Steph Curry with the big shot

Introduction: Stephen Curry of the Golden State Warriors has been one of the NBA's best shooters since coming into the league. Since his rookie year, he has been in the top 5 for 3 point shooting almost every single year. Using his 3 point percentages since coming into the league, we are going to calculate his  Instantaneous rate of change (IRC) to meausure his career path as the greatest three point shooter of all time.

The ARC is calculated by using the slope formula: (y2-y1)/(x2-x1)

Shooting Percentage v season

(1, 43.7) (2, 44.2) = (44.2-43.7)/(2-1) = 1/2
(2, 44.2) (3, 45.5) = (45.5-44.2)/(3-2) = 1.3
(3, 45.5) (4, 44.9) = (44.9- 45.5)/(4-3) = -0.6
(4, 44.9) (5, 42.4) = (42.4-44.9)/(5-4) -2.5

IRC:

(1, 43.7) (3, 45.5) = (45.5-43.7)/(3-1) = 0.9
Stephen Curry's shooting percentage is increasing 0.9 during the 2010-2011 season (year 2). The slope at year 2 is the average of the slopes directly to the left and right of year 2. Also as the secant lines inch toward year 2, the slopes get closer to rate of change at year 2.

Mohammed Noor (IRC)

• Noor, is a footballer from Saudi Arabia. He currently plays for Saudi Professional League side Al Ittihad. Noor is considered to be one of the best players ever to play in the Asian Champions League. What does this have to do with Applied Calculus?  Were going to look at the instantaneous rate of change using his goals per season. Also, Noor had a very well-known rough path in his career in 2011-2012; so were going to use the IRC to determine his performance throughout the years.

Season	Goals
1990-1991	20
1992-1993	33
1994-1995	27
1996-1997	34
1998-1999	31

Average Rate of Change:

(3, 33) to (4, 20) = 3-4/ 33-20 = -1/13

(2, 27) to (4, 33) = 4-2/ 33-27 = 2/6

(1, 31) to 2,34) = 2-1/ 34-31= 1/3

The calculations were done by using the slope of the line. If you pay attention you see that Mohammed Noor has had a rough path in his career because one of the slopes are negative. this shows that his performance was affected in that year and we found that out using the IRC and slope.

Instantaneous Rate of Change:

We use the IRC to get closer and closer to a relationship we want to analyze. In this case mohammed noor's performance as a soccer player. It is done also by taking the slope of the points we want to analyze.

IRC Calculations:

(2, 33) to (3,27) 

We take the slope so: 3-2/ 33-27 = 6

(1, 20) to (2,33) 

We take the slope so: 2-1/ 33-20 = 8

We take the IRC by analyzing the two slope points we just calculated. Therefore we have

(6=8) / 2 = 7

The number 7 shows the IRC for Mohammed Noor during his second year as a player in Al Itihad. This shows that he has been improving from his first season to the next but taking into consideration that we had a negative slope of in one of the points we must but regardless of that he has had improvement. Therefore even though Noor did well in his second season and the ones that follow his increase in amount of goals may not seem so impressive when applying the IRC to be more specific

Big Ben's Throw

As a die-hard Steelers fan, Juan Jr. Jr. is interested in everything Ben Roethlisberger. He is watching a Steelers Game on Sunday night with his family and he notices how amazing Ben’s throwing is. Then all of a sudden, Great Grandpa Juan collapses and falls to the ground. As he is taking his last breathe, mathematician, Great Grandpa Juan asks Jr. to find the IRC of Roethlisberger’s winning pass at 7 seconds (t=7). A few weeks later, Juan Jr. Jr. decides to fulfill his great grandfather's last wish and finds the IRC at t=7. These are the steps he takes

Time (t) in seconds	height (ft)
0	6
1	21
2	44
3	53
4	45
5	33
6	26
7	19.5
8	12
9	5

Average Rate of Change

ARC: (y2-y1)/(x2-x1)

(7,19.5) and (9,5)

(5-19.5)/(9-7)= -7.25  (R)

(7, 19.5) and (8,12)

(12-19.5)/(8-7)= -7.5  (R)

(7, 26) and (6,26)

(27.5-19.5)/(6-7)= -8 (L)

(7, 19.5) and (5,33)

(33-19.5)/(5-7)= -6.75 (L)

In order to find the ARC I had to use the slope formula. Juan Jr. Jr. used the point at for which he was trying to find the IRC, (7,19.5), and then used other points from the chart. From the calculations he saw that the ARC was negative on both the right and left side of t=7. From the right side the numbers are getting smaller and smaller and from the left side the numbers are getting bigger and bigger.

Instantaneous Rate of Change

To find the IRC he drew a tangent line to t=7 on the graph. He picked a point on this tangent line in order to calculate the IRC. He chose the point (7.5,17.3). Using the original point (7,19.5) and this new point Juan Jr. Jr. was able to find the IRC.

