# MAT - Paper 2014 - Q3

> Source: https://ollybritton.com/notes/maths/problems/mat/2014/q3/ · Updated: 2021-09-18 · Tags: mat, notes, maths

## Flashcards
### 2021-09-18
##### What is the trapezium rule for estimating $T_n$, the integral of a function $f(x)$ between $a$ and $b$ and using $n$ strips??
$$
T_n \frac{\Delta X}{2}\left( f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n) \right)
$$

where

$$
\Delta X = \frac{b - a}{n}
$$

##### What is magical about $$\frac{1}{2n}\left(1 + 2b + 2b^2 + 2b^3 + ... + 2b^{n-1} + b^n\right)$$??
The middle bit is the sum of a geometric sequence.

##### Why do you have to be careful taking the reciprocal of both sides of an inequality??
It can flip the inequality.

##### How can you simplify $$\frac{b + 1}{b - 1}$$??
Use algebraic long division.

##### If you get a quotient $q$ and remainder $r$ when doing the algebraic long division $$\frac{f(x)}{g(x)}$$ what is the overall result of the division??
$$
q + \frac{r}{g(x)}
$$

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