. Introduction: In this section, you will briefly describe the mathematical theorem you selected, what you are trying to answer with your report, the approach for how you arrived at this selection, and the data points related to the rate of change of the object and how you will use them to arrive at your final results. 1 A. Briefly describe the mathematical theorems you selected and what you are trying to answer with your report. [MAT-223-01] B. Describe the approach for determining how you arrived at this selection. [MAT-223-01] C. Explain mathematically how the provided data will be used to arrive at your final results. [MAT-223-01] II. Analysis of Data: Applying Derivatives A. Using the given data, calculate the average acceleration of the changing object over given time intervals. [MAT-223-02] B. Using the given data, calculate the instantaneous acceleration at specific time values. [MAT-223-02] III. Analysis of Data: Applying Integrals A. Using the data provided, estimate the total change in the object. [MAT-223-02] 1. Use a right-endpoint estimate. 2. Use a left-endpoint estimate to approximate the total change of the object. 3. Calculate the best estimate for the total change of the object. B. Graph the model using the behavior of the functions represented by the data. [MAT-223-03] Note: To complete your graph, you can print graph paper at the Incompetech website, or you can complete a graph in Excel. The Microsoft Office website offers help with creating graphs in Excel. C. After completing your graph, discuss the relevance of the solution and how this graph represents it, using the calculus terminology of curve sketching. (Is the graph decreasing or increasing? How is this related to the data presented?) [MAT-223-04] IV. Analysis of the Model: Calculate Parts I and II. A. Given the model for each set of data, calculate the acceleration of the object using rules for differentiation. [MAT-223-03] B. Given the model for each set of data, calculate the total change of the object using rules for integration. [MAT-223-03] V. Final Results and Recommendations: In this section, you will conclude your report with your recommendations for a solution based on your findings. A. Compare the results obtained in Section II and Section III (Applying Derivatives and Applying Integrals) with the results in Section IV (Analysis of the Model). Discuss the accuracy of each method and explain the application of each method in a real-world context. [MAT-223-04] B. Defend your process of solving the problem by explaining a rationale for each process step. What does each step contribute to the ability to solve the problem and make recommendations? [MAT-223-04] C. Use calculus terminology to clearly explain your results and recommendations. Be sure to explain your answers using real-world terminology relevant to your topic in a way that is clear and understood. [MAT-223-04] 2 Milestones Milestone One: Draft of Part II In Module Five, you will submit part II of your project. This is an important opportunity to ensure your report is accurate and gain feedback prior to submitting your final project. This milestone will be graded with the Milestone One Rubric. Final Submission: Final Report In Module Seven, you will submit your final project. It should be a complete, polished artifact containing all of the critical elements of the final project prompt. It should reflect the incorporation of feedback gained throughout the course. This submission will be graded with the Final Project Rubric. Final Project Rubric Guidelines for Submission: Your final problem walkthroughs should be a 3- to 4-page Microsoft Word document with double spacing, 12-point Times New Roman font, and one-inch margins. Critical Elements Exemplary Introduction: Mathematical Theorems [MAT-223-01] Meets “Proficient” criteria, and the description illustrates an in-depth grasp of the mathematical theorems selected (100%) Proficient Needs Improvement Describes the mathematical theorems that were selected and the question being answered in the report (85%) Not Evident Value Describes the mathematical theorems, but does not describe the question being answered with the report, or the description is incomplete or contains inaccuracies (55%) Does not describe the mathematical theorems that were selected (0%) 7 Introduction: Approach [MAT-223-01] Meets “Proficient” criteria, and Describes the approach for the description demonstrates a determining mathematical keen insight into the process of theorems selected (85%) selecting appropriate theorems (100%) Describes the approach for determining mathematical theorems selected, but the description is illogical or incomplete or contains inaccuracies (55%) Does not describe the approach for determining mathematical theorems selected (0%) 7 Introduction: Explain Data [MAT-223-01] Meets “Proficient” criteria, and Explains mathematically how the explanation illustrates a the provided data was used to comprehensive application of arrive at the final results (85%) the mathematics used (100%) Explains mathematically how the provided data was used to arrive at the final results, but the explanation is illogical or incomplete or contains inaccuracies (55%) Does not explain mathematically how the provided data was used to arrive at the final results (0%) 7 3 Analysis of Data: Applying Derivatives: Average Acceleration [MAT-223-02] Correctly calculates the average acceleration of the changing object over all given time intervals (100%) Incorrectly calculates the average acceleration of the changing object over some of given time intervals (55%) Does not calculate the average acceleration of the changing object over all given time intervals (0%) 8 Analysis of Data: Applying Correctly calculates the Derivatives: instantaneous acceleration at