A-level Mathematics/AQA/MFP3

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Series and limits[edit]

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[edit]

( 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[edit]

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[edit]

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[edit]

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. \,

Numerical methods for the solution of first order differential equations[edit]

Euler's formula[edit]

 y_{r + 1} = y_r + hf( x_r, y_r )\,

The mid-point formula[edit]

 y_{r + 1} = y_{r - 1} + 2 h f( x_r , y_r )\,

The improved Eular formula[edit]

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 ).\,

Second order differential equations[edit]

Euler's identity[edit]

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

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

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

Further reading[edit]

The AQA's free textbook [1]