4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.\\
Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 20.\\
(i) A function $f ( x )$ is said to be even if $f ( - x ) = f ( x )$ for all $x$. A function is said to be odd if $f ( - x ) = - f ( - x )$ for all $x$.\\
(a) What symmetry does the graph $y = f ( x )$ of an even function have? What symmetry does the graph $y = f ( x )$ of an odd function have?\\[0pt]
(b) Use these symmetries to show that the derivative of an even function is an odd function, and that the derivative of an odd function is an even function. [You should not use the chain rule.]\\
(ii) For $- 45 ^ { \circ } < \theta < 45 ^ { \circ }$, the line $L$ makes an angle $\theta$ with the line $y = x$ as drawn in the figure below. Let $A ( \theta )$ denote the area of the triangle which is bounded by the $x$-axis, the line $x + y = 1$ and the line $L$.\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-16_675_659_1363_719}\\
(a) Let $0 < \theta < 45 ^ { \circ }$. Arguing geometrically, explain why

$$A ( \theta ) + A ( - \theta ) = \frac { 1 } { 2 } .$$

(b) For $0 < \theta < 45 ^ { \circ }$, determine a formula for $A ( \theta )$.\\
(c) Sketch the graph of $A ( \theta )$ against $\theta$ for $- 45 ^ { \circ } < \theta < 45 ^ { \circ }$.\\
(d) In light of the identity in part (ii)(a), what symmetry does the graph of $A ( \theta )$ have?\\
(e) Without explicitly differentiating, explain why $\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} \theta ^ { 2 } } = 0$ when $\theta = 0$.



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