Maths Formula Sheet

Fact card

18/9/06

11:16 am

y

Page 1

eg tiv osi

ie rad

nt

(x2, y2)

P

y = f(x)

dy = f’(x) dx

k, constant

0

x

1

y

y ve

siti Po

2x

x3

3x2

xn, any constant n

nxn–1

ex

ex = y

e kx

ke kx = ky

e f (x)

f’(x)e x

ln x

1/x

c

nt die

(x2, y2)

gra

Ne

m=

(x1, y1)

x

Ney = e x gat ive g

x –1 = 1/x ex e

kx

nt

x

1

x

x

1

(1)

1

x

if a is positive

(2)

–1

(3)

1

x

Graph of y = e –x showing exponential decay x –b / 2a

Quadratic functions y = ax2 + bx + c 1

(1) (2)

if a is positive

–1

–b / 2a

(3) x

1

–b / 2a

if a is negative

x

(1) b2 – 4ac < 0;

a

x

–b / 2a

TC = a + bq – cq2 + dq3

(3) (2) (1)

ln x + c a

(3) ba2=–fixed 4ac cost >0

y = a/x = ax–1

y

Example: Unit price elasticity of demand

q = a/p = ap–1

a = fixed cost Output (q )

y

PRINCIPLES AND FORMULAE

Inverse functionsOutput (q )

Total cost functions

if a is negative

(2)

Maths for Economics

(1)

(2) b2 – 4ac = 0;

TC

x (3)

TC

ex + c

Supporting economics in higher education

(x2, y2)

die

1

= e x showing Graph of y –x y=e exponential growth

x3 +c 3

e kx +c k

gra

y 2 – y1 x 2 – x1

Economics Network

(x2, y2)

ent

–1

kx + c

x n+1 +c n+1

x

y = e–x rad i

1

x +c 2

x n, (n = –1)

ve

y Exponential functions y – y1 m= 2 –1 e ≈ 2.1783 is the exponential x 2 – x1 constant (x1, y1) c

2

2

yg=atei x

y 2 – y1 x 2 – x1

Integration

x

c

m=

(x1, y1)

x

ln kx = logdukx dv d k/x d du dv (u( x) ± v( x )) = e ± (u( x) ± v( x )) = ± dx dx dx dx f’(x)/x dx dx ln f(x) d du dv (u( x) ± v( x )) = ± dx dx dx d d u d v d f d d f d (u( x) ± v((xk)) =f ( x )) ±rule =k The sum–difference Constant ( k f ( xmultiples )) = k dx dx dx dx dx dx dx df d du dv d ( k f ( x )) = k (u( x) ± v( x )) = ± d x d x dx d x d x df d dv du dfor k constant ( k f ( x ))d =(uv k ) = u dv + v du (uv) = u +v dx dx dx dx dx dx dx dx df v d du d Thedproduct The quotient rule ( k f ( x ))rule =k v uv u ( ) = + dx dxdu x dvdx d dx d v d u d dv dv u – u v –u (uv) = u d  u+ v d  u dx dx dx dx  = dxdx 2 dx  = du 2 dv dx dvv dx  v  du v d v v–u (uv) = u +v d  u dx dx dx The chain rule  = dudx dvdx dx  v  v –u v2 d  u dy dy . du dx dx dy dy . du If y = y(=u), where u( x ), then =), where u = u( x ), then If y = y(u = 2 ud= du v dx  v  v d x d u d x dx du dx v –u d  u dy dy . du d x d x If y = y(u), where u = u( x ), then =  = dx  v  v2 dx du dx dy dy . du If y = y(u), where u = u( x ), then = dx du dx d y dy . du f ( x ) dx If y = y(u), where u = uf((xx)), then = ∫ dx du dx k, (any) constant c

1

y = mx + c; m = gradient; c = vertical intercept

Linear

1

x2

y 2 – y1

c (x , y ) m= Graphs of Common Functions x 2 – x1

Differentiation

Example: Unit price

This leaflet has been produced in conjunction with mathcentre

x

www.mathcentre.ac.uk

Fact card

18/9/06

11:16 am

Page 2

Arithmetic

Algebra

When multiplying or dividing positive and negative numbers, the sign of the result is given by:

Removing brackets

+ and + gives + – and + gives – + and – gives – – and – gives +

(a + b)(c + d) = ac + ad + bc + bd

e.g. 6 x 3 = 18; e.g. (–6) x 3 = –18 e.g. 6 x (–3) = –18 e.g. (–6) x (–3) = 18

21 ÷ 7 = 3 (–21) ÷ 7 = –3 21 ÷ (–7) = –3 (–21) ÷ (–7) = 3

Order of calculation First: Second: Third:

brackets x and ÷ + and –

Fractions Fraction =

numerator denominator

Adding and subtracting fractions To add or subtract two fractions, first rewrite each fraction so that they have the same denominator. Then, the numerators are added or subtracted as appropriate and the result is divided by the common denominator: e.g. 4 3 16 15 31 + = + = 5 4 20 20 20 15 3 5 fractions Multiplying = 4 3 716 1115 77 31 To + multiply = + two = fractions, multiply their numerators and 5 4 20 20 20

then multiply their denominators: e.g. 3 2 3 3 9 = 35 ÷ 53 = 15 5 2 10 4 3 =16 15 31 7+ 11= 77+ = 5 4 20 20 20

Dividing 3 2 3 fractions 3 9 ÷

=3

5

= 15

5 divide 3 7 5 11 2= 77 10 To two fractions, invert the second and then multiply: e.g. 3 2 3 ÷ = 5 3 5

3 9 = 2 10

Series (e.g. for discounting) 1 + x + x2 + x3 + x4 + …

= 1/(1–x )

