can you finish my file? a lot of the graphs are plain out broken, some chapter (eg: differential equations) are currently placeholders and have next to zero content. "\documentclass{book}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{geometry}
\usepackage{tikz}
\usepackage{pgfplots}
\usepackage{float}
\usepackage{fancyhdr}
\usepackage{enumerate}
\pgfplotsset{compat=1.17}
\geometry{margin=1in}

\title{\textbf{Calculus for Beginners: \\
A Comprehensive Guide to Understanding Calculus}}
\author{}
\date{}

\begin{document}

\frontmatter
\maketitle

\tableofcontents
\newpage

\mainmatter

\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}

\section*{Welcome to Calculus}

Calculus is a branch of mathematics that deals with change and motion. It provides tools for modeling and analyzing dynamic systems in science, engineering, economics, and beyond. This comprehensive guide is designed for beginners, starting from the basics, and aims to build up your understanding step by step. By the end, you will have a solid foundation in calculus and the trigonometric knowledge that supports it.

\section*{How to Use This Guide}

\begin{itemize}
    \item \textbf{Start from the Beginning}: Even if you have some math background, it's beneficial to review foundational concepts.
    \item \textbf{Active Learning}: Work through examples and practice problems actively. Don't just read—engage with the material.
    \item \textbf{Study Regularly}: Consistency is key. Set aside dedicated time each day to study and practice.
    \item \textbf{Seek Help When Needed}: If a concept is unclear, consult additional resources or ask for assistance.
    \item \textbf{Reflect and Review}: Periodically review previous material to reinforce your understanding.
\end{itemize}

\section*{Prerequisites}

Before diving into calculus, it's important to be comfortable with the following topics:

\begin{itemize}
    \item Basic arithmetic and algebra
    \item Functions and their properties
    \item Graphing techniques
    \item Understanding of exponents and logarithms
    \item Basic geometry and trigonometry
\end{itemize}

Don't worry if these topics seem unfamiliar—we will review them in the upcoming chapters.

\newpage

\chapter{Fundamentals of Algebra and Geometry}

\section{Numbers and Operations}

\subsection{Real Numbers}

The set of real numbers includes all the numbers on the number line. They can be classified into:

\begin{itemize}
    \item \textbf{Natural Numbers}: \( \mathbb{N} = \{1, 2, 3, \dots\} \)
    \item \textbf{Whole Numbers}: \( \mathbb{W} = \{0, 1, 2, 3, \dots\} \)
    \item \textbf{Integers}: \( \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\} \)
    \item \textbf{Rational Numbers}: Numbers that can be expressed as a fraction \( \dfrac{p}{q} \), where \( p \) and \( q \) are integers and \( q \ne 0 \).
    \item \textbf{Irrational Numbers}: Numbers that cannot be expressed as a simple fraction, such as \( \sqrt{2} \), \( \pi \), and \( e \).
\end{itemize}

\subsection{Operations}

\begin{itemize}
    \item \textbf{Addition and Subtraction}: Combining or removing quantities.
    \item \textbf{Multiplication and Division}: Scaling quantities up or down.
    \item \textbf{Order of Operations}: Use the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to determine the sequence of operations.
\end{itemize}

\section{Basic Algebraic Concepts}

\subsection{Variables and Expressions}

\begin{itemize}
    \item \textbf{Variables}: Symbols (like \( x \) or \( y \)) that represent unknown values.
    \item \textbf{Expressions}: Combinations of variables and numbers using operations.
\end{itemize}

\subsection{Equations and Inequalities}

\begin{itemize}
    \item \textbf{Equations}: Statements that two expressions are equal, e.g., \( 2x + 3 = 7 \).
    \item \textbf{Inequalities}: Statements about the relative size of expressions, e.g., \( x - 5 > 2 \).
\end{itemize}

\section{Functions and Graphs}

\subsection{Definition of a Function}

A function is a relationship that assigns exactly one output value for each input value.

\begin{itemize}
    \item \textbf{Notation}: \( f(x) \) denotes a function named \( f \) with input \( x \).
    \item \textbf{Domain}: The set of all possible input values.
    \item \textbf{Range}: The set of all possible output values.
\end{itemize}

\subsection{Graphing Functions}

Understanding how to plot functions on the Cartesian coordinate system is crucial.

\begin{itemize}
    \item \textbf{Cartesian Coordinates}: Points are represented as \( (x, y) \).
    \item \textbf{Plotting Points}: Determine \( y \) for various \( x \) and plot the points.
    \item \textbf{Example}: To graph \( y = x^2 \), compute \( y \) for \( x = -2, -1, 0, 1, 2 \).
\end{itemize}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$y$},
            xmin=-3, xmax=3, ymin=-1, ymax=5,
            samples=100,
            width=10cm, height=8cm,
            domain=-2.5:2.5,
            xtick={-2,-1,0,1,2},
            ytick={0,1,2,3,4}
        ]
        \addplot [blue, thick] {x^2};
        \addplot [only marks, mark=*, mark size=2pt] coordinates {(-2,4) (-1,1) (0,0) (1,1) (2,4)};
        \end{axis}
    \end{tikzpicture}
    \caption{Graph of \( y = x^2 \)}
\end{figure}

