Arithmetic series General (kth) term, last (nth) term, l = Sum to n terms,

W

or

uk = a + (k – 1)d un = a + (n – l)d Sn = 1–2 n(a + l) = 1–2 n[2a + (n – 1)d]

Geometric series General (kth) term, Sum to n terms, Sum to infinity

ld

uk = a r k–1 a(1 – r n) a(r n – 1) Sn = –––––––– = –––––––– 1–r r–1

Of

()

M

()

at

() ( ) (

2

h) e m

(x – a)2 (x – a)rf(r)(a) = f(a) + (x – a)f'(a) + –––––– f"(a) + ... + –––––––––– + ... r! 2!

f(a + x)

x2 xr = f(a) + xf'(a) + –– f"(a) + ... + –– f(r)(a) + ... 2! r!

ln(1 + x)

x2 x3 xr = x – –– + –– – ... + (–1)r+1 –– + ... , – 1 < x < 1 2 3 r

sin x

x3 x5 x 2r+1 = x – –– + –– – ... + (–1)r –––––––– + ... , all x 3! 5! (2r + 1)!

cos x

x2 x4 x 2r = 1 – –– + –– – ... + (–1)r –––– + ... , all x 2! 4! (2r)!

arctan x

x3 x5 x 2r+1 = x – –– + –– – ... + (–1)r –––––– + ... , – 1 < x < 1 3 5 2r + 1

at

sinh x

General case n(n – 1) 2 n(n – 1) ... (n – r + 1) ––––––––––––––––– xr + ... , |x| < 1, (1 + x)n = 1 + nx + ––––––– x + ... + 2! 1.2 ... r n∈R Logarithms and exponentials exln a = ax

f(x)

logbx loga x = ––––– logba

Numerical solution of equations f(xn) Newton-Raphson iterative formula for solving f(x) = 0, xn+1 = xn – –––– f'(xn) Complex Numbers {r(cos θ + j sin θ)}n = r n(cos nθ + j sin nθ) ejθ = cos θ + j sin θ 2πk j) for k = 0, 1, 2, ..., n – 1 The roots of zn = 1 are given by z = exp( –––– n Finite series n n 1 1 ∑ r2 = – n(n + 1)(2n + 1) ∑ r3 = – n2(n + 1)2 4 6 r=1 r=1

ics

ALGEBRA

n n n (a + b)n = an + 1 an – 1 b + 2 an –2 b2 + ... + r an–r br + ... bn , n ∈ N where n n n n+1 n! n + r+1 = r+1 r = Cr = –––––––– r r!(n – r)!

()

x2 xr = f(0) + xf'(0) + –– f"(0) + ... + –– f (r)(0) + ... 2! r!

x2 xr ex = exp(x) = 1 + x + –– + ... + –– + ... , all x 2! r!

a ,–1
Binomial expansions When n is a positive integer

()

Infinite series f(x)

x3 x5 x 2r+1 = x + –– + –– + ... + –––––––– + ... , all x 3! 5! (2r + 1)!

cosh x

x2 x4 x 2r = 1 + –– + –– + ... + –––– + ... , all x 2! 4! (2r)!

artanh x

x3 x5 x 2r+1 = x + –– + –– + ... + –––––––– + ... , – 1 < x < 1 3 5 (2r + 1)

&

Hyperbolic functions

St

at

cosh2x – sinh2x = 1, sinh2x = 2sinhx coshx, cosh2x = cosh2x + sinh2x arcosh x = ln(x + x 2 + 1 ), 1 + x 1 artanh x = –2 ln ––––– 1 – x , |x| < 1

arsinh x = ln(x +

(

)

ist ics(

Matrices Anticlockwise rotation through angle θ, centre O: Reflection in the line y = x tan θ :

x 2 – 1 ), x > 1

cos θ sin θ

2θ ( cos sin 2θ

–sin θ cos θ

)

sin 2θ –cos 2θ

)

Cosine rule

A

b2 + c2 – a2 (etc.) cos A = ––––––––––

W

2bc

c

a2 = b2 + c2 –2bc cos A (etc.)

