Edexcel A-Level Maths Pure Y2

Some is in formula sheet some is not

difference between rational and irrational numbers

rational numbers can be written in form a/b where a and b are integers, irrational numbers cannot

prove by contradiction that 2^1/2 is an irrational number

prove by contradiction that there are infinitely many prime numbers

types of mapping

• one to one - a function

• many to one - a function

• one to many - not a function

graph of y = If(x)I

reflect any parts below x axis in x axis

graph of y = f(IxI)

remove any parts where x<0, reflect x>0 in y axis

formula for nth term in arithmetic

formula for S_n in arithmetic

proof of S_n in arithmetic sequence

formula for nth term in geometric

formula for S_n in geometric

proof for S_n in geometric

S_infinity in geometric

convergent series

series where Sn gets closer and closer to a finite value (difference between terms decreases)

divergent series

series where difference between each term increases from term to term

sigma notation meaning

sum to n terms, starting from term r=1, following the rule next to it - uses an nth term rule

increasing sequence

if u_n+1 > u_n for all n

decreasing sequence

if u_n+1 < u_n for all n

periodic sequence

terms repeat in a cycle

order of a periodic sequence

the value of k such that u_n+k = u_n

recurrence relation

term to term rule

binomial expansion for negative or fractional values of n (infinite series obtained)

validity of binomal expansion for negative/fractional

• (1 + x)ⁿ → IxI<1

• (1 + bx)^{n →} IbxI<1

• (a + bx)ⁿ → Ibx/aI<1

validity of approximation based on binomial expansion

• more valid when more terms of expansion are use

• the values of x substituted in are closer to 0

360 =

2 pi rad

180 =

pi rad

arc length of sector using radians

l = rx

area of sector using radians

A = ½ r²x

area of segment using radians

A = ½ r² (x-sinx)

if x is small + radians, sin x =

x

if x is small + radians, tan x =

x

if x is small + radians, cos x =

1 - x²/2

sec x =

1/cos x

cosec x =

1/sin x

cot x =

1/tan x; cos x/sin x

sec x graph

cosec x graph

cot x graph

sec² x =

1 + tan²x

cosec² x =

1+ cot² x

arcsin x graph

arccos x graph

arctan x graph

sin2x =

2sinxcosx

cos2x =

cos²x-sin²x = 2cos²x - 1 = 1 - 2sin²x

tan 2x =

2tanx/1-tan²x

a sinx + b cos x =

• R sin (x + c)

• R sin c = b; R = (a² + b²)^1/2

a cos x + b sin x =

• R cos (x-c)

• R cos c = a

• R = (a² + b²)^1/2

prove the derivative of sin x is cos x using first principles

similar proof for cos x → -sinx

dy/dx; y = sinkx

k coskx

dy/dx; y = coskx

-ksinkx

dy/dx; y = lnx

1/x

dy/dx; y = a^kx

a^kx k ln a

chain rule

product rule

quotient rule

dy/dx; y = tan kx

k sec² kx

dy/dx; y = cosec kx

-k cosec kx cot kx

dy/dx; y = sec kx

k sec kx tan kx

dy/dx; y = cot kx

-k cosec² kx

concave function on [a,b]

f’’ <= 0 for a < x < b

convex function on [a,b]

f’’ >= 0 for a < x < b

point of inflection in terms of f’’(x)

when f’’(x) changes sign - from concave to convex (points of inflection can be stationary but do not have to be)

change of sign method

if f (x) is continuous on [a,b] and f(a) and f(b) have opposite signs there will be at least one root in that interval

iteration method

f(x) = 0, rearrange to x = g(x) and use formula x_n+1 = g(x_n)

newton raphson method

x_n+1 = x_n - f(x_n)/f’(x_n)

integration by parts

area bounded by two curves

trapezium rule

trapezium rule + convex curve

overestimate as line connecting 2 endpoints above curve

trapezium rule + concave curve

underestimate as line connecting 2 endpoints is below curve

unit vector on z axis

k