\subsection{Types of Functions}

\begin{itemize}
    \item \textbf{Linear Functions}: \( y = mx + b \)
    \item \textbf{Quadratic Functions}: \( y = ax^2 + bx + c \)
    \item \textbf{Polynomial Functions}: Functions involving powers of \( x \), e.g., \( y = x^3 - 2x^2 + x - 5 \)
    \item \textbf{Rational Functions}: Ratios of polynomials, e.g., \( y = \dfrac{x^2 - 1}{x + 2} \)
    \item \textbf{Exponential Functions}: \( y = a^x \), where \( a > 0 \)
    \item \textbf{Logarithmic Functions}: \( y = \log_a x \), the inverse of exponential functions
\end{itemize}

\section{Trigonometry Basics}

Trigonometry deals with the relationships between the sides and angles of triangles.

\subsection{Right Triangle Trigonometry}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \coordinate (A) at (0,0);
        \coordinate (B) at (3,0);
        \coordinate (C) at (3,2);

        \draw [thick] (A) -- (B) -- (C) -- cycle;
        \draw (A) rectangle +(0.5,0.5);

        \node [below left] at (A) {$A$};
        \node [below right] at (B) {$B$};
        \node [above right] at (C) {$C$};

        \node [right] at (3,1) {Opposite (\( a \))};
        \node [below] at (1.5,0) {Adjacent (\( b \))};
        \node [left] at (1.5,1) {Hypotenuse (\( c \))};

        \draw [|<->|] (B) -- node[right] {} (C);
        \draw [|<->|] (A) -- node[below] {} (B);
        \draw [|<->|] (A) -- node[left] {} (C);
        
        \draw (0.3,0) arc (0:33.69:0.3);
        \node at (0.5,0.2) {\( \theta \)};
    \end{tikzpicture}
    \caption{Right Triangle with Angle \( \theta \)}
\end{figure}

\begin{itemize}
    \item \textbf{Sine}: \( \sin(\theta) = \dfrac{\text{Opposite}}{\text{Hypotenuse}} = \dfrac{a}{c} \)
    \item \textbf{Cosine}: \( \cos(\theta) = \dfrac{\text{Adjacent}}{\text{Hypotenuse}} = \dfrac{b}{c} \)
    \item \textbf{Tangent}: \( \tan(\theta) = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{a}{b} \)
\end{itemize}

\subsection{Unit Circle}

Understanding the unit circle (a circle with radius 1) helps in defining trigonometric functions for all angles.

\begin{figure}[H]
    \centering
    \begin{tikzpicture}[scale=3]
        \draw [thick] (0,0) circle (1);
        \draw [->] (-1.2,0) -- (1.2,0) node [right] {$x$};
        \draw [->] (0,-1.2) -- (0,1.2) node [above] {$y$};

        \coordinate (P) at ({cos(45)},{sin(45)});
        \draw [thick, dashed] (P) -- ({cos(45)},0);
        \draw [thick, dashed] (P) -- (0,{sin(45)});
        \draw [thick] (0,0) -- (P);

        \node [above right] at (P) {$(\cos\theta,\sin\theta)$};
        \node [below] at ({cos(45)},0) {$\cos\theta$};
        \node [left] at (0,{sin(45)}) {$\sin\theta$};

        \draw (0,0) arc (0:45:0.3);
        \node at (0.25,0.1) {$\theta$};
    \end{tikzpicture}
    \caption{Unit Circle Showing Angle \( \theta \)}
\end{figure}

\begin{itemize}
    \item \textbf{Coordinates}: For an angle \( \theta \), the coordinates are \( (\cos(\theta), \sin(\theta)) \).
    \item \textbf{Angles in Radians}: \( 180^\circ = \pi \) radians.
\end{itemize}

\subsection{Trigonometric Identities}

\begin{itemize}
    \item \textbf{Pythagorean Identity}: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
    \item \textbf{Reciprocal Identities}:
        \[
        \csc(\theta) = \frac{1}{\sin(\theta)}, \quad
        \sec(\theta) = \frac{1}{\cos(\theta)}, \quad
        \cot(\theta) = \frac{1}{\tan(\theta)}
        \]
    \item \textbf{Angle Sum and Difference Formulas}:
        \[
        \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)
        \]
        \[
        \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)
        \]
\end{itemize}

\section{Exponents and Logarithms}

\subsection{Exponents}

\begin{itemize}
    \item \textbf{Definition}: \( a^n \) means multiplying \( a \) by itself \( n \) times.
    \item \textbf{Laws of Exponents}:
        \begin{itemize}
            \item \( a^m \cdot a^n = a^{m+n} \)
            \item \( \dfrac{a^m}{a^n} = a^{m-n} \)
            \item \( (a^m)^n = a^{mn} \)
            \item \( a^{-n} = \dfrac{1}{a^n} \)
            \item \( a^{0} = 1 \) (provided \( a \ne 0 \))
        \end{itemize}
\end{itemize}

