MAT - Paper 2014 - Q3
Flashcards
The trapezium rule
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.
Inequalities and algebraic long division
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?
\[q + \frac{r}{g(x)}\]