Trigonometry

B

or

Perpendicular distance of a point from a line and a plane b

a

Line: (x1,y1) from ax + by + c = 0 :

a2 + b2 n1α + n2β + n3γ + d Plane: (α,β,γ) from n1x + n2y + n3z + d = 0 : –––––––––––––––––– √(n12 + n22 + n32)

C

sin (θ ± φ) = sin θ cos φ ± cos θ sin φ

ld

cos (θ ± φ) = cos θ cos φ 7 sin θ sin φ

Vector product

i a1 b1 a2b3 – a3b2 a × b = |a| |b| sinθ ^n = j a2 b2 = a3b1 – a1b3 k a3 b3 a1b2 – a2b1

tan θ ± tan φ tan (θ ± φ) = –––––––––––– , [(θ ± φ) ≠ (k + W)π] 1 7 tan θ tan φ

cos θ + cos φ = 2 cos 1–2 (θ + φ) cos cos θ – cos φ = –2 sin 1–2 (θ + φ) sin

|

M

at

sin θ – sin φ = 2 cos 1–2 (θ + φ) sin 1–2 (θ – φ)

a × (b × c) = (c . a) b – (a . b) c

he m

1– 2 (θ – φ) 1– 2 (θ – φ)

3 Vectors and 3-D coordinate geometry (The position vectors of points A, B, C are a, b, c.)

The position vector of the point dividing AB in the ratio λ:µ µa + λb is ––––––– (λ + µ) Line:

)

a1 b1 c1 a. (b × c) = a2 b2 c2 = b. (c × a) = c. (a × b) a3 b3 c3

(1 + t )

sin θ + sin φ = 2 sin 1–2 (θ + φ) cos 1–2 (θ – φ)

| |( |

Cartesian equation of line through A in direction u is x – a1 y – a2 z – a3 =t = –––––– = –––––– –––––– u1 u2 u3

( )

Conics Ellipse

Parabola

Hyperbola

Rectangular hyperbola

Standard form

y2 x2 + –– –– =1 b2 a2

y2 = 4ax

y2 x2 – –– –– =1 b2 a2

x y = c2

Parametric form

(acosθ, bsinθ)

(at2, 2at)

(asecθ, btanθ)

(ct, –c–) t

e<1 b2 = a2 (1 – e2)

e=1

e>1 b2 = a2 (e2 – 1)

e = √2

(± ae, 0)

(a, 0)

(± ae, 0)

(±c√2, ±c√2)

x = ± –a e

x = –a

x = ± –ae

x + y = ±c√2

none

none

y x a– = ± –b

x = 0, y = 0

at

ics

Eccentricity

Foci

Directrices Asymptotes

&

St

at

a.u The resolved part of a in the direction u is ––––– |u|

Any of these conics can be expressed in polar coordinates (with the focus as the origin) as: where l is the length of the semi-latus rectum.

Plane: Cartesian equation of plane through A with normal n is n1 x + n2y + n3z + d = 0 where d = –a . n

Mensuration

The plane through non-collinear points A, B and C has vector equation r = a + s(b – a) + t(c – a) = (1 – s – t) a + sb + tc The plane through A parallel to u and v has equation r = a + su + tv

Cone :

Sphere : Surface area = 4π r2

l – = 1 + e cos θ r

ist ics

Curved surface area = π r × slant height

TRIGONOMETRY, VECTORS AND GEOMETRY

Of

(1 – t2) 2t , cos θ = –––––– For t = tan 1–2 θ : sin θ = –––––– 2 2 (1 + t )

ax1 + by1 + c

Differentiation f(x) tan kx sec x cot x cosec x arcsin x

W

or

arccos x

ld

arctan x sinh x cosh x tanh x

Of

f'(x) ksec2 kx sec x tan x –cosec2 x –cosec x cot x 1 ––––––– √(1 – x2) –1 ––––––– √(1 – x2) 1 ––––– 1 + x2 cosh x sinh x sech2 x 1 ––––––– √(1 + x2)

Integration f(x) sec2 kx tan x cot x cosec x

M

at

arsinh x

artanh x 4

he m

du dv v ––– – u ––– dx dx u dy Quotient rule y = – , ––– = 2 v v dx b ––– a ∫a ydx ≈ 1–2 h{(y0 + yn) + 2(y1 + y2 + ... + yn –1)}, where h = ––– n b