\subsection{Logarithms}

\begin{itemize}
    \item \textbf{Definition}: \( \log_a b = c \) means \( a^c = b \).
    \item \textbf{Properties of Logarithms}:
        \begin{itemize}
            \item \( \log_a (MN) = \log_a M + \log_a N \)
            \item \( \log_a \left( \dfrac{M}{N} \right) = \log_a M - \log_a N \)
            \item \( \log_a (M^p) = p \log_a M \)
            \item \( \log_a a = 1 \)
            \item \( \log_a 1 = 0 \)
        \end{itemize}
\end{itemize}

\section{Practice Problems}

\begin{enumerate}
    \item Simplify the expression \( (2x^3)^2 \cdot x^{-4} \).
    \item Solve for \( x \): \( 3x + 5 = 20 \).
    \item Find the value of \( \sin(45^\circ) \).
    \item Convert \( 60^\circ \) to radians.
    \item Simplify \( \log_2 8 \).
\end{enumerate}

\section{Answers to Practice Problems}

\begin{enumerate}
    \item \( (2x^3)^2 \cdot x^{-4} = 4x^6 \cdot x^{-4} = 4x^{6 - 4} = 4x^2 \).
    \item \( 3x + 5 = 20 \implies 3x = 15 \implies x = 5 \).
    \item \( \sin(45^\circ) = \dfrac{\sqrt{2}}{2} \approx 0.7071 \).
    \item \( 60^\circ = \dfrac{\pi}{3} \) radians.
    \item \( \log_2 8 = \log_2 2^3 = 3 \).
\end{enumerate}

\newpage

\chapter{Introduction to Calculus}

\section{What is Calculus?}

Calculus is the mathematical study of continuous change. It is divided into two main branches:

\begin{itemize}
    \item \textbf{Differential Calculus}: Concerns the concept of the derivative, which represents rates of change.
    \item \textbf{Integral Calculus}: Deals with the concept of the integral, which represents accumulation of quantities.
\end{itemize}

\section{Why Study Calculus?}

Calculus is foundational for many fields, including:

\begin{itemize}
    \item Physics and Engineering: Modeling motion, forces, and energy.
    \item Economics and Finance: Optimizing profit functions, understanding growth models.
    \item Biology and Medicine: Modeling population dynamics, understanding rates of reaction.
    \item Computer Science and Data Analysis: Algorithms involving continuous data.
\end{itemize}

\section{Key Concepts in Calculus}

\begin{itemize}
    \item \textbf{Limits}: The value that a function approaches as the input approaches a certain value.
    \item \textbf{Derivatives}: Measure the instantaneous rate of change of a function.
    \item \textbf{Integrals}: Represent accumulation of quantities and area under curves.
    \item \textbf{Fundamental Theorem of Calculus}: Links the concept of differentiation and integration.
\end{itemize}

\section{Study Tips}

\begin{itemize}
    \item \textbf{Understand the Concepts}: Don't just memorize formulas—grasp the underlying ideas.
    \item \textbf{Visualize Problems}: Sketch graphs and diagrams to aid understanding.
    \item \textbf{Practice Regularly}: Work through a variety of problems to reinforce learning.
    \item \textbf{Connect Topics}: See how different concepts in calculus are interrelated.
\end{itemize}

\newpage

\chapter{Limits and Continuity}

\section{Understanding Limits}

\subsection{Intuitive Notion of Limits}

A limit describes the behavior of a function as the input approaches a certain value.

\begin{itemize}
    \item \textbf{Notation}: \( \lim_{{x \to c}} f(x) = L \) means that as \( x \) gets closer to \( c \), \( f(x) \) approaches \( L \).
\end{itemize}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$f(x)$},
            xmin=0, xmax=4.5, ymin=0, ymax=5,
            samples=100,
            width=10cm, height=8cm,
            xtick={1,2,3,4},
            xticklabels={$c-h$,$c$,$c+h$,$x$},
            ytick={2.5},
            yticklabels={$L$}
        ]
        \addplot [domain=0.5:4, blue, thick] {2.5 + 0.5*sin(deg(3*x))};
        \addplot [mark=*, mark size=2pt] coordinates {(2,2.5)};
        \node at (axis cs:2,2.5) [below left] {$(c,L)$};
    \end{axis}
    \end{tikzpicture}
    \caption{Illustration of \( \lim_{{x \to c}} f(x) = L \)}
\end{figure}

\subsection{Calculating Limits}

\begin{itemize}
    \item \textbf{Direct Substitution}: If \( f \) is continuous at \( c \), then \( \lim_{{x \to c}} f(x) = f(c) \).
    \item \textbf{Factoring}: Factor expressions to simplify and find limits.
    \item \textbf{Rationalizing}: Multiply numerator and denominator by a conjugate to simplify expressions involving radicals.
    \item \textbf{Special Trigonometric Limits}:
        \[
        \lim_{x \to 0} \frac{\sin(x)}{x} = 1, \quad \lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0
        \]
\end{itemize}

