MAT 223 Milestone One 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?
Scenario One: Motion Problem
Prompt
You have been hired by a firm that is designing a runway for an airport. Your job is to confirm that the runway is long enough for an airplane to land, but not unnecessarily long. You are given the velocity data for the largest aircraft that will land at the airport. The velocity data will be used to calculate the distance required for the aircraft to safely land and come to a stop.
Below is a set of data that represents the velocity (in feet per second) of the final 45 seconds of the landing. At t = 0, the plane is on its final descent.
Table I
| t in seconds |
0 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
| v(t) in feet per second |
274.27 |
223.19 |
179.23 |
141.4 |
108.83 |
80.80 |
56.68 |
35.91 |
18.04 |
2.65 |
Table II
| t in seconds |
4 |
5 |
14 |
15 |
24 |
25 |
34 |
35 |
44 |
45 |
| v(t) in feet per second |
232.8 |
223.19 |
148.52 |
141.4 |
86.08 |
80.80 |
39.82 |
35.91 |
5.55 |
2.65 |
Part II: Analysis of Data – Applying Derivatives
- Calculating average
Using the data in Table I, calculate the average acceleration for the following intervals:
- From t = 0 to t = 45
- From t = 25 to t = 45
- Calculating instantaneous
- Using the data in Table II, calculate the instantaneous acceleration at the following intervals:
- t = 5
- t = 15
- t = 35
- t = 45
- Explain how you used the limit definition of a derivative to calculate the instantaneous Use your results to explain why the limit definition of a derivative is true.
- At what point is the acceleration at a maximum? How is this relevant to the landing aircraft?
For each of the questions above, provide supporting solutions and calculus terminology to explain how you arrived at your answers. Also, explain in detail what your answer represents in a real-world context and why this is useful.
Scenario Two: Decay Problem
Prompt
You have been hired by a company that has recently developed a medication designed to reduce the size of benign tumors. Your role is to
confirm that the medication does reduce the size of the tumor, given the rate-of-change data. There are many factors to consider, and the goal is to determine the total change in the size of the tumor. Using this data, can you confirm that there is a change in the size of the tumor?
Table I
| t in days |
0 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
| r(t) in mm
per day |
0 |
-0.0105 |
-0.02093 |
-0.03134 |
-0.04171 |
-0.05204 |
-0.06234 |
-0.07261 |
-0.08283 |
-0.09303 |
Table II
| t in days |
4 |
5 |
14 |
15 |
24 |
25 |
34 |
35 |
44 |
45 |
| r(t) in mm
per day |
-0.00839 |
-0.0105 |
-0.02926 |
-0.03134 |
-0.04998 |
-0.05204 |
-0.07056 |
-0.07261 |
-0.09099 |
-0.09303 |
Part II: Analysis of Data – Applying Derivatives
- Calculating average change in the rate of
Using the data in Table I, calculate the average change in the rate of change data for the following intervals:
- From t = 0 to t = 45
- From t = 25 to t = 45
- Calculating instantaneous change in the rate of
- Using the data in Table II, calculate the instantaneous acceleration at the following intervals:
- t = 5
- t =15
- t = 35
- t = 45
- Explain how you used the limit definition of a derivative to calculate the instantaneous rate of change. Use your results to explain why the limit definition of a derivative is true.
- At what point is the rate of change at a maximum? How is this relevant to the size of the tumor?
For each of the questions above, provide supporting solutions and calculus terminology to explain how you arrived at your answers. Also, explain in detail what your answer represents in a real-world context and why this is useful.
Rubric
Guidelines for Submission: Your final problem walkthroughs should be a 1- to 2-page Microsoft Word document with double spacing, 12-point Times New Roman font, and one-inch margins.
| Critical Elements |
Proficient (100%) |
Needs Improvement (75%) |
Not Evident (0%) |
Value |
| Calculate the Average Rate of Acceleration/Change |
Correctly calculates the average rate acceleration/change over all given time intervals |
Applies correct calculus techniques to calculate average acceleration of each time interval, with minor errors in some calculations |
Does not accurately calculate the average rate of acceleration/change for a majority of time intervals |
40 |
| Instantaneous Acceleration / Rate of Change |
Correctly calculates the instantaneous acceleration / rate of change at all specific time values |
Applies correct calculus techniques in calculating instantaneous acceleration at all specific time values, with minor errors in calculation |
Does not accurately calculate the instantaneous acceleration
/ rate of change for a majority of time values |
40 |
| Application of Calculus Terminology in Explanation With Supporting Examples |
Correctly applies supporting solutions and calculus terminology to explain how answers were determined |
Applies correct calculus terminology in a majority of steps to explain how answers were determined, with some supporting solutions |
Does not apply calculus terminology in explanation, or does not support with examples to explain solutions |
20 |
| Total |
100% |
Related; MAT 350 Project Two Guidelines and Rubric.
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