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\section{Calculus}

Calculus is the branch of mathematics concerned with continually changing
quantities. The basic laws of science and engineering are expressed in terms
of concepts of calculus, and the deduction of knowledge about the physical
phenomena makes intensive use of the techniques of calculus. The two basic
operational ideas of calculus are differentiation (together with its
inverse, antidifferentiation) and the definite integral. Underlying each of
these is a concept of limit. The derivative arises from the notion of a
limit of a function and the definite integral from the notion of a limit of
a sequence.

Integral calculus is used to find areas and volumes. Differential calculus
is concerned with how one quantity changes in relation to another quantity.
Although both concepts have been recognized in some form for many centuries,
the fact that these concepts are closely related was only formalized in the
late seventeenth century. This connection is the content of the \hyperref{%
Fundamental Theorem of Calculus}{}{}{Cal05_04.tex#fundamental theorem of
calculus}.

\subsection{\ Calculus Contents}

\subsubsection{Working With Functions}

\hyperref{+\textbf{\quad }Functions and Their Graphs}{}{}{cal00_01.tex}

\hyperref{+\textbf{\quad }Types of Functions: Shifting and Scaling}{}{}{%
cal00_02.tex}

\hyperref{+\textbf{\quad }Graphing Calculators and Computers}{}{}{%
cal00_03.tex}

\hyperref{\textbf{+\quad }Principles of Problem Solving}{}{}{cal00_04.tex}

\subsubsection{Limits and Rates of Change}

\hyperref{+\textbf{\quad }The Tangent and Velocity Problems}{}{}{cal01_01.tex%
}

\hyperref{+\textbf{\quad }The Limit of a Function}{}{}{cal01_02.tex}

\hyperref{+\textbf{\quad }Calculating Limits Using the Limit Laws}{}{}{%
cal01_03.tex}

\hyperref{+\textbf{\quad }The Precise Definition of a Limit}{}{}{cal01_04.tex%
}

\hyperref{+\textbf{\quad }Continuity}{}{}{cal01_05.tex}

\hyperref{+\textbf{\quad }Limits at Infinity; Horizontal Asymptotes}{}{}{%
cal01_06.tex}

\hyperref{+\textbf{\quad }Tangents, Velocities, and Other Rates of Change}{}{%
}{cal01_07.tex}

\subsubsection{Derivatives}

\hyperref{+\textbf{\quad }\hyperref{Derivatives}{}{}{cal02_01.tex}\textbf{%
\qquad }}{}{}{cal02_01.tex}

\hyperref{+\textbf{\quad }Differentiation Formulas}{}{}{cal02_02.tex}

\hyperref{+\textbf{\quad }Rates of Change in the Natural and Social Sciences%
}{}{}{cal02_03.tex}

\hyperref{+\textbf{\quad }Derivatives of Trigonometric Functions}{}{}{%
cal02_04.tex}

\hyperref{+\textbf{\quad }The Chain Rule}{}{}{cal02_05.tex}

\hyperref{+\textbf{\quad }Implicit Differentiation}{}{}{cal02_06.tex}

\hyperref{+\textbf{\quad }\hyperref{Higher Derivatives}{}{}{cal02_07.tex}%
\textbf{\qquad }}{}{}{cal02_07.tex}

\hyperref{+\textbf{\quad }Related Rates}{}{}{cal02_08.tex}

\hyperref{+\textbf{\quad }Differentials; Linear and Quadratic Approximations%
}{}{}{cal02_09.tex}

\hyperref{+\textbf{\quad }Newton's Method}{}{}{cal02_10.tex}

\subsubsection{Inverse Functions}

\hyperref{+\textbf{\quad }Exponential Functions and Their Derivatives}{}{}{%
cal03_01.tex}

\hyperref{+\textbf{\quad }Inverse Functions}{}{}{cal03_02.tex}

\hyperref{+\textbf{\quad }Logarithmic Functions}{}{}{cal03_03.tex}

\hyperref{+\textbf{\quad }Derivatives of Logarithmic Functions}{}{}{%
cal03_04.tex}