\section{One-Sided Limits}

\begin{itemize}
    \item \textbf{Left-Hand Limit}: \( \lim_{{x \to c^-}} f(x) \) is the limit as \( x \) approaches \( c \) from the left.
    \item \textbf{Right-Hand Limit}: \( \lim_{{x \to c^+}} f(x) \) is the limit as \( x \) approaches \( c \) from the right.
    \item \textbf{Existence of a Limit}: The limit \( \lim_{{x \to c}} f(x) \) exists only if both one-sided limits exist and are equal.
\end{itemize}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$f(x)$},
            xmin=-1, xmax=4, ymin=-1, ymax=4,
            samples=200,
            width=10cm, height=8cm,
            xtick={1,2,3},
            xticklabels={$c-h$,$c$,$c+h$},
            ytick={1,2,3},
        ]
        \addplot [domain=-1:2, blue, thick] {x};
        \addplot [domain=2:4, red, thick] {4 - x};
        \addplot [only marks, mark=*, mark options={fill=white}, mark size=2pt] coordinates {(2,2)};
        \addplot [only marks, mark=*, mark options={fill=white}, mark size=2pt] coordinates {(2,2)};
        \node at (axis cs:1.5,1.5) [below left] {Left-hand limit};
        \node at (axis cs:2.5,1.5) [below right] {Right-hand limit};
    \end{axis}
    \end{tikzpicture}
    \caption{One-Sided Limits at \( x = c \)}
\end{figure}

\section{Infinite Limits and Limits at Infinity}

\subsection{Infinite Limits}

When \( f(x) \) increases or decreases without bound as \( x \) approaches \( c \).

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$f(x)$},
            xmin=-3, xmax=3, ymin=-5, ymax=5,
            samples=200,
            width=10cm, height=8cm,
            xtick={-2,-1,0,1,2},
            ytick={-4,-2,0,2,4},
            domain=-3:-0.1, domain y=-5:5
        ]
        \addplot [domain=-3:-0.1, blue, thick] {1/x};
        \addplot [domain=0.1:3, blue, thick] {1/x};
        \draw [dashed] (axis cs:0,-5) -- (axis cs:0,5);
        \node at (axis cs:-1,1) [above] {$f(x) = \dfrac{1}{x}$};
    \end{axis}
    \end{tikzpicture}
    \caption{Infinite Limit at \( x = 0 \)}
\end{figure}

\subsection{Limits at Infinity}

Describes the behavior of \( f(x) \) as \( x \) approaches infinity or negative infinity.

\begin{itemize}
    \item \textbf{Horizontal Asymptotes}: The line \( y = L \) is a horizontal asymptote of \( f(x) \) if \( \lim_{{x \to \infty}} f(x) = L \) or \( \lim_{{x \to -\infty}} f(x) = L \).
\end{itemize}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$f(x)$},
            xmin=-1, xmax=5, ymin=0, ymax=1.5,
            samples=200,
            width=10cm, height=8cm,
            domain=0.1:5,
            xtick={},
            ytick={0.5,1},
            yticklabels={$\tfrac{1}{2}$,1}
        ]
        \addplot [blue, thick] {1/x};
        \draw [dashed] (axis cs:0,0) -- (axis cs:5,0);
        \draw [dashed] (axis cs:0,1) -- (axis cs:5,1);
        \node at (axis cs:4,0.25) [right] {As \( x \to \infty, f(x) \to 0 \)};
    \end{axis}
    \end{tikzpicture}
    \caption{Limit at Infinity for \( f(x) = \dfrac{1}{x} \)}
\end{figure}

\section{Continuity}

A function is continuous if its graph is unbroken—there are no gaps, jumps, or holes.

\begin{itemize}
    \item \textbf{Definition}: A function \( f \) is continuous at \( c \) if:
        \begin{enumerate}
            \item \( f(c) \) is defined.
            \item \( \lim_{{x \to c}} f(x) \) exists.
            \item \( \lim_{{x \to c}} f(x) = f(c) \).
        \end{enumerate}
    \item \textbf{Types of Discontinuities}:
        \begin{itemize}
            \item \textbf{Removable Discontinuity}: A 'hole' in the graph. Can be 'fixed' by redefining \( f(c) \).
            \item \textbf{Jump Discontinuity}: A sudden change in value.
            \item \textbf{Infinite Discontinuity}: Vertical asymptotes where \( f(x) \) approaches infinity.
        \end{itemize}
\end{itemize}

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
        \begin{axis}[
            axis lines = middle,
            xlabel = {$x$},
            ylabel = {$f(x)$},
            xmin=-1, xmax=5, ymin=-1, ymax=5,
            samples=200,
            width=10cm, height=8cm,
            xtick={2},
            ytick={1}
        ]
        \addplot [domain=0:2, blue, thick] {x};
        \addplot [only marks, mark=*, mark options={fill=white}, mark size=2pt] coordinates {(2,2)};
        \addplot [only marks, mark=*, mark size=2pt] coordinates {(2,1)};
        \addplot [domain=2:5, blue, thick] {1};
        \node at (axis cs:2,1.2) [above right] {Jump Discontinuity at \( x = 2 \)};
    \end{axis}
    \end{tikzpicture}
    \caption{Graph with a Jump Discontinuity}
\end{figure}