Integration by parts Area of a sector

Arc length

dv du dx = uv – ∫ v ––– dx ∫ u ––– dx dx A = 1–2 ∫ r dθ (polar coordinates) A = 1–2 ∫ (x˙y – yx˙) dt (parametric form) s = ∫ √ (x˙ + y˙ ) dt (parametric form) dy s = ∫ √ (1 + [ –––] ) dx (cartesian coordinates) dx dr s = ∫ √ (r + [ –––] ) dθ (polar coordinates) dθ 2

2

2

2

2

2

at

ics

Surface area of revolution

Curvature

&

| (

)|

| | x arcsin ( – a ) , |x| < a 1 x – a arctan( – a) 1 1 x a+x –– ln ––––– = – artanh ( – a ) , |x| < a 2a | a – x | a

cosh x sinh x ln cosh x x 2 2 arsinh – a or ln (x + x + a ),

() x arcosh ( – a ) or ln (x + x S = 2π∫y ds = 2π∫y√(x˙ S = 2π∫x ds = 2π∫x√(x˙

2

– a 2 ), x > a , a > 0

2

+ y˙ 2) dt

2

+ y˙ 2) dt

x

y

d2y ––– dψ x˙ ÿ – x¨ y˙ dx2 κ = ––– = –––––––– 3/2 = ––––––––––––– 2 2 ds dy 2 3/2 (˙x + y˙ ) 1 + –– dx

St ( [ ] ) at ist i

1 Radius of curvature ρ = –– κ, L'Hôpital’s rule

Centre of curvature c = r + ρ n^

If f(a) = g(a) = 0 and g'(a) ≠ 0 then

f(x) f'(a) Lim –––– = –––– x ➝a g(x) g'(a)

cs

Multi-variable calculus ∂g/∂x ∂w ∂w ∂w grad g = ∂g/∂y For w = g(x, y, z), δw = ––– δx + ––– δy + ––– δz ∂x ∂y ∂z ∂g/∂z

( )

CALCULUS

1 ––––––– √(x2 – 1) 1 ––––––– (1 – x2)

arcosh x

Trapezium rule

sec x 1 –––––– x2 – a2 1 ––––––– √(a2 – x2) 1 –––––– a2 + x2 1 –––––– a2 – x2 sinh x cosh x tanh x 1 ––––––– √(a2 + x2) 1 ––––––– √(x2 – a2)

∫f(x) dx (+ a constant) (l/k) tan kx ln |sec x| ln |sin x| x –ln |cosec x + cot x| = ln |tan – 2| x π ln |sec x + tan x| = ln tan – + – 2 4 1 x–a –– ln ––– x +–– a 2a

Centre of mass (uniform bodies)

Moments of inertia (uniform bodies, mass M)

W

Triangular lamina: Solid hemisphere of radius r:

or

Hemispherical shell of radius r:

ld

Solid cone or pyramid of height h:

Of

Sector of circle, radius r, angle 2θ:

2 – along median from vertex 3 3 – r from centre 8 1 – r from centre 2 1 – h above the base on the 4

Thin rod, length 2l, about perpendicular axis through centre: Rectangular lamina about axis in plane bisecting edges of length 2l: Thin rod, length 2l, about perpendicular axis through end: Rectangular lamina about edge perpendicular to edges of length 2l:

line from centre of base to vertex

Rectangular lamina, sides 2a and 2b, about perpendicular 1 – M(a2 + b2) axis through centre: 3

2r sin θ ––––––– from centre 3θ

M

Hoop or cylindrical shell of radius r about perpendicular axis through centre:

r sin θ from centre Arc of circle, radius r, angle 2θ at centre: –––––––

θ

at

Hoop of radius r about a diameter:

1 – h above the base on the 3 line from the centre of

Conical shell, height h:

Solid sphere of radius r about a diameter:

5

at

Spherical shell of radius r about a diameter:

Motion in a circle Transverse velocity: v = rθ˙ v2 Radial acceleration: –rθ˙ 2 = – –– r Transverse acceleration: v˙ = rθ¨ General motion Radial velocity: Transverse velocity: Radial acceleration: Transverse acceleration:

Disc of radius r about a diameter:

r˙ rθ˙

r¨ – rθ˙ 2 1 d rθ¨ + 2r˙θ˙ = – –– (r2θ˙) r dt

Moments as vectors The moment about O of F acting at r is r × F

ics

1 – Mr2 2 1 – Mr2 2 1 – Mr2 4 2 – Mr2 5 2 – Mr2 3

IA = IG + M(AG)2

Parallel axes theorem:

