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 v(t) in feet per second 274.27 30 223.19 35 179.23 40 45 141.4 108.83 80.80 56.68 35.91 18.04 2.65 Table II t in seconds 4 5 14 15 24 v(t) in feet per second 25 232.8 34 223.19 35 14 148.52 45 141.4 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