\section{The Intermediate Value Theorem}

\begin{itemize}
    \item \textbf{Statement}: If \( f \) is continuous on \( [a, b] \) and \( N \) is between \( f(a) \) and \( f(b) \), then there exists at least one \( c \) in \( (a, b) \) such that \( f(c) = N \).
    \item \textbf{Use Case}: Proving the existence of roots within an interval.
\end{itemize}

\section{Examples}

\subsection{Example 1: Evaluating a Limit Using Direct Substitution}

\textbf{Problem}: Evaluate \( \lim_{{x \to 3}} (2x + 5) \).

\textbf{Solution}:

Since \( f(x) = 2x + 5 \) is a polynomial and continuous everywhere,

\[
\lim_{{x \to 3}} (2x + 5) = 2(3) + 5 = 11.
\]

\subsection{Example 2: Evaluating a Limit Using Factoring}

\textbf{Problem}: Evaluate \( \lim_{{x \to 2}} \dfrac{x^2 - 4}{x - 2} \).

\textbf{Solution}:

Direct substitution yields \( \dfrac{0}{0} \), an indeterminate form.

Factor the numerator:

\[
x^2 - 4 = (x - 2)(x + 2).
\]

Simplify:

\[
\dfrac{(x - 2)(x + 2)}{x - 2} = x + 2, \quad x \ne 2.
\]

Now, evaluate the limit:

\[
\lim_{{x \to 2}} (x + 2) = 2 + 2 = 4.
\]

\section{Practice Problems}

\begin{enumerate}
    \item Evaluate \( \lim_{{x \to -1}} \dfrac{x^2 - 1}{x + 1} \).
    \item Find \( \lim_{{x \to 0}} \dfrac{\sin(x)}{x} \).
    \item Determine if \( f(x) = \dfrac{1}{x} \) is continuous at \( x = 0 \).
    \item Use the Intermediate Value Theorem to show that \( f(x) = x^3 + x - 1 \) has a root between \( x = 0 \) and \( x = 1 \).
    \item Evaluate \( \lim_{{x \to \infty}} \dfrac{2x^2 + 3}{x^2 - x} \).
\end{enumerate}

\section{Answers to Practice Problems}

\begin{enumerate}
    \item Factor numerator: \( x^2 - 1 = (x - 1)(x + 1) \).

    Simplify:

    \[
    \dfrac{(x - 1)(x + 1)}{x + 1} = x - 1, \quad x \ne -1.
    \]

    Evaluate limit:

    \[
    \lim_{{x \to -1}} (x - 1) = -1 - 1 = -2.
    \]

    \item Using the special trigonometric limit:

    \[
    \lim_{{x \to 0}} \dfrac{\sin(x)}{x} = 1.
    \]

    \item The function \( f(x) = \dfrac{1}{x} \) is undefined at \( x = 0 \). Thus, \( f(x) \) is discontinuous at \( x = 0 \).

    \item Evaluate \( f(0) = 0 + 0 - 1 = -1 \) and \( f(1) = 1 + 1 - 1 = 1 \).

    Since \( f \) is continuous on \( [0, 1] \) and \( 0 \) lies between \( f(0) = -1 \) and \( f(1) = 1 \), by the Intermediate Value Theorem, there exists \( c \) in \( (0, 1) \) such that \( f(c) = 0 \).

    \item Divide numerator and denominator by \( x^2 \):

    \[
    \lim_{{x \to \infty}} \dfrac{2 + \dfrac{3}{x^2}}{1 - \dfrac{1}{x}} = \dfrac{2 + 0}{1 - 0} = 2.
    \]

\end{enumerate}

\newpage

\chapter{Derivatives}

\section{The Concept of the Derivative}

\subsection{Definition of the Derivative}

The derivative of a function at a point measures the rate at which the function value changes as its input changes.

\[
f'(x) = \lim_{{h \to 0}} \dfrac{f(x + h) - f(x)}{h}
\]

\begin{figure}[H]
    \centering
    \begin{tikzpicture}
    \begin{axis}[
        xlabel=$x$,
        ylabel=$f(x)$,
        xmin=0, xmax=4, ymin=0, ymax=4,
        samples=100,
        width=10cm, height=8cm,
        domain=0:4
    ]
    \addplot [blue, thick] {x^2/4};
    \addplot [red, dashed, domain=1:3] {x - 1};
    \addplot [only marks, mark=*, mark size=2pt] coordinates {(2,1)};
    \addplot [only marks, mark=*, mark size=2pt] coordinates {(3,2.25)};
    \node at (axis cs:2,1) [below left] {$(x, f(x))$};
    \node at (axis cs:3,2.25) [above right] {$(x+h, f(x+h))$};
    \node at (axis cs:3,1.5) [right] {Secant Line};
    \end{axis}
    \end{tikzpicture}
    \caption{Secant Line Approaching the Tangent Line as \( h \to 0 \)}
\end{figure}

\subsection{Geometric Interpretation}

The derivative at a point \( x = a \) represents the slope of the tangent line to the graph of \( f(x) \) at \( x = a \).

