### AN ANALYTICAL STUDY ON BEHAVIORAL SIMILARITY BETWEEN ROLLE’S THEOREM AND JUNE SOLSTICE

M.Sc, Department of Statistics
Gauhati University, Guwahati, Assam, India
&
M.Tech, Department of Information Technology
Gauhati University, Guwahati, Assam, India

Abstract
Rolle’s Theorem which is a special case of Mean value theorem has its immense applicability across disciplines. It is evident that many classical results of real analysis are the consequences of Rolle’s Theorem. It states that if a function f(x) is continuous on [a,b] and derivable on (a,b) including the condition that f(a)=f(b) then there exist at least one point in between a and b where the function is parallel to the x axis. This theoretical concept can be used extensively in many real life situations apart from its mathematical applications. Distinct from the idealized world, in this paper an attempt has been made to show the implementation of Rolle’s theorem in one of the most obvious fact of earth, called season change i.e. how all the conditions of the Rolle’s theorem is contended by the different attributes of  season change.
Keywords: Solstice, June Solstice, Rolle’s Theorem, Daylength duration
1.Introduction
Rolle’s Theorem is a multidisciplinary concept which is a special case of Lagrange’s mean value theorem of differential calculus. It is well known that it has enormous use in mathematical field. This interdisciplinary notion was first profound by the French mathematician Michel Rolle’s in 1691 though it was not proved until 12th century, when a formal proof was given by Indian Mathematician Bhaskara II. The geometrical interpretation of Rolle’s Theorem stipulates that if f(x) is continuous function whose domain is a closed interval and the function has tangents at every point excluding possibly the end points, then there will be at least one tangent which will be parallel to the x-axis (W A Lampi).In real analysis, scores of results are derived from this concept. From mathematical perspective it is a versatile theorem because of its wide range of exercises. But in real life also, the theoretical concept of Rolle’s Theorem has immense applicability. One of those is in “Season Change”. By the term season change, here only summer solstice is considered.  It occurs in between June 20thto June 22nd in northern hemisphere when longest day of the year is observed and after this day summer season starts (Solstice, 2015). In this paper an effort has been made to visualize the impact of Rolle’s Theorem on season change by showing how all the conditions of Rolle’s Theorem are satisfied by the various aspects of summer solstice. This will simultaneously show the relationship between Rolle’s Theorem and summer season.
2.Objective
Through this paper, an attempt has been made to show the implementation of Rolle’s Theorem on June solstice, and with this respect a comparative Analysis of June Solstice and Rolle’s Theorem is carried out.
3.Methodology
This Paper has been prepared based on Secondary data sources collected from books, journals, periodicals, research articles, websites. Analytical approach is used and discussed the topic with the concerned resource persons. Personal observations and interpretation were also added here.
4.Results and Discussion
4.1Rolle’s Theorem: Rolle’s theorem states that if a function f(x) defined on [a,b] is
1.   Continuous on [a,b]
2.   Differentiable on (a,b) and
3.  f(a)=f(b)

then there exists at least one point say c in between a and b (a<c<b) where f’(c)=0 (W A Lampi; S C Malik, 1982). Following is the diagrammatic representation of the theorem.

In the above diagram we have considered a curve f(x) which is continuous on [a, b] and differentiable on (a,b). Also it takes equal ordinates on y axis i.e. f(a)=f(b). If we draw different tangents on the curve, then there will at least on tangent such that it is parallel to x-axis. Which implies that, at the point where the tangent is parallel to x-axis, will have zero derivatives.

4.2Summer season as a result of June solstice: Solstice is an astronomical event that occurs twice a year popularly known as summer solstice and winter solstice. It is that time of year when sun reaches either its highest or lowest point in the sky at noon which also signifies longest or shortest day respectively (Solstice, 2015). Summer solstice also known as June solstice takes place when leaning of earth towards the sun is maximum. Therefore on the day of summer solstice, sun appears for the longest time with a noontime position. The changes of day light duration after and before this longest day are seen to be very little. In general June 21st is called the day of summer solstice in northern hemisphere and simultaneously winter solstice in the southern hemisphere. But the date of summer solstice varies because of calendar system. Also the facts like exact orbital, and daily rotational motion of earth contributes to the changing of solstice date. As for example in 2015, 21st June is the day of summer solstice but contradictorily for the year 2016, 20th June is the day when summer solstice occurs. The summer season is the result of June solstice. The observation of summer season is due to the fact that this hemisphere is receiving more direct sun rays than that of the opposite hemisphere (Earth Sky, 2015). Since from the day of midsummer, popularly known as the day of summer solstice, earth’s inclination towards the sun achieves its zenith, hence the day of midsummer is known as starting day of summer season. Therefore we can interpret summer season as the consequence of June solstice (Solstice, 2015; June Solstice, 2015).

