MAT - Paper 2014 - Q3
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?
\[q + \frac{r}{g(x)}\]