# Project one .

Can you help me understand this Calculus question?

Your final project consists of five parts that all contribute to your final calculations and resolution of the scenario. In this milestone, you will calculate average acceleration and instantaneous acceleration based on the data provided. This milestone provides you with an opportunity to begin an important part of the final project, and to ensure that you are properly progressing toward your resolution. ( PLEASE A CLEAR COPY OF THE REQUIREMENTS WILL BE ATTACHED)

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 A. Calculating average acceleration. Using the data in Table I, calculate the average acceleration for the following intervals: i. From t = 0 to t = 45 ii. From t = 25 to t = 45 iii. From t = 40 to t = 45 B. Calculating instantaneous acceleration. 1. Using the data in Table II, calculate the instantaneous acceleration at the following intervals: i. t = 5 ii. t = 15 iii. t = 25 iv. t = 35 v. t = 45 2. Explain how you used the limit definition of a derivative to calculate the instantaneous acceleration. Use your results to explain why the limit definition of a derivative is true. 3. 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 A. Calculating average change in the rate of change. Using the data in Table I, calculate the average change in the rate of change data for the following intervals: i. From t = 0 to t = 45 ii. From t = 25 to t = 45 iii. From t = 40 to t = 45 B. Calculating instantaneous change in the rate of change. 1. Using the data in Table II, calculate the instantaneous acceleration at the following intervals: i. t = 5 ii. t =15 iii. t = 25 iv. t = 35 v. t = 45 2.

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. 3. 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.