\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}

\begin{document}
\centerline{MATH 136 -- Calculus 2}
\centerline{Global Climate Change and Integration}
\centerline{February 3, 2020}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
This worksheet is inspired by materials developed by Prof. Diana Davis of Swarthmore
College.  Prof. Davis used them last week in a class meeting of Prof. Roberts' 
MATH 134 class here at Holy Cross and has given us permission to use and adapt them.  
\vskip 10pt
As you should be aware, a major driver of global climate change is emissions of
\emph{greenhouse gases} such as ${\rm CO}_2$ into the atmosphere from human activities
involving burning of fossil fuels for transportation, electric power generation, 
manufacturing, home heating, and so forth.  Global average temperatures are now
slightly higher than $1^\circ\ {\rm C}$ above their pre-Industrial Revolution
levels.  Damaging sea level rise and extreme weather events (such as the 
recent Australian wildfires) will become a major problem around the world if
global average temperature rises more than $1.5^\circ\ {\rm C}$.  Scientists
estimate that the \emph{total additional amount} of ${\rm CO}_2$ that we can put into the 
atmosphere without reaching the $1.5^\circ\ {\rm C}$ level is \emph{about $400$ Gigatons (Gt) of 
carbon dioxide}.  Let's call that our \emph{carbon budget}.  

In this discussion, we want to use calculus to analyze the predictions of various
model scenarios regarding the \emph{rate of carbon dioxide emissions} (in Gt per year)
and the accumulated \emph{total emissions} in each case.  
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[(A)] (A ``warmup.'') Let $a$ be a constant.  Using your knowledge of derivative rules,
find a general antiderivative for $f(t) = e^{at}$.
\item[(B)]  
Let $t$ be the number of years after 1/1/2020, so $t = 0$ is close
to the present.  If current trends continue, we will be emitting 
${\rm CO}_2$ into the atmosphere at a rate of about 
$$f_{\rm current}(t) = 34 + 1.3\cdot e^{0.065\cdot t}$$
Gt per year. 
\begin{enumerate}
\item[(1)] Explain why the \emph{total additional amount of carbon dioxide} emitted
between the present time and year $t = b$ in the future will be 
$$F(b) - F(0) = \int_0^b f_{\rm current}(t)\ dt$$
where $F$ is an antiderivative of $f_{\rm current}(t)$.  
Plot $y = f_{\rm current}(t)$ to visualize the value of this integral as an area.  
\item[(2)]  How much carbon dioxide will be emitted between now and 2050?  
\item[(3)]  Estimate when we will exceed the carbon dioxide emissions budget of 400 Gt.
\end{enumerate}
\item[(C)]  A better plan would be to gradually ``wean ourselves off'' from 
fossil fuel burning.  If we gradually reduce our emissions, we might come
up with a plan to make our rate of emissions at time $t = $ number of years after 2020
go according to this function:
$$f_{\rm better}(t) = 34 + 1.3\cdot e^{0.065\cdot t} - 0.024\cdot e^{0.25\cdot t}\ 
{\rm Gt}/{\rm yr}.$$
\begin{enumerate}
\item[(1)]  What is $f_{\rm better}(30)$?  What does this number mean?
\item[(2)]  What is the total carbon dioxide emitted between 2020 and 2050
under the better plan?  (Hint:  Compute this with an integral like the one in question
B, part 1).  
\item[(3)]  When will the world exceed its carbon budget with this plan?
\end{enumerate}
\item[(D)]  A \emph{carbon sequestration} plan, in which we actively take 
carbon dioxide out of the atmosphere (either by planting many trees world-wide, 
or through other means), might yield an emissions rate like this:
$$f_{\rm sequestration} = 34 + 1.3\cdot e^{0.065\cdot t} - 4\cdot e^{0.12\cdot t}\ 
{\rm Gt}/{\rm yr}.$$
\begin{enumerate}
\item[(1)]  Plot $y = f_{\rm sequestration}(t)$ for $0 \le t \le 30$.
\item[(2)]  Under this plan, when will carbon dioxide in the atmosphere
reach its \emph{maximum level}?  (Use your graph to estimate.)
\item[(3)] Will the world exceed its carbon budget with this plan?
\end{enumerate}
\item[(E)]  Write a short paragraph explaining the conclusions you 
derive from these calculations.
\end{enumerate}
\vskip 10pt
\noindent
{\it Assignment}
\vskip 10pt
Group writeups due at the start of class on Wednesday, February 5.
\end{document}