\hyperref{+\textbf{\quad }Exponential Growth and Decay}{}{}{cal03_05.tex}

\hyperref{+\textbf{\quad }Inverse Trigonometric Functions}{}{}{cal03_06.tex}

\hyperref{+\textbf{\quad }Hyperbolic Functions}{}{}{cal03_07.tex}

\hyperref{+\textbf{\quad }Indeterminate Forms and l'Hospital's Rule}{}{}{%
cal03_08.tex}

\subsubsection{The Mean Value Theorem and Curve Sketching}

\hyperref{+\textbf{\quad }Maximum and Minimum Values}{}{}{cal04_01.tex}

\hyperref{+\textbf{\quad }The Mean Value Theorem}{}{}{cal04_02.tex}

\hyperref{+\textbf{\quad }Monotonic Functions and the First Derivative Test}{%
}{}{cal04_03.tex}

\hyperref{+\textbf{\quad }Concavity and Points of Inflection}{}{}{%
cal04_04.tex}

\hyperref{+\textbf{\quad }Curve Sketching}{}{}{cal04_05.tex}

\hyperref{+\textbf{\quad }Graphing with Calculus and Calculators}{}{}{%
cal04_06.tex}

\hyperref{+\textbf{\quad }Applied Maximum and Minimum Problems}{}{}{%
cal04_07.tex}

\hyperref{+\textbf{\quad }Applications to Economics}{}{}{cal04_08.tex}

\hyperref{+\textbf{\quad }Antiderivatives}{}{}{cal04_09.tex}

\subsubsection{Integrals}

\hyperref{+\textbf{\quad }Sigma Notation}{}{}{cal05_01.tex}

\hyperref{+\textbf{\quad }Area}{}{}{cal05_02.tex}

\hyperref{+\textbf{\quad }The Definite Integral}{}{}{cal05_03.tex}

\hyperref{+\textbf{\quad }The Fundamental Theorem of Calculus}{}{}{%
cal05_04.tex}

\hyperref{+\textbf{\quad }The Substitution Rule}{}{}{cal05_05.tex}

\hyperref{+\textbf{\quad }The Logarithm Defined as an Integral}{}{}{%
cal05_06.tex}

\subsubsection{Applications of Integration}

\hyperref{+\textbf{\quad }Areas Between Curves}{}{}{cal06_01.tex}

\hyperref{+\textbf{\quad }Volume}{}{}{cal06_02.tex}

\hyperref{+\quad Volumes by Cylindrical Shells}{}{}{cal06_03.tex}

\hyperref{+\textbf{\quad }Work}{}{}{cal06_04.tex}

\hyperref{+\textbf{\quad }Average Value of a Function}{}{}{cal06_05.tex}

\hyperref{+\textbf{\quad }Differential Equations}{}{}{cal06_06.tex}

\hyperref{+\textbf{\quad }Arc Length}{}{}{cal06_07.tex}

\hyperref{+\textbf{\quad }Area of a Surface of Revolution}{}{}{cal06_08.tex}

\hyperref{+\textbf{\quad }Moments and Centers of Mass}{}{}{cal06_09.tex}

\hyperref{+\textbf{\quad }Hydrostatic Pressure and Force}{}{}{cal06_10.tex}

\hyperref{+\textbf{\quad }Applications to Economics and Biology}{}{}{%
cal06_11.tex}

\subsubsection{Techniques of Integration}

\hyperref{+\textbf{\quad }Integration by Parts}{}{}{cal07_01.tex}

\hyperref{+\textbf{\quad }Trigonometric Integrals}{}{}{cal07_02.tex}

\hyperref{+\textbf{\quad }Trigonometric Substitution}{}{}{cal07_03.tex}

\hyperref{+\textbf{\quad }Integration of Rational Functions by Partial
Fractions}{}{}{cal07_04.tex}

\hyperref{+\textbf{\quad }Rationalizing Substitutions}{}{}{cal07_05.tex}

\hyperref{+\textbf{\quad }Strategy for Integration}{}{}{cal07_06.tex}

\hyperref{+\textbf{\quad }Using Tables of Integrals and Computer Algebra
Systems}{}{}{cal07_07.tex}

