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\tableofcontents

\section{Limits and Continuity}
\subsection{Examples of Velocity, Growth Rate, and Area}
\subsubsection{Average Velocity and Instantaneous Velocity}
\subsubsection{The Growth of an Algal Culture}
\subsubsection{The Area of a Circle}
\subsection{Limits of Functions}
\subsubsection{One-Sided Limits}
\subsubsection{Rules for Calculating Limits}
\subsubsection{The Squeeze Theorem}
\subsection{Limits at Infinity and Infinite Limits}
\subsubsection{Limits at Infinity}
\subsubsection{Limits at Infinity for Rational Functions}
\subsubsection{Infinite Limits}
\subsubsection{Using Maple to Calculate Limits}
\subsection{Continuity}
\subsubsection{Continuity at a Point}
\subsubsection{Continuity on an Interval}
\subsubsection{There Are Lots of Continuous Functions}
\subsubsection{Continuous Extensions and Removable Discontinuities}
\subsubsection{Continuous Functions on Closed, Finite Intervals}
\subsubsection{Finding Roots of Equations}
\subsection{The Formal Definition of Limit}
\subsubsection{Using the Definition of Limit to Prove Theorems}
\subsubsection{Other Kinds of Limits}
\subsubsection{Chapter Review}
\section{Differentiation}
\subsection{Tangent Lines and Their Slopes}
\subsubsection{Normals}
\subsection{The Derivative}
\subsubsection{Some Important Derivatives}
\subsubsection{Leibniz Notation}
\subsubsection{Differentials}
\subsubsection{Derivatives Have the Intermediate-Value Property}
\subsection{Differentiation Rules}
\subsubsection{Sums and Constant Multiples}
\subsubsection{The Product Rule}
\subsubsection{The Reciprocal Rule}
\subsubsection{The Quotient Rule}
\subsection{The Chain Rule}
\subsubsection{Finding Derivatives with Maple}
\subsubsection{Building the Chain Rule into Differentiation Formulas}
\subsubsection{Proof of the Chain Rule (Theorem 6)}
\subsection{Derivatives of Trigonometric Functions}
\subsubsection{Some Special Limits}
\subsubsection{The Derivatives of Sine and Cosine}
\subsubsection{The Derivatives of the Other Trigonometric Functions}
\subsection{Higher-Order Derivatives}
\subsection{Using Differentials and Derivatives}
\subsubsection{Approximating Small Changes}
\subsubsection{Average and Instantaneous Rates of Change}
\subsubsection{Sensitivity to Change}
\subsubsection{Derivatives in Economics}
\subsection{The Mean-Value Theorem}
\subsubsection{Increasing and Decreasing Functions}
\subsubsection{Proof of the Mean-Value Theorem}
\subsection{Implicit Differentiation}
\subsubsection{Higher-Order Derivatives}
\subsubsection{The General Power Rule}
\subsection{Antiderivatives and Initial-Value Problems}
\subsubsection{Antiderivatives}
\subsubsection{The Indefinite Integral}
\subsubsection{Differential Equations and Initial-Value Problems}
\subsection{Velocity and Acceleration}
\subsubsection{Velocity and Speed}
\subsubsection{Acceleration}
\subsubsection{Falling Under Gravity}
\subsubsection{Chapter Review}
\section{Transcendental Functions}
\subsection{Inverse Functions}
\subsubsection{Inverting Non-One-to-One Functions}
\subsubsection{Derivatives of Inverse Functions}
\subsection{Exponential and Logarithmic Functions}
\subsubsection{Exponentials}
\subsubsection{Logarithms}
\subsection{The Natural Logarithm and Exponential}
\subsubsection{The Natural Logarithm}
\subsubsection{The Exponential Function}
\subsubsection{General Exponentials and Logarithms}
\subsubsection{Logarithmic Differentiation}
\subsection{Growth and Decay}
\subsubsection{The Growth of Exponentials and Logarithms}
\subsubsection{Exponential Growth and Decay Models}
\subsubsection{Interest on Investments}
\subsubsection{Logistic Growth}
\subsection{The Inverse Trigonometric Functions }
\subsubsection{The Inverse Sine (or Arcsine) Function}
\subsubsection{The Inverse Tangent (or Arctangent) Function}
\subsubsection{Other Inverse Trigonometric Functions}
\subsection{Hyperbolic Functions}
\subsubsection{Inverse Hyperbolic Functions}
\subsection{Second-Order Linear DEs with Constant Coefficients}
\subsubsection{Recipe for Solving ay" + by' + cy = 0}
\subsubsection{Simple Harmonic Motion}
\subsubsection{Damped Harmonic Motion}
\subsubsection{Chapter Review}
\section{More Applications of Differentiation}
\subsection{Related Rates}
\subsubsection{Procedures for Related-Rates Problems}
\subsection{Finding Roots of Equations}
\subsubsection{Discrete Maps and Fixed-Point Iteration}
\subsubsection{Newton’s Method}
\subsubsection{Solve" Routines}
\subsection{Indeterminate Forms}
\subsubsection{l'Hospital's Rules}
\subsection{Extreme Values}
\subsubsection{Maximum and Minimum Values}
\subsubsection{Critical Poi., Singular Points, and Endpoints}
\subsubsection{Finding Absolute Extreme Values}
\subsubsection{The First Derivative Test}
