math ia word doc - shloka shetty · ! 178! 16593224·"!sin"100"5" + 25" !!...

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Mathematics HL Internal Assessment, Shloka Shetty

Examining the Increase in Carbon Dioxide Levels In India(2001-‐2010) using Statistics and Calculus

Shloka Shetty, Dhirubhai Ambani International School

Math HL Internal Assessment

Abstract

India, being the third-highest emitter of CO2 in the world, has experienced the harmful consequences of rising carbon dioxide levels, primarily during the last decade between 2001 and 2010, which has been reported to be the warmest for India. In this Internal Assessment, I aim to deduce a function for CO2 levels in India(2001-10), examine this function, and mathematically analyze the rate of change of CO2 levels between 2001 and 2010. In order to find a function for atmospheric CO2 levels in India, I will be using a first order linear ODE, which expresses the derivative of CO2 levels in India(2001-10) in terms of the difference between the derivatives of CO2 absorption and emission in India(2001-10). . Upon solving this ODE, I will deduce a function, C(t), of CO2 level in India with respect to time. I will then model this function and examine it for the years 2001-10.

Mathematics HL Internal Assessment Shloka Shetty

Table of Contents

Introduction and Rationale 3 Mathematical Concepts Used 3 Outline 3 Modelling CO2 Emissions in India(2001-10) and Finding its First Derivative 4 Function for Rate of Change of CO2 Absorption in India(2001-‐-‐-‐10) 6 Deducing the First Order Linear ODE for Rate of Change of CO2 Levels in 6 India(2001-10) Solving the First Order Linear ODE for Rate of Change of CO2 Levels in 8 India(2001-10) Examining the Function for CO2 Levels in India(2001-10) 12 Conclusion 13 Evaluation 14 Bibliography 15

Page Count: 12 Pages( From Pages 3 to 14, both included)

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Introduction and Rationale

The grave effects of increasing atmospheric levels of carbon dioxide are felt by all living beings. Being one of the primary greenhouse gases, its most profound effect is global warming. Rising temperatures in turn have innumerable health and environmental effects such as rising sea levels, increased spread of diseases, agricultural adversity and damage of the ozone layer. India in particular, being the third-highest emitter of CO2 in the world, has experienced the harmful consequences of rising carbon dioxide levels, primarily during the last decade between 2001 and 2010, which has been reported to be the warmest for India, and for the rest of the world. 1

Having such profound and wide-ranging effects on the country I live in, I was highly intrigued by this occurrence. Therefore, in this Internal Assessment, I aim to deduce a function for CO2 levels in India(2001-10), examine this function, and mathematically analyze the rate of change of CO2 levels between 2001 and 2010.

Mathematical Concepts Utilised The primary area of Statistics that I will be using in this Internal Assessment is Mathematical Modelling. Specifically, I will be employing Geogebra, the graphing software to perform Two Variable Regression Analysis to statistically model CO2 emissions (2001-10).

Additionally, I will be using both, Differential and Integral Calculus. In order to determine the function for CO2 levels, I will also be using first order linear ordinary differential equations(ODE). ODEs are widely used to model the rates of change(i.e how a particular variable changes with respect to time) of real-world phenomena. A differential equation is essentially an “equation that involves the derivatives of a function as well as the function itself”2. The term ‘ordinary’ implies that the differential equation is with respect to only one independent variable. Also, first order differential equations are those in which the highest degree of the derivatives present is 1. Further, the term linear means that the dependent variable and its derivatives are of the first power only. In order to solve this first order linear ODE, I will be using the technique of ‘Integration by Parts’. I have learned and applied the concepts of ODEs and Integration by Parts by researching independently, as we have not yet covered these concepts in our IB curriculum of Mathematics HL.

Outline

The primary reason for rising CO2 levels during a given time period is that CO2 emissions are higher than CO2 absorption for that time period. Therefore, the annual atmospheric CO2 level and its rate of change(derivative) is given by the difference between their annual emission and absorption, and the difference between the rate of change(drivatives) of their emission and absorptions in that year, respectively.

In order to find a function for atmospheric CO2 levels in India, I will be using a first order linear ODE, which expresses the derivative of CO2 levels in India(2001-10) in terms of the difference between the derivatives of CO2 absorption and emission in India(2001-10). Therefore, I will be deducing a function that is based on the condition for rising CO2 levels, i.e. as long as the rate of change of CO2 emissions is higher than the rate

1 "Why India Is Getting Hotter by the Year." Rediff. Accessed March 17, 2016. http://www.rediff.com/news/special/why-india-is- 2 "Ordinary Differential Equation." --from Wolfram MathWorld. Accessed March 22, 2016. http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html.

