Partial Derivatives and the NBA Draft

By: Erica Boudette, Mike Davis, Matthew Jones

Topic

• What the draft would be like if the minimum age was raised

• Right now, the draft age is 18

• We will compare numbers if the draft age was changed to the completion of freshman year of college, then sophomore year, then junior year, and then senior year

Comparing the Topic to Calculus

• Use four different equations corresponding to the age minimum for the draft

• We will find the partial derivative of each equation to show the rate of change with respect to the year of the age minimum

The NBA Draft Over the Past 3 Years

• Out of 89 first round draft picks:– 48 players had not completed college– 15 players were drafted out of high school– 11 players were underclassmen in college

(freshmen and sophomores)

Partial Derivatives

• Derivative with multiple variables with respect to one of these variables with the others held constant

• Useful for locating extreme points of a function of multiple variables

• F(x,y)=c, where c is constant, implicitly define sets in the x,y-plane

• Usually takes the form of curves

General Equation

• Z(F,S,J,Sr)=T-0.05F-0.12S-0.18J-0.25Sr– Z=number of undergrad players– T=total number of players(F+S+J+Sr=6300)– F=number of freshmen players (1500)– S=number of sophomore players (1800)– J=number of junior players (1700)– Sr=number of senior players (1300)

– 0.05 is the number of players that would leave after freshmen year (5%)

– 0.12 is the number of players that would leave after sophomore year (12%)

– 0.18 is the number of players that would leave after junior year (18%)

– 0.25 is the number of players that would leave after senior year (25%)

Age Minimum of Freshmen Year

• Z=T-0.05F-0.12S-0.18J-0.25Sr

• Z=6300-0.05(1500)-0.12(1800)-0.18(1700)-0.25(1300)=5378– Which means that 922 players would leave

after freshmen year

• Partial derivative with respect to F:

• ZF=-0.05

Age Minimum of Sophomore Year

• Z=T-0.12S-0.18J-0.25Sr

• Z=6300-0.12(1800)-0.18(1700)-0.25(1300)=5453– Which means that 847 players would leave

after sophomore year

• Partial derivative with respect to S:

• ZS=-0.12

Age Minimum of Junior Year

• Z=T-0.18J-0.25Sr

• Z=6300-0.18(1700)-0.25(1300)=5669– Which means that 631 players would leave

after junior year

• Partial derivative with respect to J:

• ZJ=-0.18

Age Minimum of Senior Year

• Z=T-0.25Sr

• Z=6300-0.25(1300)=5975– Which means that 325 players would leave

after senior year

• Partial derivative with respect to Sr:

• ZSr=-0.25

Results

• By raising the age minimum, it requires players to stay in school longer

• The higher the age minimum, the less amount of players that will leave