A-level Mathematics/AQA/MFP3

Series and limits

Two important limits:

$\lim _{x\rightarrow \infty }\left(x^{k}e^{-x}\right)\rightarrow 0$ for any real number k

$\lim _{x\rightarrow 0}\left(x^{k}\ln {x}\right)\rightarrow 0$ for all k > 0

The basic series expansions

$(r=0,1,2,\cdots )$

$e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots +{x^{r} \over r!}+\cdots$

$\sin x=x-{x^{3} \over 3!}+{x^{5} \over 5!}-\cdots +\left(-1\right)^{r}{x^{2r+1} \over (2r+1)!}+\cdots$

$\cos x=1-{x^{2} \over 2!}+{x^{4} \over 4!}-\cdots +\left(-1\right)^{r+1}{x^{2r} \over (2r)!}+\cdots$

$(1+x)^{n}=1+nx+{n(n-1) \over 2!}x^{2}+\cdots +\;{\ n \choose r}\;x^{r}+\cdots$

$(r=1,2,3,\cdots )$

$\ln(1+x)=x-{x^{2} \over 2}+{x^{3} \over 3}-\cdots +(-1)^{r+1}{x^{r} \over r}+\cdots$

Improper intergrals

The integral :$\int _{a}^{b}f(x)\,dx\,$ is said to be improper if

1. the interval of integration is infinite, or;
2. f(x) is not defined at one or both of the end points x=a and x=b, or;
3. f(x) is not defined at one or more interior points of the interval $a\leq x\leq b$.

Polar coordinates

A diagram illustrating the relationship between polar and Cartesian coordinates.

$x=r\cos \theta ,\,$

$y=r\sin \theta ,\,$

$r^{2}=x^{2}+y^{2},\,$

$\tan \theta ={y \over x}$

The area bounded by a polar curve

For the curve $r=f(\theta ),\,$ $\alpha \leq \theta \leq \beta .\,$

$A=\int _{\alpha }^{\beta }{1 \over 2}r^{2}d\theta \,$

r must be defined and be non-negative throughout the interval $\alpha \leq \theta \leq \beta .\,$

Numerical methods for the solution of first order differential equations

Euler's formula

$y_{r+1}=y_{r}+hf(x_{r},y_{r})\,$

The mid-point formula

$y_{r+1}=y_{r-1}+2hf(x_{r},y_{r})\,$

The improved Eular formula

$y_{r+1}=y_{r}+{1 \over 2}(k_{1}+k_{2})\,$

where

$k_{1}=hf(x_{r},y_{r})\,$

and

$k_{2}=hf(x_{r}+h,y_{r}+k_{1}).\,$

Second order differential equations

Euler's identity

$e^{ix}=\cos x+i\sin x,\,x\in \mathbb {R} \,$

When $x=\pi ,\,$ substituting into the identity gives

 $e^{i\pi }=-1\,$

Further reading

The AQA's free textbook [1]