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of change of CO2 absorptions in India, the rate of change of CO2 will increase. The opposite is true for a decrease in the rate of change of CO2 levels.

The rate of change of CO2 emissions in India will be calculated by modelling a function of CO2 emissions in India(2001-10) using collected data, and then differentiating this function. The rate of change of CO2 absorptions, however, has already been modeled. I will be using this basic model due to the lack of availability of CO2 absorption data on the internet, perhaps due to difficulty in measurement of CO2 absorption by various sources(trees, oceans etc.). The model for the rate of change of CO2 absorption that I have found is with respect to the level of atmospheric CO2. Therefore, I will deduce a linear ODE of the rate of change of atmospheric CO2 levels in India, with respect to atmospheric CO2 levels in India. Upon solving this ODE, I will deduce a function, C(t), of CO2 level in India with respect to time. I will then model this function and examine it for the years 2001-10.

Modelling CO2 Emissions in India(2001-2010) The following data table shows CO2 emissions(kilotons) in India. Only emissions between 2001 and 2010 have been used due to limitations in availability of data on the Internet. Further, the values include CO2 released from the fossil fuel combustion and cement manufacturing. It does not include CO2 released during respiration. The data has been collected from the World Bank Data Catalog, which is a reliable source of information.

Figure 1: Table Showing CO2 Emissions (kt) in India from 2001-103 Years from 2000

Mass of CO2 Emitted (kt)

1 1203843 2 1226791 3 1281914 4 1348525 5 1411128 6 1504365 7 1611199 8 1731075 9 1845820 10 1950950

Using GeoGebra 5, a function of CO2 emissions with repsect to time, in India between 2001 and 2010 can be modelled. In order to do this, two variable regression analysis has been performed. The two variables here are CO2 emissions(kt) and time(years). The following image shows the modelled function for CO2 emissions in India between 2001 and 2010, in a graphical representation.

3 "CO2 Emissions (kt)." CO2 Emissions (kt). Accessed March 15, 2016. http://data.worldbank.org/indicator/EN.ATM.CO2E.KT.

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=

17 8 16593224 · s in 100 5

+ 25 186174279

100

= 17 8

16593224 sin 100 − 5 + 186174279 100 25

16593224 · cos = 100 5 100 5 + 0

25

1 7 8 1 7 8

1 7 8 1 7 16593224 · cos 100 − 5 · 100


Figure 2: Modelled Graph of CO2 Emissions (kt) with respect to Time (years)

The equation for this function has statistically been found to be the following:

= 1861742.79 + 663728.96sin 0.17 − 1.6 where t is the number ofyears from 2000, and E(t) is the CO2 emission in India during that year.

As can be seen in the graph above, this is an increasing function for this domain(time interval).The emissions in this model are primarily from human actions(i.e. buring of fossil fuels, and cement manufacturing). Therefore, this gives insight into the fact that by curtainling the exploitation of resources, we can significantly reduce CO2 emissions. However, it is important to note that this is an approximate model for CO2 emissions, for the given domain of time(2001-10).

In order to formulate the linear ODE of atmospheric CO2 levels in India(2001-10), I will now find the first derivative of the above function. The derivative of E(t) can be found as follows:

= 25

17 8 70521202 · cos 100 − 5

= 625 Therefore, having found the derivative of CO2 Emissions in India with repsect to time(i.e. the rate of change of CO2 Emissions), we now need to find the second component of the first order linear ODE, that is, the equation for the rate of change of CO2 absorptions in India.

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= 26300(() − 598000000)


Rate of Change of CO2 Absorption in India For the aforementioned purposes, I will be using a basic pre-determined model for the rate of change of CO2 absorption instead of using data to model the equation. The equation to find the rate of change of CO2 absorption in any given year is the following:

= − 598000000 Where A is the mass(kt) of the CO2 absorbed in a particular year,

t is the time in years from 2000, C(t) is the function for the CO2 levels(kt) with respect to time(t), 598000000 is the universally accepted pre-‐-‐-‐industrial level of CO2(kt), and k is the proportionality constant, which varies for different geographical zones

This model is based on the fact that the rate of change of CO2 absorption I any given year is proportional to the difference between the atmospheric CO2 level(kt) in that year, and the pre-industrial level of atmospheric CO2(kt).

