CalculusI.tex

\part*{Calculus I}


%\apexchapter[text/01_Prerequisite]{Limits}{chapter:limits}
\apexchapter{Limits}{chapter:limits}

\textit{Calculus} means ``a method of calculation or reasoning.'' When one computes the sales tax on a purchase, one employs a simple calculus. When one finds the area of a polygonal shape by breaking it up into a set of triangles, one is using another calculus. Proving a theorem in geometry employs yet another calculus.

Despite the wonderful advances in mathematics that had taken place into the first half of the 17\textsuperscript{th} century, mathematicians and scientists were keenly aware of what they \textit{could not do.} (This is true even today.) In particular, two important concepts eluded mastery by the great thinkers of that time: area and rates of change. 

Area seems innocuous enough; areas of circles, rectangles, parallelograms, etc., are standard topics of study for students today just as they were then. However, the areas of \textit{arbitrary} shapes could not be computed, even if the boundary of the shape could be described exactly. 

Rates of change were also important. When an object moves at a constant rate of change, then ``distance = rate $\times$ time.'' But what if the rate is not constant --- can distance still be computed? Or, if distance is known, can we discover the rate of change?

It turns out that these two concepts were related. Two mathematicians, Sir Isaac Newton and Gottfried Leibniz, are credited with independently formulating a system of computing that solved the above problems and showed how they were connected. Their system of reasoning was ``a'' calculus. However, as the power and importance of their discovery took hold, it became known to many as ``the'' calculus. Today, we generally shorten this to discuss ``calculus.''

The foundation of ``the calculus'' is the \textit{limit.} It is a tool to describe a particular behavior of a function. This chapter begins our study of the limit by approximating its value graphically and numerically. After a formal definition of the limit, properties are established that make ``finding limits'' tractable. Once the limit is understood, then the problems of area and rates of change can be approached.

\inputprereq{text/01_Prerequisite}
\input{text/01_Limit_Introduction}
\input{text/01_Limit_Definition}
\input{text/01_Analytic_Limits}
\input{text/01_One_Sided_Limits}
\input{text/01_Limits_Involving_Infinity}
\input{text/01_Continuity}


\apexchapter{Derivatives}{chapter:derivatives}
%\apexchapter[text/02_Prerequisite]{Derivatives}{chapter:derivatives}

The previous chapter introduced the most fundamental of calculus topics: the limit. This chapter introduces the second most fundamental of calculus topics: the derivative. Limits describe \textit{where} a function is going; derivatives describe \textit{how fast} the function is going.

\inputprereq{text/02_Prerequisite}
\input{text/02_Derivative}
\input{text/02_Derivative_Meaning}
\input{text/02_Derivative_Rules}
\input{text/02_Product_Quotient_Rules}
\input{text/02_Chain_Rule}
\input{text/02_Implicit_Differentiation}
\iftoggle{isEarlyTrans}{\input{text/02_Derivative_Inverse_Functions}}{}


\apexchapter{The Graphical Behavior of Functions}{chapter:graphbehavior}

Our study of limits led to continuous functions, which is a certain class of functions that behave in a particularly nice way. Limits then gave us an even nicer class of functions, functions that are differentiable.

This chapter explores many of the ways we can take advantage of the information that continuous and differentiable functions  provide.

\input{text/03_Extreme_Values}
\input{text/03_Mean_Value_Theorem}
\input{text/03_Increasing_Decreasing}
\input{text/03_Concavity}
\input{text/03_Curve_Sketching}


\apexchapter{Applications of the Derivative}{chapter:deriv_apps}

In \autoref{chapter:graphbehavior}, we learned how the first and second derivatives of a function influence its graph. In this chapter we explore other applications of the derivative.

\input{text/04_Related_Rates}
\input{text/04_Optimization}
\input{text/04_Differentials}
\input{text/04_NewtonsMethod}


\apexchapter{Integration}{chapter:integration}

We have spent considerable time considering the derivatives of a function and their applications. In the following chapters, we are going to starting thinking in ``the other direction.'' That is, given a function $f(x)$, we are going to consider functions $F(x)$ such that $F\primeskip'(x) = f(x)$. There are numerous reasons this will prove to be useful: these functions will help us compute areas, volumes, mass, force, pressure, work, and much more.

\input{text/05_Antiderivatives}
\input{text/05_Definite_Integral}
\input{text/05_Riemann_Sums}
\input{text/05_FTC}
\input{text/06_Substitution}

\iftoggle{bsc}{}{

\apexchapter{Applications of Integration}{chapter:app_of_int}

We begin this chapter with a reminder of a few key concepts from \autoref{chapter:integration}. Let $f$ be a continuous function on $[a,b]$ which is partitioned into $n$ equally spaced subintervals as 
\[a=x_0 < x_1 < \cdots < x_{n-1}<x_n=b.\]
Let $\Delta x=(b-a)/n$ denote the length of the  subintervals, and let $c_i$ be any $x$-value in the $i^{\text{th}}$ subinterval. \autoref{def:rie_sum} states that the sum
\[\sum_{i=1}^n f(c_i)\Delta x\]
is a \textit{Riemann Sum.} Riemann Sums are often used to approximate some quantity (area, volume, work, pressure, etc.). The \textit{approximation} becomes \textit{exact} by taking the limit 
\[\lim_{n\to\infty} \sum_{i=1}^n f(c_i)\Delta x.\]
\autoref{thm:riemannSum} connects limits of Riemann Sums to definite integrals:
\[\lim_{n\to\infty} \sum_{i=1}^n f(c_i)\Delta x = \int_a^b f(x)\ dx.\]
Finally, the Fundamental Theorem of Calculus states how definite integrals can be evaluated using antiderivatives. 

This chapter employs the following technique to a variety of applications. Suppose the value $Q$ of a quantity is to be calculated. We first approximate the value of $Q$ using a Riemann Sum, then find the exact value via a definite integral. We spell out this technique in the following Key Idea.

\begin{keyidea}[Application of Definite Integrals Strategy]\label{idea:app_of_defint}
Let a quantity be given whose value $Q$ is to be computed.\index{integration!general application technique}
\begin{enumerate}
\item	Divide the quantity into $n$ smaller ``subquantities'' of value $Q_i$.
\item	Identify a variable $x$ and function $f(x)$ such that each subquantity can be approximated with the product $f(c_i)\Delta x$, where $\Delta x$ represents a small change in $x$. Thus $Q_i \approx f(c_i)\Delta x$.
%% A sample approximation $f(c_i)\Delta x$ of $Q_i$ is called a \textit{differential element}.
\item	Recognize that $\ds Q\approx \sum_{i=1}^n Q_i = \sum_{i=1}^n f(c_i)\Delta x$, which is a Riemann Sum.
\item	Taking the appropriate limit gives $\ds Q = \int_a^b f(x)\ dx$
\end{enumerate}
\end{keyidea}

This Key Idea will make more sense after we have had a chance to use it several times. We begin with Area Between Curves.%, which we addressed briefly in \autoref{sec:FTC}.

\input{text/07_Area_Between_Curves}
\input{text/07_Disk_Washer_Method}
\input{text/07_Shell_Method}
\input{text/07_Work}
\input{text/07_Fluid_Force}

}

\cleardoublepage