\hyperref{+\textbf{\quad }Approximate Integration}{}{}{cal07_08.tex}

\hyperref{+\textbf{\quad }Improper Integrals}{}{}{cal07_09.tex}

\subsubsection{Parametric Equations and Polar Coordinates}

\hyperref{+\textbf{\quad }Curves Defined by Parametric Equations}{}{}{%
cal08_01.tex}

\hyperref{+\quad Tangents and Areas}{}{}{cal08_02.tex}

\hyperref{+\textbf{\quad }Arc Length and Surface Area}{}{}{cal08_03.tex}

\hyperref{+\textbf{\quad }Polar Coordinates}{}{}{cal08_04.tex}

\hyperref{+\textbf{\quad }Areas and Lengths in Polar Coordinates}{}{}{%
cal08_05.tex}

\hyperref{+\textbf{\quad }Conic Sections}{}{}{cal08_06.tex}

\hyperref{+\textbf{\quad }Conic Sections in Polar Coordinates}{}{}{%
cal08_07.tex}

\subsubsection{Infinite Sequences and Series}

\hyperref{+\textbf{\quad }Sequences}{}{}{cal09_01.tex}

\hyperref{+\textbf{\quad }Series}{}{}{cal09_02.tex}

\hyperref{+\textbf{\quad }The Integral Test and Estimates of Sums}{}{}{%
cal09_03.tex}

\hyperref{+\textbf{\quad }The Comparison Tests}{}{}{cal09_04.tex}

\hyperref{+\textbf{\quad }Alternating Series}{}{}{cal09_05.tex}

\hyperref{+\textbf{\quad }Absolute Convergence and the Ratio and Root Tests}{%
}{}{cal09_06.tex}

\hyperref{+\textbf{\quad }Strategy for Testing Series}{}{}{cal09_07.tex}

\hyperref{+\textbf{\quad }Power Series}{}{}{cal09_08.tex}

\hyperref{+\textbf{\quad }Representations of Functions as Power Series}{}{}{%
cal09_09.tex}

\hyperref{+\textbf{\quad }Taylor and Maclaurin Series}{}{}{cal09_10.tex}

\hyperref{+\textbf{\quad }The Binomial Series}{}{}{cal09_11.tex}

\hyperref{+\textbf{\quad }Applications of Taylor Polynomials}{}{}{%
cal09_12.tex}

\subsubsection{3-D Analytic Geometry and Vectors}

\hyperref{+ \ \ 3-D Coordinate Systems}{}{}{cal10_01.tex}

\hyperref{+ \ \ Vectors}{}{}{cal10_02.tex}

\hyperref{+ \ \ The Dot Product}{}{}{cal10_03.tex}

\hyperref{+ \ \ The Cross Product}{}{}{cal10_04.tex}

\hyperref{+ \ \ Equations of Lines and Planes}{}{}{cal10_05.tex}

\hyperref{+ \ \ Quadric Surfaces}{}{}{cal10_06.tex}

\hyperref{+ \ \ Vector Functions and Space Curves}{}{}{cal10_07.tex}

\hyperref{+ \ \ Arc Length and Curvature}{}{}{cal10_08.tex}

\hyperref{+ \ \ Motion in Space: Velocity and Acceleration}{}{}{cal10_09.tex}

\hyperref{+ \ \ Cylindrical and Spherical Coordinates}{}{}{cal10_10.tex}

\subsubsection{Partial Derivatives}

\hyperref{+ \ Functions of Several Variables}{}{}{cal11_01.tex}

\hyperref{+ \ Limits and Continuity}{}{}{cal11_02.tex}

\hyperref{+ \ Partial Derivatives}{}{}{cal11_03.tex}

\hyperref{+ \ Tangent Planes and Differentials}{}{}{cal11_04.tex}

\hyperref{+ \ The Chain Rule}{}{}{cal11_05.tex}

\hyperref{+ \ Directional Derivatives and the Gradient Vector}{}{}{%
cal11_06.tex}

