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\center{\bf{FIRST YEAR CALCULUS: TOPICS TO BE COVERED}}

\center{Julia Yeomans}

\begin{itemize}
\item[{\bf{A.}}] \begin{center}{\bf{DIFFERENTIATION}}\end{center}

\item[{\bf{A1.}}] Standard Forms

\item[{\bf{A2.}}] Derviatives from first principles

\item[{\bf{A3.}}] Useful rules

\item[{\bf{A4.}}] Chain rule

\item[{\bf{A5.}}] Changing variables in differential equations

\item[{\bf{A6.}}] Implicit differentiation

\item[{\bf{A7.}}] Powers and inverse functions

\item[{\bf{A8.}}] Parametric differentiation

\item[{\bf{A9.}}] Leibnitz theorem

\begin{center}\item[{\bf{B.}}] {\bf{INTEGRATION}}\end{center}

\item[{\bf{B1.}}] Standard forms

\item[{\bf{B2.}}] Integration by inspection

\item[{\bf{B3.}}] Integration by change of variable

\item[{\bf{B4.}}] Integration by partial fractions

\item[{\bf{B5.}}] Integration by parts

\item[{\bf{B6.}}] Integration by reduction forumulae

\item[{\bf{B7.}}] Interpretation of an integral

\item[{\bf{B8.}}] Properties of definite integrals

\item[{\bf{B9.}}] Applications of integration

\item[{\bf{B10.}}] Line integrals

\begin{center}\item[{\bf{C.}}] {\bf{SERIES AND LIMITS}}\end{center}

\item[{\bf{C1.}}] {{Introduction and Notation}}

\item[{\bf{C2.}}] {{Taylor and Maclaurin Series}}

\item[(a)] Taylor series 
%notation 1
%\item[(b)] Taylor series notation 2
%\item [(c)] Proof
\item [(b)] Maclaurin series
%\item [(e)] eg 1: expand $\sin x$ about $x = 0$
%\item [(f)] eg 2: expand $\sqrt{x}$ about $x = 1$
\item [(c)] Common series expansions
\item [(d)]  Manipulation of Series
%eg 3:  (manipulation of series)  
%	expand $ \mbox{tanh}\; x$ about $x = 0$

\item[{\bf{C3.}}] {{Limits}}

\item[(a)] definition of a limit
\item [(b)] continuous, discontinuous and differentiable functions
\item[(c)] finding limits


\begin{center}\item[D.]{\bf{CALCULUS OF FUNCTIONS OF MORE THAN ONE VARIABLE}}\\

\end{center}

\item[{\bf{D1.}}] Co-ordinate systems
\item[{\bf{D2.}}] Graphical representation
\item[{\bf{D3.}}] Partial derivatives: introduction and notation
\\
\item[(a)] definition and notation
%\item[(b)] an e.g. from first principles
%\item[(c)] another e.g.
\item[(b)] higher order derivatives $\partial_{xy} f = \partial_{yx} f$
\item[(c)] geometrical interpretation
\item[(d)] Taylor expansion
\\
\item[{\bf{D4.}}] Total derivatives
\item[(a)] total differential
\item[(b)] small changes
\item[(c)] chain rule
%\item[(c)] $u = u(x,y);\;\; x = x(t),\;\; y=y(t);\;\; \frac{du}{dt}$?
%\item[(d)] $u = u(x,y);\;\; y = y(x),\;\; \frac{du}{dx}$?
%\item[(e)] $u = u(x,y);\;\; x = x(s,t),\;\; y = y(s,t);\;\; (\frac{du}{ds})_{t}$? (chain rule)
%\item[(f)] $u = u(x,y);\;\; y = y(x,z);\;\; (\frac{du}{dz})_{x}?\;\; (\frac{du}{dx})_{z}?$
\item[(d)] implicit differentiation
\\
\item[{\bf{D5.}}] Changing variables
\item[{\bf{D6.}}] Exact derivatives
\item[{\bf{D7.}}] Maxima, minima and saddle points

\end{itemize}
				 					
\end{document}