In India, the proportionality constant has been emperically calculated as 26300 kilotons of CO2 /year. 4

Therefore, the equation for the rate of change of CO2 absorption in India, with respect to atmospheric CO2 levels in a particular year is as follows:

!"

! "

While this, too is a linear differential equation of order 1, I will not be analysing this function as my focus is on the rate of change of CO2 levels in India, rather than that of its emission or absorption.

Deducing the First Order Linear Differential Equation of Rate of Change of CO2 Levels in India

Having found the expressions for rate of change of absorption and emission, we can now use a one-box model5 to deduce the ODE for atmospheric CO2 levels in India. This model describes the amount (in this case, mass) of an atmospheric species X(in this case CO2) inside a box representing a selected atmospheric domain (in this case, India). This amount is determined by the flow of X into the box (Emissions of CO2) and out of the box (Absorption of CO2).

Further, this model describes that the rate of change of X in the specified atmospheric domain is the difference between the rate of change of its flow into and out of that atmospheric domain within the given time. This is a general model that can be used to deduce differential equations for various phenomena. In this case, I will be using it to model the atmospheric CO2 level in India(2001-10) using the already determined rates of change of its emission and absorption.

We have already modeled and calculated the equation for the rate of change of CO2 emissions as:

4 "Pollutant Stock and Flow: Reconstructing CO2 Emissions." Wisconsin Education. Accessed March 18, 2016. http://www.bibme.org/chicago/website-citation/new. 5

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17 8 70521202 · COS 100 − 5

= 625

!" where !" is the rate of change of CO2 emissions in India, and t is the time(number ofyears after 2000). This is the rate of change of the inflow of CO2 into the Indian atmosphere.

Also, we have found that the equation for the rate of change of CO2 absorptions is as follows:

= 26300 − 598000000

Where A is the mass(kt) of the CO2 absorbed in a particular year, t is the time in years from 2000, C(t) is the function for the CO2 levels(kt) with respect to time(t), 598000000 is the pre-‐-‐-‐industrial level of CO2(kt), and 26300 is the proportionality constant for India.

This is the rate of change of the outflow of CO2 into the Indian atmosphere. Therefore, using the one-box model, the rate of change of CO2 levels in the Indian atmosphere is as follows:

=

!" where !" is the rate of change of CO2 levels in the Indian atmosphere,

!" !" !" !"

is the rate of change of the inflow(i.e. emission) of CO2 into the Indian atmosphere, and is the the rate of change of the outflow(i.e. absorption) of CO2 into the Indian atmosphere.

This equation implies that as long as the difference between the rate of change of CO2 emission and CO2 absorption is increasing, the rate of change of atmospheric CO2 levels will also be increasing. This is assuming that CO2 emissions is always greater than CO2 absorption, which in India, has been true since 19006, according to a study by the Ministry of Environment, Forests and Climate Change in 2013. This study also states the rate of CO2 emissions and absorption in India is increasing since 1980. This ODE therefore implies that as long as the difference between the rates of increase of CO2 emissions and CO2 absorptions is increasing, the rate of change of atmospheric CO2 levels will also be increasing. Therefore, this ODE lends insight into how the rate of change of CO2 levels can decrease if society makes an effort to slow down the rate of increase of CO2 emissions and accelerate the rate of increase of CO2 absorptions. Therefore, this ODE can be used to analyse the rate of change of CO2 levels in India(2001-10), and can also be used to model a function for CO2 levels in India(2001-10), as will be done below.Upon differentiating this function, the rate of change of CO2 levels will be determined and linked back to this ODE.

Now, to deduce the ODE for the rate of change of CO2 levels in India(2001-10), we need to substitute the already deduced expressions for the rates of change of CO2 emissions and absorptions, as follows:

1 7 8 70521202 · cos 100 − 5

=

625

26300 − 598000000

7

6 "Atmospheric Concentrations of Fossil Fuels." Ministry of Environment, Forest and Climate Change Government of India. Accessed March 20, 2016. http://www.moef.nic.in/keywords.

