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 $$a \le x \le b$$.

Polar 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 $$r = f(\theta),\,$$ $$\alpha \le \theta \le \beta.\,$$

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

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

Euler's formula
$$ y_{r + 1} = y_r + hf( x_r, y_r )\,$$

The mid-point formula
$$ y_{r + 1} = y_{r - 1} + 2 h f( x_r, y_r )\,$$

The improved Euler formula
$$y_{r + 1} = y_r + {1 \over 2} ( k_1 + k_2 )\,$$

where

$$k_1 = h f ( x_r, y_r)\,$$

and

$$k_2 = h f ( x_r + h, y_r + k_1 ).\,$$