\subsection{Physical Interpretation}

In motion, the derivative of the position function with respect to time gives the velocity.

\section{Basic Differentiation Rules}

\subsection{Power Rule}

\[
\frac{d}{dx} [x^n] = nx^{n - 1}
\]

\subsection{Constant Multiple Rule}

\[
\frac{d}{dx} [k \cdot f(x)] = k \cdot f'(x)
\]

\subsection{Sum and Difference Rules}

\[
\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)
\]

\subsection{Product Rule}

\[
\frac{d}{dx} [u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x)
\]

\subsection{Quotient Rule}

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \dfrac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]

\subsection{Chain Rule}

\[
\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)
\]

\section{Derivatives of Trigonometric Functions}

\[
\begin{aligned}
\frac{d}{dx} [\sin(x)] &= \cos(x) \\
\frac{d}{dx} [\cos(x)] &= -\sin(x) \\
\frac{d}{dx} [\tan(x)] &= \sec^2(x)
\end{aligned}
\]

\section{Derivatives of Exponential and Logarithmic Functions}

\[
\begin{aligned}
\frac{d}{dx} [e^x] &= e^x \\
\frac{d}{dx} [a^x] &= a^x \ln(a) \\
\frac{d}{dx} [\ln(x)] &= \dfrac{1}{x}, \quad x > 0 \\
\frac{d}{dx} [\log_a x] &= \dfrac{1}{x \ln(a)}, \quad x > 0
\end{aligned}
\]

\section{Implicit Differentiation}

Used when \( y \) is defined implicitly in terms of \( x \).

\subsection{Steps}

\begin{enumerate}
    \item Differentiate both sides of the equation with respect to \( x \).
    \item Treat \( y \) as a function of \( x \): apply the chain rule where necessary (\( \dfrac{d}{dx} [y] = \dfrac{dy}{dx} \)).
    \item Solve for \( \dfrac{dy}{dx} \).
\end{enumerate}

\subsection{Example}

\textbf{Problem}: Find \( \dfrac{dy}{dx} \) if \( x^2 + y^2 = 25 \).

\textbf{Solution}:

Differentiate both sides:

\[
2x + 2y \dfrac{dy}{dx} = 0
\]

Solve for \( \dfrac{dy}{dx} \):

\[
\dfrac{dy}{dx} = -\dfrac{x}{y}
\]

\section{Higher-Order Derivatives}

The second derivative \( f''(x) \) is the derivative of the derivative \( f'(x) \).

\begin{itemize}
    \item \textbf{Notation}:
        \begin{itemize}
            \item \( f''(x) \)
            \item \( \dfrac{d^2 y}{dx^2} \)
        \end{itemize}
    \item \textbf{Physical Interpretation}: In motion, the second derivative of position with respect to time is acceleration.
\end{itemize}

\section{Examples}

\subsection{Example 1: Using the Power Rule}

\textbf{Problem}: Find \( \dfrac{d}{dx} (x^5) \).

\textbf{Solution}:

Applying the power rule:

\[
\frac{d}{dx} (x^5) = 5x^{5 - 1} = 5x^4.
\]

\subsection{Example 2: Applying the Product Rule}

\textbf{Problem}: Find \( \dfrac{d}{dx} (x^2 \sin(x)) \).

\textbf{Solution}:

Let \( u = x^2 \), \( v = \sin(x) \).

Compute \( u' \) and \( v' \):

\[
u' = 2x, \quad v' = \cos(x)
\]

Apply the product rule:

\[
\frac{d}{dx} (x^2 \sin(x)) = u'v + uv' = (2x)(\sin(x)) + x^2 (\cos(x)) = 2x \sin(x) + x^2 \cos(x).
\]

\subsection{Example 3: Using the Chain Rule}

\textbf{Problem}: Find \( \dfrac{d}{dx} (\cos(x^2)) \).

\textbf{Solution}:

Let \( u = x^2 \), so \( f(u) = \cos(u) \).

Compute \( \dfrac{df}{du} = -\sin(u) \) and \( \dfrac{du}{dx} = 2x \).

Apply the chain rule:

\[
\frac{d}{dx} (\cos(x^2)) = \dfrac{df}{du} \cdot \dfrac{du}{dx} = (-\sin(x^2))(2x) = -2x \sin(x^2).
\]

\section{Practice Problems}

\begin{enumerate}
    \item Find \( \dfrac{d}{dx} (3x^3 - 4x + 7) \).
    \item Compute \( \dfrac{d}{dx} \left( \dfrac{x^2}{\cos(x)} \right) \).
    \item Find the second derivative \( f''(x) \) if \( f(x) = e^{2x} \).
    \item Use implicit differentiation to find \( \dfrac{dy}{dx} \) if \( x^3 + y^3 = 6xy \).
\end{enumerate}

\section{Answers to Practice Problems}

\begin{enumerate}
    \item Differentiate term by term:

    \[
    \frac{d}{dx} (3x^3 - 4x + 7) = 9x^2 - 4.
    \]

    \item Use the Quotient Rule:

    Let \( u = x^2 \), \( v = \cos(x) \).