17 8 70521202 · cos 100 − 5 + 15727400000000

625

___________ + 26300 =

17 8 70521202 cos 100 − 5 + 15727400000000

625

=

∙ !"#$$! + 26300 ∙ !"#$$! =

17 8

70521202cos 100 − 5 + 15727400000000 ∙ !"#00! 625

∙ !"#$$! + 26300 ∙ !"#$$! =

17 8

70521202cos 100 − 5 + 15727400000000 ∙ !"#00! 625

__ ∙ !"#$$! + 26300 ∙ !"#$$! = ______________________ · !"!""#

∴


On further simplification, we have:

We now satisfy the condition of a differential equation which is an “equation that involves the derivative ( !" !") of a function as well as the function itself (())”7.Additionally, since the equation involves only first derivatives and the power of dependent variable (()), and its derivative (!"

!") is 1, this is a first order, linear ODE.

Solving the First Order Linear Differential Equation: This equation follows the general form of a first order linear ODE, which is:

+ = In this case, = , ( ) = ( ) ,

= 26300 and ,

In order to solve any ODE of this form, its Integrating Factor(I(t)), needs to be found. The Integrating Factor for the above general equation for an ODE is:

( ) = ! ! ! " s ince = 26300 I(t) for the differential equation of atmospheric CO2 levels in India is as follows:

() = ! " # 0 0 ! " Upon Integrating, we have,

() = !"#$$! I(t) now needs to be multiplied throughout the above mentioned ODE of atmospheric CO2 levels in India with respect to time, in order to make it integrable, as follows.

The LHS can be simplified by equating its terms to those of the product rule.

7 "Ordinary Differential Equation." --from Wolfram MathWorld. Accessed March 22, 2016. http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html.

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We now have, · !"#$$% =

17 8

70521202cos 100 − 5 + 15727400000000 ∙ !"#$$!

625

17 8 70521202cos 100 − 5 + 15727400000000 ∙ !"#$$! 625

∴ · !"#°°% =

17 8

70521202cos 100 − 5 + 15727400000000 ∙ !"#°°! 625

2!"#$$! 35260601cos 17 − 160 100 + 4914812500000000 =

625

= 35260601 cos 17 160 100 + 4914812500000000

∴ ′ =

160 599430217 sin 17 − 100

100

263$$t 35260601 cos 17 − !"# !00 + 4914812500000000 ∴ !

!"#

= 625 2

26300t 35260601 cos 17 − 160 100 + 4914812500000000

26300

599430217 · 263$$t sin 17 − 160 100 dt

2630000 −

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Integrating the RHS **Throughout the process of integrating the RHS, it will be assumed that the constant of integration(c) is 0. This is because in order to find this constant, a coordinate of the function C(t) is needed. These coordinates are not yet known, as the function C)t)will only be determined upon solving this ODE. This will be taken into account in the evaluation of this IA.

We now have,

2 17 − 160 · !"#$$% = _____________ 263$$t 35260601 cos __________________ + 4914812500000000 625 100

Integrating by Parts using the following formula:

′ = f g − ′

We have, ′ = ! " # $ $ !

! " # $ $ ! ∴ = 26300

= 625 2

263$$ t 35260601 cos 17 − 160 100 + 4914812500000000

26300

599430217 17− 160

263$$t sin ___________________ 2630000 100

+

17 − 160 !"#$$! sin _________________ dt 100

Integrating By Parts, we have: ′ = − ′

Also,

= sin 17 − 160 100

∴ ′ =

17 − 160 17cos 100

100

= !"#$$!sin 17 − 160

100 26300

17!"#$$!cos 17 − 160 100 dt 2630000

17 − 160 ∴ !"#$$! sin ________________ dt 100

17263$$tcos 17 − 160 100

17 − 160 17cos 100

100 =

1 0


Now Solving:

′ = ! " # $ $

!"#00 ∴ = 26300

Now Solving:

= 2630000

Integrating by Parts, we have: ! = – !

