Clay Bucholz ERA/Year

This table represents Clay Bucholz’s (pitcher for the Boston Red Sox) ERA averages year over year. The value t represents the year at which the ERA was averaged. How would we find the Instantaneous Rate of change at t=1?

T(year)	0	1	2	3	4	5	6
ERA	0	1.59	6.75	4.21	2.33	3.48	4.56

e. (6.75-1.59)   = 5.16

          2-1

    (4.21-1.59)   = 1.31

          3-1

    (2.33-1.59)   = .25

          4-1

e. The slope of these secant lines tend to fluctuate and are not at all linear. This shows that in the MLB, Clay Bucholz tends to fluctuate with his performance year over year. The lower the ERA, the better because that means that there were few runs let up the entire season for each pitch the pitcher threw. However, the slopes show that his performance from his first year to the performance in his third year was worse because ERA increased. However, from his first year to his fourth year, he increases performance from previous years because ERA decreases compared to years after his first year.

f. Based on finding the line of the tangent, I found that points (1,1.59) and (1.2,3) would best represent this line. The tangent line would be a slope (1.59-3)/(1.2-1)=   7.05. The slope would be 7.05. So, Bucholz’s ERA is increasing at an instanteuous rate of change of 7.05 Era/year.

g. In part e, we realize that the steepness of the secant line slopes decreases. This shows that the values, as years increase, are moving away from the tangent line.