&

Mr2

Iz = Ix + Iy (for a lamina in the (x, y) plane)

Perpendicular axes theorem:

St

at

ist ics

MECHANICS

he m

Disc or solid cylinder of radius r about axis:

base to the vertex

Motion in polar coordinates

1 – Ml2 3 1 – Ml2 3 4 – Ml2 3 4 – Ml2 3

Probability

Product-moment correlation: Pearson’s coefficient ∑ xi yi –xy Sxy Σ( xi – x )( yi – y ) n r= = = 2 2 Sxx Syy  ∑ xi 2   ∑ yi 2  Σ( xi – x ) Σ( yi – y ) − x2 − y2     n  n

P(A∪B) = P(A) + P(B) – P(A∩B) P(A∩B) = P(A) . P(B|A) P(B|A)P(A) P(A|B) = –––––––––––––––––––––– P(B|A)P(A) + P(B|A')P(A')

W

[

or

P(Aj)P(B|Aj) Bayes’ Theorem: P(A j |B) = –––––––––––– ∑P(Ai)P(B|Ai)

ld

Populations Discrete distributions

Of

Rank correlation: Spearman’s coefficient 6∑di2 rs = 1 – –––––––– n(n2 – 1)

X is a random variable taking values xi in a discrete distribution with P(X = xi) = pi Expectation: µ = E(X) = ∑xi pi σ2 = Var(X) = ∑(xi – µ)2 pi = ∑xi2pi – µ2 Variance: For a function g(X): E[g(X)] = ∑g(xi)pi

Regression

M

at

Cumulative x distribution function F(x) = P(X < x) = ∫–∞f(t)dt

n

, Syy = ∑(yi – y )2 = ∑yi2 –

(∑xi)(∑yi) Sxy = ∑(xi – x )(yi – y ) = ∑xi yi – ––––––––– n Covariance

Sxy = ∑( xi – x )( yi – y ) = ∑ xi yi – x y –––– n n n

ics

1 ∑(x – x )2f S2 for population variance σ 2 where S2 = –––– i i n–1

&

Probability generating functions For a discrete distribution

For a sample of n pairs of observations (xi, yi) (∑xi)2 –––––

at

(∑yi)2 ––––– n

,

G(t) = E(tX)

St

at

E(X) = G'(1); Var(X) = G"(1) + µ – µ2

ist ics

GX + Y (t) = GX (t) GY (t) for independent X, Y Moment generating functions: MX(θ) =

E(eθX)

E(X) = M'(0) = µ;

E(Xn) = M(n)(0)

Var(X) = M"(0) – {M'(0)}2

MX + Y (θ) = MX (θ) MY (θ) for independent X, Y

STATISTICS

6

X is a continuous variable with probability density function (p.d.f.) f(x) Expectation: µ = E(X) = ∫ x f(x)dx Variance: σ2 = Var (X) = ∫(x – µ)2 f(x)dx = ∫x2 f(x)dx – µ2 For a function g(X): E[g(X)] = ∫g(x)f(x)dx

Sxx = ∑(xi – x )2 = ∑xi2 –

Least squares regression line of y on x: y – y = b(x – x ) ∑ xi yi –xy Sxy ∑(xi – x) (yi – y ) n = ––––––––––––––– = b = ––– Sxx ∑ xi 2 ∑(xi – x )2 – x2 n Estimates Unbiased estimates from a single sample σ2 X for population mean µ; Var X = –– n

he m

Continuous distributions

Correlation and regression

]

Regression

Markov Chains pn + 1 = pnP Long run proportion p = pP

W

or

Bivariate distributions Covariance Cov(X, Y) = E[(X – µX)(Y – µY)] = E(XY) – µXµY

ld

Cov(X, Y) ρ = –––––––– Product-moment correlation coefficient σX σY Sum and difference Var(aX ± bY) = a2Var(X) + b2Var(Y) ± 2ab Cov (X,Y) If X, Y are independent: Var(aX ± bY) = a2Var(X) + b2Var(Y) E(XY) = E(X) E(Y) Coding X = aX' + b ⇒ Cov(X, Y) = ac Cov(X', Y') Y = cY ' + d