(7,19.5) and (7.5, 17.3)

(19.5-17.3)/(7-7.5) = -4.4

From this calculation the IRC is concluded to be 4.4sec/ft. The IRC is the same thing as saying the derivative of the slope at a point. The ARC were getting smaller and smaller or larger and larger from both sides to each other. So this is the slope or Instantaneous rate of change at 7 seconds, meaning that is how fast the football is going at that certain point.

The Inflation Rate

Synopsis:
Great Grandpa Juan died on May 23, 2013. Juan Jr. Jr. purchased a coffin for him to be buried in. The next year Juan Jr. Jr. was learning about inflation rates, and was interested in knowing how much more the coffin would have cost if Great Grandpa Juan had died later. To find this out, Juan must calculate the instantaneous rate of change in May in order to ascertain the change in the inflation rate. What was the instantaneous rate of change in may?

Table:

Graph:

Average Rate of Change: when x=5
(5, 1.4) and (4, 1.1)   (1.4-1.1)/ (5-4)=     .3
(5,1.4) and (6,1.8) (1.4-1.8)/(5-6)=               .4
(5,1.4) and (7,2) (1.4-2)/(5-7)=         .3
The average rates of change calculated show the approximate change in the inflation rate over a certain period of time, and will be helpful to Juan Jr. Jr. when trying to figure out the average change in inflation over time. These results will give him a better idea as to what the instantaneous rate of change is and a greater overall view of the monthly changes in inflation.

Instantaneous Rate of Change:
(5,1.4) and (5.5,1.6) (1.4-1.6)/(5-5.5)=     .4
By calculating the Instantaneous rate of change, Juan Jr. Jr. learned that the inflation rate in May was increasing by .4% at that given moment, which means that the price of the coffin would have been about. 4% higher if Great Grandpa Juan had passed away at a later date.

Explanation:
Juan Jr. Jr. knows that the value he calculated as the IRC is in fact correct because as the intervals decreased when calculating the ARC they were approaching .4. Because these match up, Juan knows that his calculations are correct and that in the month of May, the inflation rate was increasing by .4 % when Juan found this out, he thanked his Great Grandpa for dying when he did, and consequently saving him money.

Messi and IRC- Mohamed Salahuddin

Lionel Messi and IRC

A-  Has anyone thought about why footballers retired early, considering 35 years old is early, well now were going to look at the progress of a footballer during his years, and see if his activity does decrease as he grows older. The stats will be based on his 7^th year, which was when he was at his peak.

B-   I am a huge fan of Lionel Messi, since he first started playing for Barcelona, he has been called the next Maradona who was also a football legend, and not only did Messi follow Diego Maradona’s footsteps, but he’s also surpassing his own icon’s record. Now to get to the point, we all no that skills, speed, and all of the things that involve football involve science, but it also does involve math, the number of goals, the number of minutes played, everything to do with number in football ofcourse relates to math. For the IRC what I will do, is take Lionel Messi’s goal from 2008-2014 and see how progress he has made during his time with Barcelona, what we are about to find out is how much has he improved, and to find out if he had a slip up during one season, or if his performance is decreasing as the years progress, and his age increases. As we all know age is a huge factor on how well a footballer plays.

C-   So now were going to look at his statics in Barcelona, he joined Barcelona’s First team during 2004-2005 season, but it was until 2005-2006 where he was actually a starting player, so we will look at the stats from 2005-2006 but we will be excluding his current year with Barcelona as the season has just begun.

Goals	Years in Barcelona
14	1
16	2
38	3
47	4
53	5
73	6
46	7
41	8

D- 

As we can see the goals are on the left, and the years are on the bottom.

E-   Average Rate of Change

(6,73) & (7,46)

(7-6)/(46-73)= -0.14

(6,73) & (8,41)

(8-6)/(41-73)= -0.0625

(5,53) & (6,73)

(6-5) & (73-53)= 0.05

The calculations as shown for the average rate of change, show that by using the slope of the line between the points, as you can see during his second year his performance started decreasing, but during the third year the slopes started increasing again, but this shows us that since the line went up, between the last year, that means even though he did decrease in terms of goals, at the end of it did increase.

F-   Now as we are heading to the IRC, we chose the points that we want to analyze, so here is the graph.

calculate the IRC

(6,73) & (7,46)

To get the slope we know the formula so, its 46-73/7-6= -27

(5,53) & (6,73)

73-53/6-5= 20

We then get the IRC by adding the following points together and dividing by 2, 

(-27+20)/2 = -3.5

     So we now know that during Lionel Messi’s years as player, he did not improve, in fact his performance decreased, so as we can tell that age does make a difference in football, as Messi did not score his record of 73 goals in the past two years.