! = 263$$ t 26300

263$$ t ∴ =

Also,

691690000

∴ ! = 17− 160

289 sin 100 1000

=

=


289 17 − 160 !"#$$!sin ________________

6916900000000 100

Solving for we have: 17!"#00cos 17 − 160

10000 26300!"#00 sin 17 − 160 100 100 − 100 17 − 160

= !"#$$!sin __________________ 100 ___________________________________________________________________________________ = ________________________________ 6916900000289

= 625 2

!"#$$! 35260601 cos 17 − 160 100 + 4914812500000000

26300

+

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599430217 17− 160 !"#$$! sin ________________ dt )

2630000 100

17!"#$$!cos 17 − 160 1 0 0 2630000 ∴

17!"#$$!cos 17 − 160 100 =

69169000000 289!"#$$!sin 17 − 160 100 dt

6916900000000

17 − 160 ∴ !"#$$! sin ______________ 100 =

e!"#$$!sin 17 − 160 100

26300

17!"#$$!cos 17 − 160 100 _________________________ +

289!"#$$!sin 17 − 160 100 dt 6916900000000 69169000000

!"#$$!sin 17 − 160 100

26300

17!"#$$!cos 17 − 160 100

69169000000

17− 160 If we take !"#$$!sin

100 _________ = , we have

!"#$$!sin 17 − 160 100

26300

17!"#$$!cos 17 − 160 100

69169000000 289

6916900000000

The original equation was: · !"#$$%

· !"#$$% = _________________________________________________________________________________

! "#$$! 35260601 cos 17 − 160 100 + 4914812500000000

+ 821750

On simplifying the RHS, we have:

· ! " # $ $ !

∴ =

1 2


Having found , we now have,

17!"#$$!cos 17 − 160 1198860434 26300263$$t sin 17 − 160 100

100 − 100

1136965437547504375

17 − 160

8 !"#$$! 599430217sin 17 − 160 100 + 92735380630000cos 100 + 12925956875540068750000

= 172922500007225

17− 160 17− 160

8 599430217sin 100 + 92735380630000cos 100 + 12925956875540068750000 172922500007225

Examining the Function for CO2 Levels(kt) in India(2001-10), with respect to Time We now have the function for CO2 levels(kt) in India, from 2001-10. This fulfils one of the objectives for my IA. In order to examine the function and the rate of change of CO2 levels, which was part of my original objective, I have graphed the function using my graphing calculator( fx-CG 20), and then transmitted it to my computer.

Figure 3: Graph of CO2 Levels(kt) in India(2001-10) with respect to Time

This graph shows that C(t) is an increasing function. This indicates that from 2001-2010, the difference between CO2 emissions and absorptions was increasing. By decreasing this difference, CO2 levels can be reduced significantly. Therefore, this function can be used to find the atmospheric CO2 level(kt) in India between the years 2001-10.

17 8

70521202 · cos 100 − 5 =

625 26300 − 598000000


Further, as can be seen by the decreasing slope of the tangents at each value of time(year), it can be said that the function is increasing, but at a decreasing rate. This can be further validated by analysing the trend in the instantaneous rate of change of atmospheric CO2. Although the derivative of the function C(t) was found earlier(Page 7), it wasn’t analysed then, as this would detract the IA from the deduction of the function C(t). Therefore, after having found and graphed the fucntion C(t), its derivative will be analysed here. This will be done by examining the trend in the instantaneous rate of change of atmospheric CO2 levels in India between 2001 and 2010. The derivative of C(t), already deduced on Page 7, was as follows:

Using the function C(t), I have evaluated the CO2 levels for each year, inputted these into an Excel Spreadsheet, and then applied the above function to these values to determine the instantaneous rate of change in each year. A screenshot of this spreadhseet is shown below:

Figure 4: Spreadsheet showing CO2 levels(kt) and their instantaneous rates of change in Indian for each year(2001-‐-‐-‐10)

With reference to this spreadhsheet, between the years 2001 and 2009, the CO2 levels in India were increasing. However, their instantaneour rates of change were decreasing. This is because although CO2 emissions were higher than CO2 absorptions during these years, the difference between the rates of increase of CO2 absorptions and emissions was decreasing. Therefore, with reference to the box model and the following ODE explained on Page 9, the rate of increase of atmospheric CO2 levels in India were decreasing for this time period. However, 9.41 years after 2000(during 2009), the CO2 level was maximum, as shown by Figure 3 on the previous page. The function C(t) reaches a stationary point(maxima) at t=9.41 . At this point, the instantaneous rate of change is 0. Therefore, after this point the CO2 level starts to decrease and the instantaneous rate of change becomes negative, as shown by its value at t=10. This is because after the year 2009(specifically after the stationary point), CO2 absorptions are greater than its emissions.

