We consider the trapezoidal rule on $I = [a,b]$ with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$, where $f$ is of class $\mathcal{C}^2$.
Deduce the error bound $$\left|e_n(f)\right| \leqslant \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} |f''(x)|.$$

It is given that $f ( x ) = - 2 x ^ { 2 } + 10$
Consider the following three curves:
(1) $y = f ( x )$
(2) $y = f ( x + 1 )$
(3) the curve $y = f ( x + 1 )$ reflected in the line $y = 6$
The trapezium rule is used to estimate the area under each of these three curves between $x = 0$ and $x = 1$.
State whether the trapezium rule gives an overestimate or underestimate for each of these areas.

	(1)	(2)	(3)
A	underestimate	underestimate	underestimate
B	underestimate	underestimate	overestimate
C	underestimate	overestimate	underestimate
D	underestimate	overestimate	overestimate
E	overestimate	underestimate	underestimate
F	overestimate	underestimate	overestimate
G	overestimate	overestimate	underestimate
H	overestimate	overestimate	overestimate

The function f is such that $0 < f(x) < 1$ for $0 \leq x \leq 1$. The trapezium rule with $n$ equal intervals is used to estimate $\int_0^1 f(x) \, dx$ and produces an underestimate.
Using the same number of equal intervals, for which one of the following does the trapezium rule produce an overestimate?

A student approximates the integral $\int _ { a } ^ { b } \sin ^ { 2 } x \mathrm {~d} x$ using the trapezium rule with 4 strips. The resulting approximation is an overestimate.
Which of the following is/are necessarily true?
I If the student approximates $\int _ { - b } ^ { - a } \sin ^ { 2 } x \mathrm {~d} x$ in the same way, the result will be an overestimate.
II If the student approximates $\int _ { a } ^ { b } \cos ^ { 2 } x \mathrm {~d} x$ in the same way, the result will be an underestimate.

Which one of $\mathbf { A } - \mathbf { F }$ correctly completes the following statement? Given that $a < b$, and $\mathrm { f } ( x ) > 0$ for all $x$ with $a < x < b$, the trapezium rule produces an overestimate for $\int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x \ldots$
A ... if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$
B ... only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$
C ... if and only if $\mathrm { f } ^ { \prime } ( x ) > 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) < 0$ for all $x$ with $a < x < b$
D ... if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$
E $\ldots$ only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$ F ... if and only if $\mathrm { f } ^ { \prime } ( x ) < 0$ and $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ for all $x$ with $a < x < b$