Tim Linthicums ERA and Instantaneous Rate of Change

Introduction: Tim Linthicum is one of Baseball's top pitchers. Since you are a huge fan of baseball and math you want to find a way to relate the two. You think to yourself and realize that you can further your understanding of instaneous rate of change by finding the instaneous rate of change in Tim's ERA during his 5th year in the major. So, what is the instantaneous rate of change in Tim's ERA during his fifth year in the majors. Use the chart below to figure it out.



 Tim Linticum's ERA from 2007-2014

   

Year	ERA
2007	4.00
2008	2.62
2009	2.48
2010	3.43
2011	2.74
2012	5.18
2013	4.37
2014	4.74

Average Rate of Change:

(5, 2.74) to (6, 5.18)

(6 - 5) / (5.18 - 2.74) = .41

(5, 2.74) to (7, 4.37)

(7-5) / (4.37 - 2.74) = 1.23

(4, 3.43) to (5, 2.74)

(5-4) to (2.74- 3.43) = -1.45


The calculations of the ARC show the slope of the line between two points on the graph. If the slope of this line is positive that means that Tim's ERA went up between those two years and if the slope is negative, it means that Tim's ERA went down between those years.


Instantaneous Rate of Change:

To find the instantaneous rate of change during Linthicum's 5th year in the major you have to take the two points above and below the 5th years ERA and that will narrow down the selection to IRC. You find this by determining the slope of the tangent line.

(5, 2.74) to (6, 3.45)

(6-5) / (3.45 -2.74) =1.408

(4, 2.13) to (5, 2.74)

(5-4) / (2.74 - 2.13) =1.639

The IRC with most likely be directly in between these two numbers.

(1.408 + 1.639) / 2 = 1.523 ERA

 The IRC in this situation shows the instant rate of change during Tim's 5th year in the majors for his ERA. Basically it is means there was around a 1.523 ERA decrease in TIm's ERA in his 5th year in the majors.