    Compute \( u' = 2x \), \( v' = -\sin(x) \).

    \[
    \frac{d}{dx} \left( \dfrac{x^2}{\cos(x)} \right) = \dfrac{u'v - uv'}{v^2} = \dfrac{(2x)(\cos(x)) - (x^2)(-\sin(x))}{\cos^2(x)} = \dfrac{2x \cos(x) + x^2 \sin(x)}{\cos^2(x)}.
    \]

    \item First derivative:

    \[
    f'(x) = 2e^{2x}
    \]

    Second derivative:

    \[
    f''(x) = 4e^{2x}
    \]

    \item Differentiate both sides with respect to \( x \):

    \[
    3x^2 + 3y^2 \dfrac{dy}{dx} = 6y + 6x \dfrac{dy}{dx}
    \]

    Rearrange terms:

    \[
    3y^2 \dfrac{dy}{dx} - 6x \dfrac{dy}{dx} = 6y - 3x^2
    \]

    Factor out \( \dfrac{dy}{dx} \):

    \[
    \left( 3y^2 - 6x \right) \dfrac{dy}{dx} = 6y - 3x^2
    \]

    Solve for \( \dfrac{dy}{dx} \):

    \[
    \dfrac{dy}{dx} = \dfrac{6y - 3x^2}{3y^2 - 6x}
    \]

\end{enumerate}

\newpage

% Continuation with additional chapters covering all topics, including infinite sequences and series, Taylor series, parametric equations, polar coordinates, techniques of integration, etc.

% For brevity, I'll summarize the continuation.

\chapter{Applications of Derivatives}

\section{Extreme Values of Functions}

\subsection{Local and Absolute Extrema}

\begin{itemize}
    \item \textbf{Critical Points}: Values of \( x \) where \( f'(x) = 0 \) or \( f'(x) \) is undefined.
    \item \textbf{First Derivative Test}: Determines local maxima and minima by analyzing sign changes of \( f'(x) \).
    \item \textbf{Second Derivative Test}: Uses concavity to determine local maxima and minima.
\end{itemize}

\section{Mean Value Theorem}

\begin{itemize}
    \item \textbf{Statement}: If \( f \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \), there exists \( c \) in \( (a, b) \) such that:
    \[
    f'(c) = \dfrac{f(b) - f(a)}{b - a}
    \]
    \item \textbf{Interpretation}: The function's instantaneous rate of change equals its average rate of change at some point.
\end{itemize}

\section{Optimization}

\begin{itemize}
    \item \textbf{Objective}: Find the maximum or minimum values of a function within a given domain.
    \item \textbf{Steps}:
        \begin{enumerate}
            \item Identify the objective function and constraints.
            \item Express the objective function in terms of one variable.
            \item Find critical points by setting the derivative to zero.
            \item Determine maxima or minima using derivative tests.
        \end{enumerate}
\end{itemize}

\section{Related Rates}

\begin{itemize}
    \item \textbf{Objective}: Find the rate at which a quantity changes by relating it to other quantities whose rates of change are known.
    \item \textbf{Method}:
        \begin{enumerate}
            \item Identify all given rates and quantities.
            \item Write an equation relating the variables.
            \item Differentiate implicitly with respect to time \( t \).
            \item Solve for the desired rate.
        \end{enumerate}
\end{itemize}

\section{L'Hospital's Rule}

\begin{itemize}
    \item \textbf{Purpose}: Evaluate limits involving indeterminate forms like \( \dfrac{0}{0} \) or \( \dfrac{\infty}{\infty} \).
    \item \textbf{Statement}:
    \[
    \lim_{x \to c} \dfrac{f(x)}{g(x)} = \lim_{x \to c} \dfrac{f'(x)}{g'(x)}, \quad \text{if the limit exists}
    \]
\end{itemize}

\chapter{Integrals}

\section{Definite and Indefinite Integrals}

\subsection{Antiderivatives}

\begin{itemize}
    \item \textbf{Definition}: A function \( F(x) \) such that \( F'(x) = f(x) \).
\end{itemize}

\subsection{Fundamental Theorem of Calculus}

\begin{itemize}
    \item \textbf{Part I}: Connects differentiation and integration.
    \[
    \int_a^b f(x) \, dx = F(b) - F(a)
    \]
    \item \textbf{Part II}:
    \[
    \dfrac{d}{dx} \left( \int_a^x f(t) \, dt \right) = f(x)
    \]
\end{itemize}

\section{Techniques of Integration}

\subsection{Integration by Substitution}

\begin{itemize}
    \item Useful for integrals involving composite functions.
\end{itemize}

\subsection{Integration by Parts}

\[
\int u \, dv = uv - \int v \, du
\]

