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\begin{document}

\begin{center}{CALCULUS ANSWERS}

\end{center}
~\\
\begin{itemize}
\item[A1(a)]
$
(i)\; (\cos x + 3x^{2} \sin x)e^{x^{3}}\;\;\;\;\;\;
(ii)\; (x^{3} \cos x + 3x^{2} \sin x) e^{x^{3}\sin x}\;\;\;\;\;\;
(iii)\;  \frac{-\tanh (1/x)}{x^{2}}
$
\item[(b)]
$
(i)\; - \frac{1}{\sqrt{1-x^{2}}}\;\;\;\;\;\;
(ii\;) \frac{1}{(1 + 2 x)}
$
\item[(c)]
$
(i)\; x^{\cos x} (\frac{\cos x}{x} - \sin x \ln x)\;\;\;\;\;\;
(ii)\; \frac{2}{x \ln 10}
$
\item[(d)]
$
(i)\; \frac{2x - (y^{2}/x) e^{y \ln x}}{(y ln x + 1)e^{y ln x}-2y}
$
\item[(e)]
$
\frac{dy}{dx} = \mbox{coth} \theta;\;\;\;\;\;\; 
\frac{d^{2}y}{dx^{2}} = -\mbox{cosech}^{3} \theta$

\item[A6]
$
\frac{d^{8}y}{dx^{8}} = x^{2} \sin x - 16 x \cos -56 \sin x
$
\item[B1(a)]
$
(i)\; \frac{-1}{(1 + 2 ax + x^{2})^{1/2}} + C\;\;\;\;\;\;
(ii)\; e-1\;\;\;\;\;\;
(iii)\; 2/3\;\;\;\;\;\;
(iv)\; 4
$
\item[(b)]
$
(i)\; \sin^{-1} (\frac{x - 1}{2}) + C\;\;\;\;\;\;\;
(ii)\; \pi/4
$
\item[(c)]
$
(i)\; \ln (\frac{x}{\sqrt{1 + x^{2}}}) + C
$
\item[(d)]
$
(i)\; \sin x - x \cos x + C\;\;\;\;\;\;
(ii)\; x \ln x - x + C
$
\item[(e)]
$ 
1
$
\item[B3]
$
(a)\; \sinh 1\;\;\;\;\;\;
(b)\; \pi/2\;\;\;\;\;\;
(c)\; S.A. = 4\pi R^{2},\;\;\; V = \frac{4}{3}\pi R^{3}
$
\item[B4]
$
(a)\; 32/3\;\;\;\;\;\; (b)\; 64/3\;\;\;\;\;\; (c)\;(i)\; 1 \;\;\;\;\;\; (ii)\; 1
$
\item[C2]
$
(a)(i)\; 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + ...\;\;\;\;
(ii)\;  1 + \frac{x}{2} - \frac{x^{2}}{8} + \frac{x^{3}}{16} - ... \;\;\;\;
(iii)\; x - \frac{x^{3}}{3} + ....
$
\item[(b)] 0.515038072; error in last digit

\item[C4] 0.9461

\item[C6(a)]  (i) 1 (ii) 1 (iii) 0
\item[(c)] (i) 0 (ii) 0 (iii) -1
\item[(d)] (i) maximum (ii) $\infty$

\item[D2(a)]
$
(i)\; \frac{x}{(x^{2} + y^{2})^{1/2}}\;\;\;\;\;\;
(ii)\; - \frac{y}{x^{2} + y^{2}}\;\;\;\;\;\;
(iii)\; y^{x} \ln y
$
\item[D3] 9$\%$

\item[D4]
$ (a)\; na \cos^{n-1} at \sin^{n-1} at (\cos^{2} at - \sin^{2} at)\;\;\;\;\;\;
(b)\; x + 2 x \ln x - \frac{1}{x(\ln x)^{2}}
$
\item[D5]
$\frac{\partial w}{\partial r} = -2(x^{2} + y^{2})^{\frac{1}{2}}
e^{-x^2-y^2}, \;\;\;\;\frac{\partial w}{\partial \theta} = 0$

\item[D7]
(a) $(\frac{\partial z}{\partial x})_{y} = (\frac{\partial z}{\partial u})_{v} 2x + (\frac{\partial z}{\partial v})_{u} 2y$

(b) $(\frac{\partial z}{\partial u})_{v} = \frac{1}{2(x^{2}-y^{2})} 
\{x(\frac{\partial z}{\partial x})_{y} - y(\frac{\partial z}{\partial y})_{x}\}$

(c) $(\frac{\partial z}{\partial u})_{v} - (\frac{\partial z}{\partial v})_{u} = \frac{1}{2(x-y)} \{(\frac{\partial z}{\partial x})_{y} - (\frac{\partial z}{\partial y})_{x}\}$

\item[D8]
$f(x,y) = e^{6} + 3e^{6} (x-2) + 2e^{6}(y-3) + \frac{9e}{2}^{6}(x-2)^{2} + 6 e^{6} (x-2)(y-3) + 2 e^{6} (y-3)^{2}$

\item[D9]
(i) $x = y= 0$ (minimum)

(ii) $x = y = 0$ (maximum);  $\;\;\;x = y = 1/3$ (saddle)

(iii) $x = y = \pi/3$ (maximum)

\item[D10]
(i) exact $f = xy$

(ii) not exact

(iii) exact $f=\frac{1}{2} (x^{2} + y^{2} + z^{2})$

(b) 0

\item[E5]
(i) $- \pi$ (ii) oscillates (iii) $\sqrt{2}/4$
(iv) 0 (v) 1

\item[E6]
$y(x) = -3x + 4x^{3}$

\end{itemize}				 					
\end{document}