\hyperref{+ \ Maximum and Minimum Values}{}{}{cal11_07.tex}

\hyperref{+ \ Lagrange Multipliers}{}{}{cal11_08.tex}

\subsubsection{Multiple Integrals}

\hyperref{+ \ Double Integrals over Rectangles}{}{}{cal12_01.tex}

\hyperref{+ \ Iterated Integrals}{}{}{cal12_02.tex}

\hyperref{+ \ Double Integrals over General Regions}{}{}{cal12_03.tex}

\hyperref{+ \ Double Integrals in Polar Coordinates}{}{}{cal12_04.tex}

\hyperref{+ \ Applications of Double Integrals}{}{}{cal12_05.tex}

\hyperref{+\ \ Surface Area}{}{}{cal12_06.tex}

\hyperref{+ \ Triple Integrals}{}{}{cal12_07.tex}

\hyperref{+ \ Triple Integrals in Cylindrical and Spherical Coordinates}{}{}{%
cal12_08.tex}

\hyperref{+ \ Change of Variables in Multiple Integrals}{}{}{cal12_09.tex}

\subsubsection{Tables}

\hyperref{+ Table of Fundamental Laws and Theorems in Algebra}{}{}{%
tab_algebra_rules.tex}

\hyperref{+ Table of Algebraic Equations}{}{}{TAB_alg_equations.tex}

\hyperref{+ Table of Derivatives}{}{}{TAB_Derivatives.tex}

\hyperref{+ Table of Formulas for Length, Area, and Volume}{}{}{%
TAB_geom_formulas.tex}

\hyperref{+ Table of Integrals}{}{}{TAB_integrals.tex}

\hyperref{+ Table of Laplace Transforms}{}{}{TAB_Laplace_transforms.tex}

\hyperref{+ Table of Mathematics Symbols}{}{}{TAB_Math_symbols.tex}

\hyperref{+ Table of Trigonometric Functions and Identities}{}{}{%
TAB_trigonometry.tex}

\subsection{\ Related Reference Topics}

{\small \qquad \hyperref{SI Units}{}{}{../Physics/SIUnits.tex}}

{\small \qquad \hyperref{Algebra}{}{}{Al_contents.tex}}

{\small \qquad \hyperref{Algebraic Equations}{}{}{Tab_alg_equations.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{\qquad Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{\qquad Evaluating Calculus Expressions}{}{}{%
../../Help/DM7-1.tex#Calculus expressions}}

{\small \hyperref{\qquad Limits}{}{}{../../Help/DM7-2.tex#Limit}}

{\small \hyperref{\qquad Differentiation}{}{}{../../Help/DM7-3.tex#Derivative%
}}

{\small \hyperref{\qquad Indefinite Integration}{}{}{%
../../Help/DM7-4.tex#Indefinite integral}}

{\small \hyperref{\qquad Methods of Integration}{}{}{%
../../Help/DM7-5.tex#Methods of integration}}

{\small \hyperref{\qquad Definite Integrals}{}{}{%
../../Help/DM7-6.tex#Definite integral}}

{\small \hyperref{\qquad Sequences and Series}{}{}{%
../../Help/DM7-7.tex#Sequences}}

{\small \hyperref{\qquad Multivariable Calculus}{}{}{%
../../Help/DM7-8.tex#Multivariable calculus}}

{\small \hyperref{\qquad Sample Calculus Problems}{}{}{%
../../Help/DM7-9.tex#Exercises}}

{\small \hyperref{\qquad Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{\qquad Differential Equations}{}{}{../../Help/DM10.tex}}

\noindent \hrulefill

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\noindent \hrulefill

\CustomNote[Acknowledgement]{Margin Hint}{%
Much of the data in the calculus reference materials is an adaptation, with
permission, of information in 
\[
\begin{tabular}{l}
{\Large Calculus }Third Edition \\ 
{\Large James Stewart }McMaster Universitiy \\ 
Copyright 1995, ISBN 0 534-25158-7 \\ 
Brooks/Cole Publishing Company \\ 
An International Thomson Publishing Company
\end{tabular}
\]
}

\end{document}