4.3Problem definition: As it is known that Rolle’s Theorem is the acquisition of three conditions, hence it is obvious that the particulate on which it is going to relate must content these three circumstances. Here the specific case on which Rolle’s Theorem is going to apply is June solstice on earth. The first two conditions of the theorem are already fulfilled by earth as it possesses continuous behavioral structure (due to its approximately spherical structure). In order to satisfy the third condition by June solstice, a comparative analysis has been performed on day length duration of June solstice (21st June) with its previous and next day. If the differences of June Solstice day length duration with that of previous and next day are found to be approximately equal then we can infer that the third condition of Rolle’s Theorem is contented by June solstice.

4.4Comparative Analysis of June Solstice and Rolle’s Theorem: With an aim to analyze the duration of daylight for the June solstice (i.e. the day with longest daylight duration) and its neighboring days (i.e. the previous and next day of the longest day of the year in context of Northern Hemisphere) and then to perform a comparative study with Rolle’s Theorem we have tabulated the data as shown below (Time and Date, 2015). All data listed down are allied to Guwahati city, Assam, India. The table consists of seven columns viz. Year, Previous Day DayLength (Denoted as A), Highest DayLength (Denoted as B), Next Day DayLength (Denoted as C), Difference between previous and highest day Daylength [(B-A) denoted as D1], Difference between next and highest dayDaylength [(B-C) denoted as D2] and Remark. Since we are dealing with the difference of time in seconds which is very small amount with respect to calculation purpose, to nullify any fractional value in between, we have well-thought-out a difference of one second as no difference. All the below tabulated values are subjected to this contemplation.

 YEAR Previous day daylength(A) Longest day daylength(B) Next day daylength(C) D1=(B-A) (seconds) D2=(B-C) (seconds) Remark (Is D1-D2=0/1) 1996 13:46:46 13:46:47 13:46:45 1 2 Yes 1997 13:46:45 13:46:47 13:46:46 2 1 Yes 1998 13:46:45 13:46:47 13:46:46 2 1 Yes 1999 13:46:44 13:46:47 13:46:47 3 0 NO 2000 13:46:46 13:46:47 13:46:46 1 1 Yes 2001 13:46:46 13:46:47 13:46:46 1 1 Yes 2002 13:46:46 13:46:47 13:46:47 1 0 Yes 2003 13:46:45 13:46:47 13:46:47 2 0 NO 2004 13:46:45 13:46:47 13:46:46 2 0 NO 2005 13:46:47 13:46:48 13:46:47 1 1 Yes 2006 13:46:46 13:46:48 13:46:46 2 2 Yes 2007 13:46:46 13:46:48 13:46:48 2 0 NO 2008 13:46:47 13:46:47 13:46:46 0 1 Yes 2009 13:46:46 13:46:47 13:46:46 1 1 Yes 2010 13:46:46 13:46:47 13:46:46 1 1 Yes 2011 13:46:45 13:46:47 13:46:47 2 0 NO 2012 13:46:46 13:46:46 13:46:45 0 1 Yes 2013 13:46:45 13:46:46 13:46:45 1 1 Yes 2014 13:46:45 13:46:46 13:46:45 1 1 Yes 2015 13:46:44 13:46:46 13:46:46 2 0 NO

Table: Day Length Duration Analysis

Now, for the exploration, we have jaggedly considered 20 (from 1996 to 2015) years to study behavioral change in day length duration of the longest day of the year (Northern Hemisphere) with its next and previous day length.
From the table, by considering the Remark column we can spot that, out of 20 observations for 20 considered years, for 14 years, we have obtained the two values i.e D1 and D2 same (ignoring the difference one second). This means 70 percent of time, D1 is same as D2.
Now if we consider f(T) as function of Time, then we can take as f(T1) as function of time in previous day and f(T2) as function of time in next day. For those years with D1 and D2 difference 0 or 1, we have f(T1)=f(T2) where f(T0) is the function of time for the longest day.

5.Conclusion

From the analysis, conducted in between Roll’s theorem and June solstice, the conclusion can be made that Rolle’s Theorem has an effective impact upon June solstice which further leads to change of season. In accordance with Rolle’s Theorem, any function which goes through three of its premier conditions has a tangent on its maximum point, which becomes parallel to x-axis. As the theme here is to show the implementation of Rolle’s Theorem on June solstice (Northern hemisphere of earth), hence it is essential to show whether the fact possesses the three fundamental terms of the theorem. The first two conditions are obvious on earth (considering earth as a circle). Therefore there is an immediate need to show whether the third condition is satisfied by the fact.

For the experiment, June solstice (generally 21st June) is considered as the elite point and the changes in day length duration of the neighboring days from elite point i.e. f(T1), and f(T2) are analyzed. For 70% of cases f(T1), and f(T2) are found to be same. Hence it can be concluded that all the three conditions are satisfied by the June solstice.

References