Of

α + βf(xi) + εi α + βxi + γzi + εi

∑(yi – a –

εi ~ N(0, σ2)

Ti2 T 2 SSB = ∑ni ( x i – x )2 = ∑ ––– ni – –– n i i x )2 =

∑ ∑ xij i j

2

2 –– – T n

2 2

czi)2

3

a, b, c are estimates for α, β, γ.

b−β ~ tn – 2 σˆ 2 / Sxx

)

σ 2∑xi2 a = y – b x , a ~ N α, ––––––– nS

(

xx

RSS σ^2 = n–––– –p

)

(x0 – x )2 a + bx0 ~ N(α + βx0, σ2 1 + ––––––– – n Sxx 2 (Sxy) RSS = Syy – ––––– = Syy (1 – r2) Sxx

{

at

}

Randomised response technique

ics

–y – (1 – θ)

n E(p^) = –––––––––– (2θ – 1)

Factorial design

&

[(2θ – 1) p + (1 – θ)][θ – (2θ – 1)p]

Var(p^) = –––––––––––––––––––––––––––––– 2 n(2θ – 1)

St

Interaction between 1st and 2nd of 3 treatments (–)

{

at

(Abc – abc) + (AbC – abC) (ABc – aBc) + (ABC – aBC) ––––––––––––––––––––– – –––––––––––––––––––––– 2 2

Exponential smoothing

ist ics

}

y^n+1 = α yn + α(1 – α)yn–1 + α(1 – α)2 yn–2 + ... + α(1 – α)n–1 y1 + (1 – α)ny0 ^y = y^ + α(y – y^ ) n n n+1 n y^n+1 = α yn + (1 – α) y^n

STATISTICS

7

One-factor model: xij = µ + αi + εij, where εij ~ N(0,σ2)

∑(yi – a – bxi –

(

he m

No. of parameters, p

bf(xi))2

For the model Yi = α + βxi + εi, Sxy σ2 , b = ––– , b ~ N β, ––– Sxx Sxx

M

Analysis of variance

i j

RSS ∑(yi – a – bxi)2

at

}

SST = ∑ ∑ (xij –

Yi α + βxi + εi

W

Description

Pearson’s product moment correlation test

or

ld

∑ xi yi –xy n  ∑ xi 2    ∑ yi 2 – x2 – y2     n  n

Of

8

x–µ σ/ n

t-test for a mean

x–µ s/ n

t-test for paired sample

Normal test for the difference in the means of 2 samples with different variances

∑

(f

o

– fe )

( x – y ) – ( µ1 – µ2 ) 1 1 + s n1 n2

t-test for the difference in the means of 2 samples

at

N(0, 1)

tn – 1

2

( x1 – x2 ) – µ s/ n

χ 2v

at

Wilcoxon Rank-sum (or Mann-Whitney) 2-Sample test

ics

&

Normal test on binomial proportion t with (n – 1) degrees of freedom

χ2 test for variance

( x – y ) – ( µ1 – µ2 ) N(0, 1)

F-test on ratio of two variances

Distribution

tn

1

+ n2 – 2

(n1 – 1)s12 + (n2 – 1)s22 where s2 = ––––––––––––––––––––––– n1 + n2 – 2

Wilcoxon single sample test

he m

fe

σ 12 σ 2 2 + n1 n2

Test statistic

M

6∑di2 rs = 1 – ––––––– n(n2 – 1)

Normal test for a mean

χ2 test

Description

Distribution

A statistic T is calculated from the ranked data.

See tables

Samples size m, n: m < n Wilcoxon W = sum of ranks of sample size m Mann-Whitney T = W – 1–2 m(m + 1)

See tables

p –θ

St

 θ (1 – θ )   n 

at

(n – 1)s 2

ist ics

σ2

s12 /σ12 ––––––– s22 /σ22

, s12 > s22

N(0, 1)

χ2n – 1 Fn

1

–1, n2 –1

STATISTICS: HYPOTHESIS TESTS

Spearman rank correlation test

r=

Test statistic

W

or

Variance

p.g.f. G(t) (discrete) m.g.f. M(θ) (continuous)

ld

Binomial B(n, p) Discrete

P(X = r) = nCr qn–rpr , for r = 0, 1, ... ,n , 0 < p < 1, q = 1 – p

np

npq

G(t) = (q + pt)n

Poisson (λ) Discrete

Of

λr P(X = r) = e–λ ––– r! , for r = 0, 1, ... , λ > 0

λ

λ

G(t) = eλ(t – 1)