Conclusion Therefore, I have deduced a function for atmospheric CO2 levels in India(2001-10), and examined this function, and mathematically analysed the rate of change of CO2 levels between 2001 and 2010, I have fulfilled my aim for this Internal Assessment. In the process of developing the function for CO2 levels in India(2001-10), I also gained an in-depth understanding of the reasons for change in atmospheric CO2 levels and changes in its rate of change. The model for CO2 levels I have developed mathematically explains that the rate of change of CO2 levels in India can be greatly reduced by increasing the rate of change of CO2 absorption(by reducing the consumption of fossil fuels, the primary source of CO2 emissions, and instead

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switch to using renewable resources)and by decreasing the rate of change of CO2 absorption(by reducing deforestation and planting more trees, the main sink for CO2). The model serves yet another purpose in that it can be used to find an approximate value for CO2 levels in India during any year between 2001 and 2010, and can be graphed to analyse trends in atmospheric CO2 levels in India for the same time period. My conclusion of these trends in atmospheric CO2 levels in India(2001-10) were that they were increasing at a decreasing rate until 2009, after which CO2 levels themselves decreased. Therefore,I have mathematically adressed and analysed one of the wold’s front running issues that has wide ranging and long lasting impacts on our planet, and on our lives as well. I have focused this global issue to a national context, in which I, personally felt the effects of rising CO2 levels. Although I did face several difficulties, primarily in solving the first order linear ODE, I believe I have fulfilled my aim for this Internal Assessment. Yet, the results of my Internal Assessment do have limitations, which will be evaluated below.

Evaluation

Why this methodology was chosen: C(t) is simply the difference E(t)- A(t) i.e. the difference between CO2 emission and absorption in a particular year, plus the constant atmospheric level of CO2 which hasn’t been emitted or absorbed as part of the carbon cycle, but was naturally present in the atmosphere(Atmospheric CO2 Constant)8. This is an alternative way to deduce a function forC(t).

However, solving an ODE is a more feasible and accurate way to find a function for C(t).It is more feasible as the Atmospheric CO2 Constant is not known for India, and research on it is still underway. If this was assumed to be 0, this would lead to inaccuracy of the result. Since this value is a constant, it will not affect the rate of change of CO2 levels. Thus, solving an ODE of this rate of change is also a more accurate way to deduce a function for C(t).

Limitations: • The constant of integration was not known, and therefore was assumed to be 0. This is a limitation as if

this constant could be solved for, it would change affect the function C(t) as well. • The function I have deduced can be used only for the time period 2001-10. This is because the function is

trigonometric and the trends of a trigonometric function do not necessarily apply to atmospheric CO2 levels for other time periods.

• The function C(t) will give only approximate values for CO2 levels in India for the given time period, as the model of CO2 emission is a best fit curve, and not a completely accurate one.

• Although combustion of fossil fuels and cement production are the main emitters of CO2, there are various other sources of CO2 such as the process of respiration by plants and animals. Emission from these sources have not been included in the data values used to model the function for CO2 Emissions in India.

• The model for CO2 absorption does not take into account the CO2 sinks whose rates of absorption do not depend on the atmospheric CO2 level. Examples of such sinks are CO2 absorbed by weathering rock, and CO2 absorbed by certain types of leaves for photosynthesis.

Therefore, this Internal Assessment has made reasonable conclusions, but does have its limitations. These can be reduced with more advanced technology to pobtain more accurate models and perhaps a longer period of time to further develop the analysis and make more reliable conclusions from this analysis.

8"Earth's CO2 Home Page." CO2.Earth. Accessed March 30, 2016. http://co2.earth/.

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Bibliography

• "Why India Is Getting Hotter by the Year." Rediff. Accessed March 27, 2016. http://www.rediff.com/news/special/why-india-is-getting-hotter-by-the-year/20150526.htm.

• "Ordinary Differential Equation." -- from Wolfram MathWorld. Accessed March 27, 2016. http://mathworld.wolfram.com/OrdinaryDifferentialEquation.html.

• "Pollutant Stock and Flow: Reconstructing CO2 Emissions." Wisconsin Education. Accessed March 18, 2016. http://www.bibme.org/chicago/website-citation/new.

• "Atmospheric Concentrations of Fossil Fuels." Ministry of Environment, Forest and Climate Change Government of India. Accessed March 20, 2016. http://www.moef.nic.in/keywords.

• "Earth's CO2 Home Page." CO2.Earth. Accessed March 30, 2016. http://co2.earth/.

math ia word doc - shloka shetty · ! 178! 16593224·"!sin"100"5" + 25" !!...

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