Computer Science and Computer Science \& Philosophy applicants should turn to page 14. The function $f ( x )$ is defined for all real numbers and has the following properties, valid for all $x$ and $y$ : (A) $\quad f ( x + y ) = f ( x ) f ( y )$. (B) $\quad \mathrm { d } f / \mathrm { d } x = f ( x )$. (C) $\quad f ( x ) > 0$. Throughout this question, these should be the only properties of $f$ that you use; no marks will be awarded for any use of the exponential function. Let $a = f ( 1 )$. (i) Show that $f ( 0 ) = 1$. (ii) Let $$I = \int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x$$ Show that $I = a - 1$. (iii) The trapezium rule with $n$ steps is used to produce an estimate $I _ { n }$ for the integral $I$. Show that $$I _ { n } = \frac { 1 } { 2 n } \left( \frac { b + 1 } { b - 1 } \right) ( a - 1 )$$ where $b = f ( 1 / n )$. (iv) Given that $I _ { n } \geqslant I$ for all $n$, show that $$a \leqslant \left( 1 + \frac { 2 } { 2 n - 1 } \right) ^ { n }$$
(i) [2 marks] Setting $x = y = 0$ in (A) we get
\section*{3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.}
Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
The function $f ( x )$ is defined for all real numbers and has the following properties, valid for all $x$ and $y$ :\\
(A) $\quad f ( x + y ) = f ( x ) f ( y )$.\\
(B) $\quad \mathrm { d } f / \mathrm { d } x = f ( x )$.\\
(C) $\quad f ( x ) > 0$.
Throughout this question, these should be the only properties of $f$ that you use; no marks will be awarded for any use of the exponential function.
Let $a = f ( 1 )$.\\
(i) Show that $f ( 0 ) = 1$.\\
(ii) Let
$$I = \int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x$$
Show that $I = a - 1$.\\
(iii) The trapezium rule with $n$ steps is used to produce an estimate $I _ { n }$ for the integral $I$. Show that
$$I _ { n } = \frac { 1 } { 2 n } \left( \frac { b + 1 } { b - 1 } \right) ( a - 1 )$$
where $b = f ( 1 / n )$.\\
(iv) Given that $I _ { n } \geqslant I$ for all $n$, show that
$$a \leqslant \left( 1 + \frac { 2 } { 2 n - 1 } \right) ^ { n }$$