µ

σ2

M(θ) = exp(µθ + Wσ 2θ 2)

M

at ( ( ) ) he m 1

x–µ exp – W ––––– σ

f(x) =

Uniform (Rectangular) on [a, b] Continuous

1 f(x) = ––––– b–a

Exponential Continuous

f(x) = λe–λx

Geometric Discrete

P(X = r) = q r – 1p ,

9

Normal N(µ, σ2) Continuous

Negative binomial Discrete

σ 2π

2

–∞ < x < ∞

, a
,

0
,

at

x > 0, λ > 0 r = 1, 2, ... ,

, q=1–p

r – 1C n–1

qr – n pn , r = n, n + 1, ... ,

0
q=1–p

ics

a+b ––––– 2

&

1 –– λ

1 –– 12

ebθ – eaθ M(θ) = ––––––––– (b – a)θ

(b – a)2

λ M(θ) = ––––– λ–θ

1 –– λ2

1 –– p

St

n –– p

nq ––2 p

q ––2 p

at

pt G(t) = ––––– 1 – qt

ist ( ) ics pt G(t) = ––––– 1 – qt

n

STATISTICS: DISTRIBUTIONS

Mean

Function

Name

Numerical Solution of Equations

f(xn) The Newton-Raphson iteration for solving f(x) = 0 : xn + 1 = xn – –––– f'(xn) Numerical integration The trapezium rule b 1 b–a ydx ≈ –2 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where h = ––––– n a

W

Taylor polynomials h2 f(a + h) = f(a) + hf '(a) + ––– f"(a) + error 2! h2 f(a + h) = f(a) + hf '(a) + ––– f"(a + ξ), 0 < ξ < h 2!

or

∫

ld

The mid-ordinate rule

∫a ydx ≈ h(y b

1 – 2

1– 3 h{(y0

M

at

+ yn) + 4(y1 + y3 + ... + yn – 1) + 2(y2 + y4 + ... + yn– 2)},

(x – a)2 = f(a) + (x – a)f '(a) + –––––– f"(η) , a < η < x 2!

Numerical solution of differential equations dy For –– = f(x, y): dx Euler’s method : yr + 1 = yr + hf(xr, yr); xr+1 = xr + h

he m

b–a where h = ––––– n

The Gaussian 2-point integration rule 10

  −h   h   f ( x )dx ≈ h f   + f    –h   3  3 Interpolation/finite differences

∫

f(x)

h

Runge-Kutta method (order 2) (modified Euler method) yr + 1 = yr + 1–2 (k1 + k2)

at

where k1 = h f(xr, yr), k2 = h f(xr + h, yr + k1) Runge-Kutta method, order 4:

n x – xi Lagrange’s polynomial : Pn(x) = ∑ Lr(x)f(x) where Lr(x) = ∏ –––––– x i=0 r – xi i≠r

Newton’s forward difference interpolation formula (x – x0)(x – x1) (x – x0) –––––––––––– f(x) = f(x0) + –––––– ∆f(x ) + ∆2f(x0) + ... 0 2!h2 h Newton’s divided difference interpolation formula

ics

f(x + h) – 2f(x) + f(x – h) f"(x) ≈ ––––––––––––––––––––– h2

&

k2 = hf(xr + 1–2 h, yr + 1–2 k1)

where k1 = hf(xr, yr)

St

k3 = hf(xr + 1–2 h, yr + 1–2 k2) Logic gates

f(x) = f[x0] + (x – x0]f[x0, x1] + (x – x0) (x – x1)f[x0, x1, x2] + ... Numerical differentiation

yr+1 = yr + 1–6 (k1 + 2k2 + 2k3 + k4),

NOT

OR

k4 = hf(xr + h, yr + k3).

at

ist ics AND

NAND

NUMERICAL ANALYSIS DECISION & DISCRETE MATHEMATICS

∫a ydx ≈

(x – a)2 = f(a) + (x – a)f '(a) + –––––– f"(a) + error 2!

b–a + y1 1– + ... + yn – 1 1– + yn – 1– ), where h = ––––– n 2 2 2

Simpson’s rule for n even b

Of

f(x)