Merged math b 640 1
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Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn =
1 -r
a for r 1 1
S 3=
1 -r
Binomial series
^a + bhn = an + nC1 a n -1b + nC2 a n -2b2 +f+ nCr a n -rbr +f+ bn ^n e Nh,
JnN
n n!
where C = C = K O=
r n r r
L P r!^n - rh!
n ^n - 1 h 2 n^n - 1h f ^n - r + 1h r ^x 1 1, n e Rh
^1 + xhn = 1 + nx + x +f + x +f
2! r!
Differentiation
f^xh f l^xh
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
v du - u dv
u dy dx dx
Quotient Rule y = v , =
dx v2
Differentiation from first principles
f^x + hh- f^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
d dx = ln f^xh + c
e f^xh
n 1 n+1
; f l^xhaf^xhk dx = af^xhk + c
n+1
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i i , cos i 1 - 1 i 2 , tan
2
i i where i is measured in radians
, 3
Trigonometric identities
sin^A ! Bh = sin A cos B ! cos A sin B cos^A !
Bh = cos A cos B " sin A sin B
tan A ! tan B
tan^A ! Bh = 1 " tan A tan B aA ! B !^k + 1h2rk
Numerical methods
b -a
Trapezium rule: ; b y dx 1
h"^y + y h + 2^y + y +f+ y h,, where h =
2 0 n 1 2 n -1 n
a f ^x h
n
The Newton-Raphson iteration for solving f^xh = 0: xn +1 = xn -
f l^xnh
Probability
P^A j Bh = P^Ah +P^Bh - P^A k Bh
P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Sample variance
2 1 2 ^/xih2 2
s = n 1 Sxx where Sxx = /^xi --xh = / x2 - n = /x2 - n-x
- i i
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = rh = nCr p r q n -r where q = 1 - p Mean of X
is np
Hypothesis testing for the mean of a Normal distribution
J 2N X -n
2 v
If X + N^n, v h then X + NKn, and ~ N^0,1h
nO v n
L P
Percentage points of the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at vt - 1 at2
s = ut + 12at2 s
= 1^2u + vht v2
= u2 + 2as s =
2
,Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn =
1 -r
a for r 1 1
S 3=
1 -r
Binomial series
^a + bhn = an + nC1 a n -1b + nC2 a n -2b2 +f+ nCr a n -rbr +f+ bn ^n e Nh,
JnN
n n!
where C = C = K O=
r n r r
L P r!^n - rh!
n ^n - 1 h 2 n^n - 1h f ^n - r + 1h r ^x 1 1, n e Rh
^1 + xhn = 1 + nx + x +f + x +f
2! r!
Differentiation
f^xh f l^xh
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
v du - u dv
u dy dx dx
Quotient Rule y = v , =
dx v2
Differentiation from first principles
f^x + hh- f^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
d dx = ln f^xh + c
e f^xh
n 1 n+1
; f l^xhaf^xhk dx = af^xhk + c
n+1
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i i , cos i 1 - 1 i 2 , tan
2
i i where i is measured in radians
, 3
Trigonometric identities
sin^A ! Bh = sin A cos B ! cos A sin B cos^A !
Bh = cos A cos B " sin A sin B
tan A ! tan B
tan^A ! Bh = 1 " tan A tan B aA ! B !^k + 1h2rk
Numerical methods
b -a
Trapezium rule: ; b y dx 1
h"^y + y h + 2^y + y +f+ y h,, where h =
2 0 n 1 2 n -1 n
a f ^x h
n
The Newton-Raphson iteration for solving f^xh = 0: xn +1 = xn -
f l^xnh
Probability
P^A j Bh = P^Ah +P^Bh - P^A k Bh
P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Sample variance
2 1 2 ^/xih2 2
s = n 1 Sxx where Sxx = /^xi --xh = / x2 - n = /x2 - n-x
- i i
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = rh = nCr p r q n -r where q = 1 - p Mean of X
is np
Hypothesis testing for the mean of a Normal distribution
J 2N X -n
2 v
If X + N^n, v h then X + NKn, and ~ N^0,1h
nO v n
L P
Percentage points of the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at vt - 1 at2
s = ut + 12at2 s
= 1^2u + vht v2
= u2 + 2as s =
2