Instantaneous Acceleration all specific time values (100%) [MAT-223-02] Applies correct calculus techniques in calculating instantaneous acceleration at all specific time values with minor errors in calculations (85%) Applies correct calculus techniques in calculating instantaneous acceleration at all specific time values with critical errors in calculations (55%) Does not apply correct calculus techniques in calculating instantaneous acceleration (0%) 8 Analysis of Data: Applying Integrals: Total Change [MAT-223-02] Estimates the total change using a right-endpoint, leftendpoint, and best estimate, with minor errors in calculation (85%) Estimates the total change using a right-endpoint, leftendpoint, and best estimate, with critical errors in calculation (55%) Does not estimate the total change using a right-endpoint, left-endpoint, and best estimate (0%) 8 Graphs the model using the behavior of the functions represented by the data (100%) Graphs the model, but the graph contains errors in construction, or the behavior of the function does not accurately represent the data (55%) Does not graph the model using the behavior of the functions represented by the data (0%) 7 Discusses the relevance of the solution this graph represents using the calculus terminology of curve sketching (85%) Discusses the relevance of the solution, but discussion is incomplete, contains inaccuracies, or does not properly use the calculus terminology of curve sketching (55%) Does not discuss the relevance of the solution this graph represents using the calculus terminology of curve sketching (0%) 8 Calculates the acceleration of the object using the model for each set of data correctly and applies rules for differentiation (100%) Calculates the acceleration of the object using the model for each set of data incorrectly, or does not properly apply the rules for differentiation (55%) Does not calculate the acceleration of the object using the model for each set of data or apply rules for differentiation (0%) 8 Correctly estimates the total change using a right-endpoint, left-endpoint, and best estimate (100%) Analysis of Data: Applying Integrals: Graph the Model [MAT-223-03] Analysis of Data: Applying Integrals: Discuss the Graph [MAT-223-04] Analysis of the Model: Rules for Differentiation [MAT-223-03] Meets “Proficient” criteria, and the description illustrates an in-depth grasp of the relevance of the relationship between the graph and the behavior of the function (100%) 4 Analysis of the Model: Rules for Integration [MAT-223-03] Calculates the acceleration of the object using the model for each set of data correctly and applies rules for integration (100%) Calculates the acceleration of the object using the model for each set of data incorrectly, or does not properly apply the rules for integration (55%) Does not calculate the acceleration of the object using the model for each set of data (0%) 8 Final Results and Recommendations: Compare Results with Analysis of the Model [MAT-223-04] Meets “Proficient” criteria, and the description demonstrates a keen insight into the accuracy and application of the calculation techniques (100%) Compares the results obtained in section II and section III with the results found in section IV, and includes a description of the accuracy of each method and explains the application of each method in a real-world context (85%) Compares the results obtained in section II and section III with the results found in section IV, but does not include a description of the accuracy of each method or does not explain the application of each method in a real-world context, or the comparison contains inaccuracies (55%) Does not compare the results obtained in section II and section III with the results found in section IV (0%) 8 Final Results and Recommendations: Defend Process of Solving the Problem [MAT-223-04] Meets “Proficient” criteria, and illustrates an in-depth grasp of the mathematical processes and their significance to solving the problem (100%) Defends the problem-solving process by explaining a rationale for each process step, including a description of what each step contributes to solving the problem and making recommendations (85%) Defends the problem-solving process, but the defense does not explain a rationale for each process step or does not include a description of what each step contributes to solving the problem and making recommendations (55%) Does not defend the problemsolving process by explaining a rationale for each process step (0%) 8 Final Results and Recommendations: Explain Your Results and Recommendations [MAT-223-04] Meets “Proficient” criteria, and demonstrates an extensive grasp of the application of calculus techniques for making recommendations in realworld problems (100%) Clearly explains the results and recommendations using proper calculus and real-world terminology relevant to your topic in a way that is clear and understood (85%) Explains the results and recommendations, but the explanation lack clarity, contains inaccuracies, or does not use proper calculus and real-world terminology in a way that is clear and understood (55%) Does not explain the results and recommendations using proper calculus (0%) 8 Total 5 100% Final Project I. Introduction Write the introduction LAST. This is so that you make sure that you notice everything that you have done in the project. Look at the description of the introduction below Introduction: In this section, you will briefly describe the mathematical theorem you selected, what you are trying to answer with your report, the approach for how you arrived at this selection, and the data points related to the rate of change of the object and how you will use them to arrive at your final results. 