1 + x + x2 + x3 + … + xk

= (1–x k+1 ) /(1–x)

(where 0 < x < 1 )

a(b + c) = ab + ac (a + b)2 = a2 + b2 + 2ab;

a(b – c) = ab – ac (a - b)2 = a2 + b2 – 2ab

Sigma Notation The Greek capital letter sigma ∑ is used as an abbreviation for a sum. Suppose we have x1 + xn2values: . . . xn x1, x2, ... xn and we wish to add them together. The sum n x + x . . . xn x1 + 1x2 x. .2+ . xn is written ∑ x i 1 x2 . . . xn

(a + b)(a – b) = a2 – b2

n

i =1

n

x ∑ + xx2ni . through . . xn all integers (whole numbers) from Note that ∑ 1xii=iruns 1x ∑+ x . . . 3x i2 1 n 1 for instance Formula for solving a quadratic equation 1 to n. i =So, x ∑ i means x1 + x 2 + x 3 ni =1 3 x i n i =1 2 ∑ – b ± b – 4 ac 3 ∑ x means +xxi 2 + x 3 3i x +i =x1 x.∑ If ax2 + bx + c = 0, then x = x12 +i =.1x.125xx+ x 3x + x ∑ xi =i 1means 1 means n + 2a x ∑ 1i 2 means 2 3 12 + 2 2 + 3 2 + 4 2 + 5 2 i i =1 ∑ i =1 3 Laws of indices n Example i = 1 5 2 ∑ x i 3means x i x21 +3x22++4x23+ 5 2 5 ∑2 + =1 ∑2 x1i means ∑ am 2 i 5 imeans 2 2 + m n mn m n m+n m– n 2 3 2 i = 1 3 22+2x+ 41 2+n3+2x+ 5 24 2x+ 2 i1 ∑ ii =1means (a ) = a a a =a =a =1 + 2 1+2 + ∑ i means x i x15+ x 2 + . . . + x n n ∑ i = 1 i = 1 a 5i =1 = x= 2 53 2 2 2 2 2 n nmeans 1 + 2 + 3 n+ 4 + 5 i ∑ xx2i i means + .x.22. +inxxnstatistical 1 + x x 21 used 2 2 2calculations. The m ∑ ∑ This notation is often n 3 –m 1/ n n m/ n n 0 i = 1 1 2 i= 1 + 2 + 3 + 4 + 5 x=i1 i n xmeans + + . . . + x x =i1=1i∑ = x∑ a = m a = a a = a a =1 2x + x +n. . . + x =i =1n1x iquantities, x =mean of the 1∑ =n 2 xn is n 2 a = n1 n 2∑xn1, x(2x, ...n–and xn i= n ni =1 i x ) = ∑ i =1 x i – x 2 x 5 ∑ n x var( = ) + + . . . + x x x i i =1 n 21 22 2 n2n 2 Laws of logarithms n x∑=i 2n means x1i) 2+ 2xn 1∑++nn3x 22 + + 42. . . + x52n x ii =–= 12 x n1∑ ix=1=n(∑ =1 nx i ix = i = y – ( ) x x 2 x n 2 = = – var( ) x ∑ ∑ y = log b x means b = x and b is called the base i n x2 = xi =)1= ∑i i =n1 n( x i =– x )i =1= ∑ var( x ) var( n i =–1 x i – x 2 n n The variance is e.g. log 10 2 = 0.3010 means 100.3010 = 2.000 to 4 sig figures n nx 2x ) 2 nn x =n. . .var( ) +(x )∑ + +x ∑∑ 1 x(nxi i – xx1sd = ( x i – x=2) 2 i =1∑ ni n –x ix2 2 2 var( x )x== i i==1∑ Logarithms to base e, denoted loge, or alternatively ln, 1 = i n n n n x ) =var(x ) = i =1 –x sdvar( (x ) = are called natural logarithms. The letter e stands for the n n x sd(x ) =sd(var( ) x x ) = var( ) exponential constant, which is approximately 2.7183. i.e. the mean squares ) 2 ∑ niminus x 2 the square of the mean ∑ n of( xthe i –x x )(= = =1 i – x 2 var(sd x ) =i =1var(x ) A n var(x ) n ln AB = ln A + ln B ; ln = ln A – ln B ; ln A n = n ln A sd(x ) =deviation The standard (sd) is the square root of the B variance:

Proportion and Percentage To convert a fraction into a percentage, multiply by 100 5 and express result asisa 5percentage. An example is: as a the percentage 100 = 62.5% 8 8 5 5 as a percentage is 100 = 62.5% 8 8 Some common conversions are 1 1 1 3 = 10% = 25% = 50% = 75% 10 4 2 4 1 1 1 3 = 10% = 25%an alternative = 50% way = 75% 10Ratios are simply 4 2 4 of expressing fractions. Consider dividing £200 between two people in the ratio of 3:2. This means that for every £3 the first person gets, the second person gets £2. So the first gets 3/5 of the total (i.e. £120) and the second gets 2/5 (i.e. £80).

Generally, to split a quantity in the ratio m : n, the quantity is divided into m/(m + n) and n/(m + n) of the total.

sd(x ) = var(x ) Note that the standard deviation is measured in the same units as x.

The Greek Alphabet Α α alpha

Ι

Β β

beta

Γ γ

ι

ρ

iota

Ρ

Κ κ

kappa

Σ σ sigma

gamma

Λ λ

lambda

Τ τ

∆ δ

delta

Μ µ mu

Υ υ upsilon

Ε ε

epsilon

Ν ν

nu

Φ φ

phi

Ζ ζ

zeta

Ξ ξ

xi

Χ χ

chi

Η η eta

Ο ο

omicron

Ψ ψ psi

Θ θ

Π π

pi

Ω ω omega

theta

rho tau