\subsubsection{Functions Not Defined on Closed, Finite Intervals}
\subsection{Concavity and Inflections}
\subsubsection{The Second Derivative Test}
\subsection{Sketching the Graph of a Function}
\subsubsection{Asymptotes}
\subsubsection{Examples of Formal Curve Sketching}
\subsection{Graphing with Computers}
\subsubsection{Numerical Monsters and Computer Graphing}
\subsubsection{Floating-Point Representation of Numbers in Computers}
\subsubsection{Machine Epsilon and Its Effect on Figure 4.45}
\subsubsection{Determining Machine Epsilon}
\subsection{Extreme-Value Problems}
\subsubsection{Procedure for Solving Extreme-Value Problems}
\subsection{Linear Approximations}
\subsubsection{Approximating Values of Functions}
\subsubsection{Error Analysis}
\subsection{Taylor Polynomials}
\subsubsection{Taylor's Formula}
\subsubsection{Big-O Notation}
\subsubsection{Evaluating Limits of Indeterminate Forms}
\subsection{Roundoff Error, Truncation Error, and Computers}
\subsubsection{Taylor Polynomials in Maple}
\subsubsection{Persistent Roundoff Error}
\subsubsection{Truncation, Roundoff, and Computer Algebra}
\subsubsection{Chapter Review}
\section{Integration}
\subsection{Sums and Sigma Notation}
\subsubsection{Evaluating Sums}
\subsection{Areas as Limits of Sums}
\subsubsection{The Basic Area Problem}
\subsubsection{Some Area Calculations}
\subsection{The Definite Integral}
\subsubsection{Partitions and Rieman Sums}
\subsubsection{The Definite Integral}
\subsubsection{General Rieman Sums}
\subsection{Properties of the Definite Integral}
\subsubsection{A Mean-Value Theorem for Integrals}
\subsubsection{Definite Integrals of Piecewise Continuous Functions}
\subsection{The Fundamental Theorem of Calculus}
\subsection{The Method of Substitution}
\subsubsection{Trigonometric Integrals}
\subsection{Areas of Plane Regions}
\subsubsection{Areas Between Two Curves}
\subsubsection{Chapter Review}
\section{Techniques of Integration}
\subsection{Integration by Parts}
\subsubsection{Reduction Formulas}
\subsection{Integrals of Rational Functions}
\subsubsection{Linear and Quadratic Denominators}
\subsubsection{Partial Fractions}
\subsubsection{Completing the Square}
\subsubsection{Denominators with Repeated Factors}
\subsection{Inverse Substitutions}
\subsubsection{The Inverse Trigonometric Substitutions}
\subsubsection{Inverse Hyperbolic Substitutions}
\subsubsection{Other Inverse Substitutions}
\subsubsection{The tan(x/2) Substitution}
\subsection{Other Methods for Evaluating Integrals}
\subsubsection{The Method of Undetermined Coefficients}
\subsubsection{Using Maple for Integration}
\subsubsection{Using Integral Tables}
\subsubsection{Special Functions Arising from Integrals}
\subsection{Improper Integrals}
\subsubsection{Improper Integrals of Type I}
\subsubsection{Improper Integrals of Type II}
\subsubsection{Estimating Convergence and Divergence}
\subsection{The Trapezoid and Midpoint Rules}
\subsubsection{The Trapezoid Rule}
\subsubsection{The Midpoint Rule}
\subsubsection{Error Estimates}
\subsection{Simpson's Rule}
\subsection{Other Aspects of Approximate Integration}
\subsubsection{Approximating Improper Integrals}
\subsubsection{Using Taylor's Formula}
\subsubsection{Romberg Integration}
\subsubsection{The Importance of Higher-Order Methods}
\subsubsection{Other Methods}
\subsubsection{Chapter Review}
\section{Applications of Integration}
\subsection{Volumes by Slicing, Solids of Revolution}
\subsubsection{Volumes by Slicing}
\subsubsection{Solids of Revolution}
\subsubsection{Cylindrical Shells}
\subsection{More Volumes by Slicing}
\subsection{Arc Length and Surface Area}
\subsubsection{Arc Length}
\subsubsection{The Arc Length of the Graph of a Function}
\subsubsection{Areas of Surfaces of Revolution}
\subsection{Mass, Moments, and Centre of Mass}
\subsubsection{Mass and Density}
\subsubsection{Moments and Centres of Mass}
\subsubsection{Two- and Three-Dimensional Examples}
\subsection{Centroids}
\subsubsection{Pappus's Theorem}
\subsection{Other Physical Applications}
\subsubsection{Hydrostatic Pressure}
\subsubsection{Work}
\subsubsection{Potential Energy and Kinetic Energy}
\subsection{Applications in Business, Finance, and Ecology}
\subsubsection{The Present Value of a Stream of Payments}
\subsubsection{The Economics of Exploiting Renewable Resources}
\subsection{Probability}
\subsubsection{Discrete Random Variables}
\subsubsection{Expectation, Mean, Variance, and Standard Deviation}
\subsubsection{Continuous Random Variables}
\subsubsection{The Normal Distribution}
\subsubsection{Heavy Tails}
\subsection{First-Order Differential Equations}
\subsubsection{Separable Equations}
\subsubsection{First-Order Linear Equations}
\subsubsection{Chapter Review}


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