2 A. Briefly describe the mathematical theorems you selected and what you are trying to answer with your report. [MAT-223-01] B. Describe the approach for determining how you arrived at this selection. [MAT-22301] C. Explain mathematically how the provided data will be used to arrive at your final results. [MAT-223-01] • Do every piece of the introduction. I am looking for all elements of this in your introduction. II. Analysis of Data: Applying Derivatives Show all of your calculations. • • • • Explain the steps that you take and explain why you are using whichever formulas you use. Please do NOT change the functions you are given unless you define them. For example, there is a function called r(t) in one of the options. You can’t introduce f(t) in the calculations because it is not defined. This is done a lot in this project and it is incorrect. Please note that the instantaneous acceleration calculation is an approximation. That means you are not literally going to take a derivative. This process was discussed in Week 2. When calculation the instantaneous acceleration do NOT use a limit. There a question that would make you think that you need to use one, however, what you need to do is to explain the relationship between the average acceleration and the instantaneous acceleration. That’s the link between the calculations performed and the limit definition of the derivative. III. Analysis of Data: Applying Integrals • Show all of your steps. I mean all of them. Pretend like I don’t know anything about endpoint estimation. • Graph the model is NOT referring to the left and right hand endpoints. There is a table that needs to be completed. It is found on the actual project options. Please do not graph the estimations. • Don’t forget to do Part C where you explain the relevance of the graphs. IV. Analysis of the Model • Show all of your steps when calculating the derivative and integral. Please do NOT use an online calculator to do this. If you need help with taking derivatives and integrals, let me know. • Explain and compare your results. This is how I will understand how well you know the material. Please be sure to be thorough. V. Final Results and Recommendations • I will look for each piece of the directions below. Again, this is how I will gauge your understanding of the concepts in this class. A. Compare the results obtained in parts II and III with the results in part IV. Discuss the accuracy of each method, and explain the application of each method in a real-world context. B. Defend your process by identifying the appropriate explanation for each process step. C. Use calculus terminology to clearly explain your results and recommendations. Be sure to explain your answers using real-world terminology relevant to your topic in a way that is clear and understood. Final project milestone one 1 Elsie Udo Milestone One MAT 223 Application of Calculus February 09, 2020 Final project milestone one Scenario One: Motion Problem Part II: Analysis of Data – Applying Derivatives A. Average Acceleration is found by finding the slope as the change in velocity over the change of time. i. From t = 0 to t = 45 acceleration = (2.65 - 274.27) / (45 – 0) = -271.62 / 45 = -6.036 ft / sec^2 ii. From t = 25 to t = 45 acceleration = (2.65 - 80.80) / (45 – 25) = -78.15 / 20 = -3.9075 ft / sec^2 iii. From t = 40 to t = 45 acceleration = (2.65 - 18.04) / (45 – 40) = -15.29 / 5 = -3.078 ft / sec^2 B. Instantaneous Acceleration is found by finding the slope over the smallest interval possible around the time in question. 1. i. t = 5 Acceleration = (223.19 - 232.8) / (5 – 4) = -9.61 / 1 = -9.61 ft / sec^2 ii. t = 15 Acceleration = (141.4 - 148.52) / (15 – 14) = -7.12 / 1 = -7.12 ft / sec^2 iii. t = 25 2 Final project milestone one 3 Acceleration = (80.80 - 86.08) / (25 – 24) = -5.28 / 1 = -5.28 ft / sec^2 iv. t = 35 Acceleration = (35.91 - 39.82) / (35 – 34) = -3.91 / 1 = - 3.91 ft / sec^2 v. t = 45 Acceleration = (2.65 - 5.55) / (45 – 44) = -2.90 / 1 = -2.90 ft / sec^2 2. The limit definition of the derivative states that the derivative of a function is equal to the limit as h approaches zero of the slope between 2 points: (x+h, f(x+h)) and (x, f(x)). If you get an h value that is small enough, then the slope between the 2 points would be equal to the slope of the tangent line and also the derivative. In this case, finding the instantaneous acceleration is controlled by the amount of data that is available. The data given in Table II, allows for an h value of –1 second. The trend in the average accelerations from part A, shows that approaching 45 seconds, the average acceleration is decreasing to a value of less that –3 ft / sec^2. The instantaneous acceleration calculated in part B for 45 seconds was determined to be –2.90 ft / sec^2. As the interval for h decreased, you get increasingly accurate measurements for the derivative based on the limit definition. To get even better representation of the instantaneous acceleration, more data points would be needed at closer time intervals to further decrease the value of h as the limit suggests. 3. The overall acceleration happens at the end of the time period. This is when the plane has slowed down sufficiently and is about to come to a stop. Once the plane is fully stopped, the acceleration and velocity would be zero. At 45 Final project milestone one 4 seconds, the acceleration –2.90 ft / sec^2, which is the maximum calculated value for the instantaneous acceleration. Since the velocity at that time is 2.65 ft / sec, if the trend continues, then the plane would be stopping in just a few seconds. By knowing when the acceleration will reach zero and therefore when the plane will come to a complete stop would allow you to determine the length of the runway required for the aircraft to land safely.
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