Problem 13 - Applied Calculus

Question

Table 1.15 shows attendance at NFL. football games. (a) Find the average rate of change in the attendance from 2003 to 2007 . Give units. (b) Find the annual increase in the attendance for each year from 2003 to \(2007 .\) (Your answer should be four numbers.) (c) Show that the average rate of change found in part (a) is the average of the four yearly changes found in part (b). $$\begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline \text { Attendance } & 21.64 & 21.71 & 21.79 & 22.20 & 22.26 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) 0.155 million/year; (b) 0.07, 0.08, 0.41, 0.06 million; (c) Verified.

Step 1: Calculate the Average Rate of Change

To find the average rate of change in attendance from 2003 to 2007, we use the formula for the average rate of change: \( \frac{f(b) - f(a)}{b - a} \), where \( f(x) \) is attendance in year \( x \), \( a = 2003 \), and \( b = 2007 \). The attendance in 2003 is 21.64, and in 2007 it is 22.26.\[ \text{Average rate of change} = \frac{22.26 - 21.64}{2007 - 2003} = \frac{0.62}{4} = 0.155 \] million per year.

Step 2: Calculate Yearly Changes in Attendance

To find the annual increase in attendance, subtract the attendance of the previous year from the current year:- From 2003 to 2004: \( 21.71 - 21.64 = 0.07 \) million- From 2004 to 2005: \( 21.79 - 21.71 = 0.08 \) million- From 2005 to 2006: \( 22.20 - 21.79 = 0.41 \) million- From 2006 to 2007: \( 22.26 - 22.20 = 0.06 \) million

Step 3: Verify the Relationship Between Average and Yearly Changes

To verify that the average rate of change from part (a) is the average of the four yearly changes from part (b), calculate the average of the yearly changes:\[ \text{Average of yearly changes} = \frac{0.07 + 0.08 + 0.41 + 0.06}{4} = \frac{0.62}{4} = 0.155 \] million per year. This matches the average rate of change found in part (a).

Key Concepts

Applied Calculus

Applied calculus is a tool that allows us to solve real-world problems by using mathematical concepts related to change and motion. One such application is calculating the "average rate of change". This concept helps us to quantify how a certain quantity evolves over time. For example, in the context of NFL attendance, applied calculus can help us evaluate how fan attendance changes from year to year.
To calculate the average rate of change, we employ the formula: \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \] where \( f(x) \) represents attendance in a given year \( x \), \( a \) is the starting year, and \( b \) is the ending year. This formula provides an insight into how rapidly or slowly attendance is increasing over a specified period.
In practical terms, applied calculus allows us to make informed predictions and strategic decisions in fields like sports management, economics, and more.

NFL Attendance

NFL attendance refers to the number of people who attend NFL games each year. Tracking NFL attendance is crucial as it gives insights into the popularity of the sport, financial health of the league, and fan engagement levels.
In our exercise, attendance is measured in millions and is recorded annually over several years. It showcases how fan attendance grows and fluctuates year on year. For instance, small increases might reflect incremental growth, while substantial jumps could indicate heightened interest.
Understanding changes in NFL attendance is significant for stakeholders such as team owners, stadium managers, and marketers. It aids them in strategizing ticket pricing, marketing efforts, and overall fan engagement activities.

Annual Increase in Attendance

The annual increase in attendance provides specific information about how much the number of attendees changes from one year to the next. It's calculated by finding the difference in attendance figures for consecutive years.
For our NFL case study, the calculation involves subtracting last year's attendance from this year's:

• 2003 to 2004: 21.71 - 21.64 = 0.07 million
• 2004 to 2005: 21.79 - 21.71 = 0.08 million
• 2005 to 2006: 22.20 - 21.79 = 0.41 million
• 2006 to 2007: 22.26 - 22.20 = 0.06 million

This information is valuable for analyzing trends over time. It can help in identifying specific years with notable increases or decreases, which could then be correlated with events such as player trades, team success, or other external factors affecting attendance.

Yearly Changes in Attendance

Yearly changes in attendance are a detailed look at how the NFL's audience size evolves each year. This element of the analysis is crucial for confirming broader trends identified by the average rate of change.
To determine yearly changes, we observe the differences over each one-year span and average them. This confirms if the migrations for each year are consistent with the average trend over a longer duration.
For example, when we add up our four yearly changes (0.07, 0.08, 0.41, and 0.06 million) and divide by 4, the result is 0.155 million per year. This validates our previous calculation of the average rate of change.
Understanding these changes helps in recognizing patterns and making predictions. Sports organizations can use this to anticipate future attendance patterns and plan accordingly.