A-level Mathematics/AQA/MPC2

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Transformations of functions[edit]

Sequences and series[edit]

Notation[edit]

 u_n \,\! — the general term of a sequence; the nth term


 a \,\! — the first term of a sequence


 l \,\! — the last term of a sequence


 d \,\! — the common difference of an arithmetic progression


 r \,\! — the common ratio of a geometric progression


 S_n \,\! — the sum to n terms:  S_n = u_1 + u_2 + u_3 + \ldots + u_n \,\!


 \sum \,\! — the sum of


 \infty \,\! — infinity (which is a concept, not a number)


 n \rightarrow \infty \,\!n tends towards infinity (n gets bigger and bigger)


 |x| \,\! — the modulus of x (the value of x, ignoring any minus signs)

Convergent, divergent and periodic sequences[edit]

Convergent sequences[edit]

A sequence is convergent if its nth term gets closer to a finite number, L, as n approaches infinity. L is called the limit of the sequence:

 \mbox{As } n \to \infty \mbox{, } u_n \to L \,\!

Another way of denoting the same thing is:

 \lim_{n \to \infty}u_n = L \,\!

Definition of the limit of a convergent sequence[edit]

Generally, the limit  L \,\! of a sequence defined by  u_{n+1} = f(u_n) \,\! is given by  L = f(L) \,\!

Divergent sequences[edit]

Sequences that do not tend to a limit as n increases are described as divergent. eg: 1, -1 , 1 -1

Periodic sequences[edit]

Sequences that move through a regular cycle (oscillate) are described as periodic.

Series[edit]

A series is the sum of the terms of a sequence. Those series with a countable number of terms are called finite series and those with an infinite number of terms are called infinite series.

Arithmetic progressions[edit]

An arithmetic progression, or AP, is a sequence in which the difference between any two consecutive terms is a constant called the common difference. To get from one term to the next, you simply add the common difference:

 u_{n+1} = u_n + d \,\!

Expression for the nth term of an AP[edit]

 u_n = a + (n-1)d \,\!

Formulae for the sum of the first n terms of an AP[edit]

The sum of an arithmetic progression is called an arithmetic series.

 S_n = \frac{n}{2} \left \lbrack 2a + (n-1)d \right \rbrack \,\!


 S_n = \frac{n}{2} (a+l) \,\!

Formulae for the sum of the first n natural numbers[edit]

The natural numbers are the positive integers, i.e. 1, 2, 3…

 S_n = \frac{n}{2} (n+1) \,\!

Geometric progressions[edit]

An geometric progression, or GP, is a sequence in which the ratio between any two consecutive terms is a constant called the common ratio. To get from one term to the next, you simply multiply by the common ratio:

 u_{n+1} = ru_n \,\!

Expression for the nth term of an GP[edit]

 u_n = ar^{n-1} \,\!

Formula for the sum of the first n terms of a GP[edit]

 S_n = a \left ( \frac{1-r^n}{1-r} \right ) \,\!


 S_n = a \left ( \frac{r^n-1}{r-1} \right ) \,\!

Formula for the sum to infinity of a GP[edit]

 S_\infty = \sum_{n=1}^\infty ar^{n-1} = \frac{a}{1-r} \qquad \mbox{where } -1 < r < 1 \,\!

Binomial theorem[edit]

The binomial theorem is a formula that provides a quick and effective method for expanding powers of sums, which have the general form (a+b)^n.

Binomial coefficients[edit]

The general expression for the coefficient of the (r+1)^{th} term in the expansion of (1+x)^n is:

{}^n\!C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

where n! = 1 \times 2 \times 3 \times \ldots \times n

n! is called n factorial. By definition, 0!=1.

Binomial expansion of (1+x)n[edit]

(1+x)^n=1+\binom{n}{1}x+\binom{n}{2}x^2+\binom{n}{3}x^3+\ldots+x^n


(1+x)^n=1+nx+\frac{n(n-1)}{2!}+\frac{n(n-1)(n-2)}{3!}+\ldots+x^n


(1+x)^n=\sum_{r=0}^n\binom{n}{r}x^r

Trigonometry[edit]

Arc length[edit]

l = r \theta \,\!

Sector area[edit]

A = \tfrac{1}{2} r^2 \theta

Trigonometric identities[edit]

\tan{\theta} \equiv \frac{\sin{\theta}}{\cos{\theta}}


\sin^2{\theta} + \cos^2{\theta} \equiv 1 \,\!

Indices and logarithms[edit]

Laws of indices[edit]

 x^m \times x^n = x^{m+n} \,\!


 x^m \div x^n = x^{m-n} \,\!


 \left ( x^m \right )^n = x^{mn} \,\!


 x^0 = 1 \,\! (for x ≠ 0)


 x^{-m} = \frac{1}{x^m} \,\!


 x^{\frac{1}{n}} = \sqrt[n]{x} \,\!


 x^{\frac{m}{n}} = \sqrt[n]{x^m} \,\!

Logarithms[edit]

10^2 = 100 \Leftrightarrow \log_{10}{100} = 2


10^3 = 1000 \Leftrightarrow \log_{10}{1000} = 3


2^5 = 32 \Leftrightarrow \log_{2}{32} = 5


\log_a{b} = c \Leftrightarrow a^c = b

Laws of logarithms[edit]

The sum of the logs is the log of the product.

\log{x} + \log{y} = \log{xy} \,\!

The difference of the logs is the log of the quotient.

\log{x} - \log{y} = \log{\left ( \frac{x}{y} \right )}

The index comes out of the log of the power.

k\log{x} = \log{\left ( x^k \right )}

Differentiation[edit]

Differentiating the sum or difference of two functions[edit]

 y = f(x) \pm g(x) \quad \therefore \quad \frac{dy}{dx} = f'(x) \pm g'(x)

Integration[edit]

Integrating axn[edit]

 \int ax^n \, dx = \frac{ ax^{n+1} }{ n+1 } + c \qquad \mbox{ for } n \neq -1 \,\!

Area under a curve[edit]

The area under the curve  y = f(x) between the limits  x = a and  x = b is given by

 A = \int_a^b y \, dx