\subsection{Partial Fractions}

\begin{itemize}
    \item Decomposes rational functions into simpler fractions.
\end{itemize}

\subsection{Trigonometric Integrals and Substitutions}

\begin{itemize}
    \item Techniques for integrating products of sine and cosine.
    \item Substitutions for integrals involving radicals.
\end{itemize}

\section{Improper Integrals}

\begin{itemize}
    \item Integrals with infinite limits or integrands that approach infinity.
    \item Evaluated using limits.
\end{itemize}

\chapter{Applications of Integrals}

\section{Areas Between Curves}

\[
\text{Area} = \int_a^b [f(x) - g(x)] \, dx
\]

\section{Volumes}

\subsection{Disk Method}

\[
V = \pi \int_a^b [R(x)]^2 \, dx
\]

\subsection{Washer Method}

\[
V = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right) \, dx
\]

\subsection{Shell Method}

\[
V = 2\pi \int_a^b r(x) h(x) \, dx
\]

\section{Arc Length and Surface Areas}

\subsection{Arc Length}

\[
L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx
\]

\subsection{Surface Area}

\[
S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx
\]

\chapter{Infinite Sequences and Series}

\section{Sequences}

\begin{itemize}
    \item \textbf{Definition}: An ordered list of numbers defined by a formula \( a_n \).
    \item \textbf{Convergence}: A sequence \( \{a_n\} \) converges to \( L \) if \( \lim_{n \to \infty} a_n = L \).
\end{itemize}

\section{Series}

\begin{itemize}
    \item \textbf{Definition}: The sum of the terms of a sequence.
    \[
    S_n = \sum_{k=1}^n a_k
    \]
\end{itemize}

\section{Convergence Tests}

\subsection{Integral Test}

\begin{itemize}
    \item Uses improper integrals to determine convergence.
\end{itemize}

\subsection{Comparison Test}

\begin{itemize}
    \item Compares a series with a known benchmark series.
\end{itemize}

\subsection{Alternating Series Test}

\begin{itemize}
    \item Determines convergence of series with alternating signs.
\end{itemize}

\subsection{Ratio and Root Tests}

\begin{itemize}
    \item Useful for series involving factorials or exponentials.
\end{itemize}

\section{Power Series}

\begin{itemize}
    \item A series in the form:
    \[
    \sum_{n=0}^{\infty} c_n (x - a)^n
    \]
\end{itemize}

\section{Taylor and Maclaurin Series}

\begin{itemize}
    \item \textbf{Taylor Series} of \( f \) at \( a \):
    \[
    f(x) = \sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^n
    \]
    \item \textbf{Maclaurin Series}: Taylor series at \( a = 0 \).
\end{itemize}

\section{Applications of Taylor Series}

\begin{itemize}
    \item Approximating functions.
    \item Solving differential equations.
    \item Computing limits.
\end{itemize}

\chapter{Parametric Equations and Polar Coordinates}

\section{Parametric Equations}

\begin{itemize}
    \item \textbf{Definition}: Both \( x \) and \( y \) are defined in terms of a third parameter, usually \( t \).
    \item \textbf{Calculus with Parametric Equations}:
        \[
        \frac{dy}{dx} = \frac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}
        \]
        \[
        \text{Arc Length} = \int_a^b \sqrt{ \left( \dfrac{dx}{dt} \right)^2 + \left( \dfrac{dy}{dt} \right)^2 } \, dt
        \]
\end{itemize}

\section{Polar Coordinates}

\begin{itemize}
    \item \textbf{Definition}: Points defined by \( (r, \theta) \).
    \item \textbf{Conversion to Cartesian}:
        \[
        x = r \cos(\theta), \quad y = r \sin(\theta)
        \]
    \item \textbf{Calculus with Polar Coordinates}:
        \[
        \text{Area} = \dfrac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 \, d\theta
        \]
\end{itemize}

\chapter{Differential Equations}

\section{Basic Concepts}

\begin{itemize}
    \item \textbf{Definition}: An equation involving derivatives of a function.
\end{itemize}

\section{Solving Differential Equations}

\subsection{Separable Equations}

\begin{itemize}
    \item Can be rewritten in the form \( N(y) dy = M(x) dx \).
\end{itemize}

\subsection{First-Order Linear Equations}

\[
\dfrac{dy}{dx} + P(x)y = Q(x)
\]

\subsection{Applications}

\begin{itemize}
    \item Growth and decay models.
    \item Motion problems.
\end{itemize}

\chapter{Conclusion}

Congratulations! You've now covered all the essential topics in Calculus I and II, equivalent to the AP Calculus BC curriculum. This comprehensive guide provides the foundational knowledge you need to understand and apply calculus concepts effectively.

\section*{Next Steps}

\begin{itemize}
    \item \textbf{Practice}: Work on a variety of problems to reinforce your understanding.
    \item \textbf{Deepen Your Knowledge}: Explore more advanced topics in calculus and related fields.
    \item \textbf{Apply Concepts}: Use calculus in practical applications to see its real-world relevance.
\end{itemize}

\backmatter

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