PURE YEAR 2 Flashcards by hannah king

how can a rational number be expressed

a/b
a and b are integers

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in an arithmetic sequence what is constant

the difference between consecutive terms

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formula for nth term of arithmetic sequence

un= a + (n-1) d

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formula for first n terms of an arithmetic sequence

Sn= n/2 (2a + (n-1) d)

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a geometric sequence has what

a common ratio between consecutive terms

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formula for nth term of geometric sequence

un = a r^(n-1)

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formula for sum of first n terms of geometric sequence

Sn= (a (1-r^n) / 1-r)

Sn= (a (r^n - 1)/ r-1)

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when is a geometric sequence convergent

|r|<1

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sum to infinity of a geometric sequence

a / 1-r

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when can you sum to infinity

when a geometric sequence is convergent

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what is a recurrence relation

u (n+1) = f (u(n))
defines each term of a sequence as a function of the previous term

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when is a sequence increasing

u(n+1) > u(n) for all n

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when is a sequence decreasing

u(n+1) < u(n) for all n

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when is a sequence periodic

if the terms repeat in a cycle
for a periodic sequence there is an integer k such that u(n+k) = u(n) for all n
the value k is called the order of the sequence

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radian to degrees

x 180/pi

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degrees to radians

x pi/180

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arc length equation

l= θr
radius x angle (in radians)

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sector area equation

1/2 r^2 θ
(in radians)

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segment area equation

1/2 r^2 (θ - sinθ)

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when θ is small and measured in radians:
approximation for sinθ

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when θ is small and measured in radians:
approximation for tanθ

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when θ is small and measured in radians:
approximation for cosθ

1 - θ^2 / 2

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sec x =

1 / cos x

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cosec x =

1 / sin x

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cot x =

1 / tan x OR cos x / sin x

equation for binomial expansion applied to negative/fractional values of n to obtain infinite series?

(1+x)^n = 1 + nx + (n(n-1)x^2)/2! + (n(n-1)(n-2)x^3)/3! + ... + (nCr) x^r

when is year 2 binomial expansion equation valid

when |x|<1

when is the expansion os (1+bx)^n (where n is negative or a fraction) valid

valid for |bx|<1

when is the expansion of (a+bx)^n (where n is negative for a fraction) valid

valid for |b/a x|<1

if y=sin kx , what is dy/dx

k cos kx

if y= cos kx , what is dy/dx

-k sin kx

if y= ln x , what is dy/dx

1/x

if y= a^(kx), what is dy/dx

a^(kx) k ln a

chain rule equation

dy/dx = dy/du x du/dx

product rule equation

dy/dx= u'v + uv'

quotient rule equation (u/v)

dy/dx= u'v - uv' / v^2

how to differentiate if x and y are given as functions of a parameter, t

dy/dx = dy/dt / dx/dt

when is the function f(x) concave

if f''(x) <0 for every value of x in the specified interval

when is the function f(x) convex

if f''(x)>0 for every value of x in that interval

what is a point of inflection

a point at which f''(x) changes sign

how to solve an equation in the form f(x)=0 by an iterative method

rearrange f(x)=0 into the form x=g(x) and use the iterative formula x(n+1)=g(x(n))

what is newton raphson formula used for

approximating the roots of a function f(x)

newton raphson formula

x(n+1) = x(n) - f(x(n)) / f'(x(n))

sign change to find roots of a function?

if the function f(x) is continuous on the interval [a,b] and f(a) and f(b) have opposite signs, then f(x) has at least one root, x, which satisfies a

integration equation for x^n

∫x^n dx= (x^(n+1))/ n+ 1 + c

integration of e^x

e^x + c

integration of 1/x

ln|x| + c

integration of cos x

sin x + c

integration of sin x

-cos x + c

integration of f'(ax+b)

1/a f(ax+b) + c

integration by parts formula

∫uv'=uv=∫u'v

trapezium rule

∫y dx= 1/2 h (y0+2(y1 + y2 + ... + y(n-1)) + yn) where h = b - a /n

graph of y=sec x: symmetry? period? vertical asymptotes? range?

symmetry in y-axis period 360 or 2pi vertical asymptotes at all the values of x for which cosx=0 (90, 270, 450) y<-1 or y>1

graph of y=cosec x: period? vertical asymptotes? range

period 360 or 2pi vertical asymptotes at values of x for which sin x= 0 (0,180,360) range y<-1 or y>1

graph of y=cot x: period? vertical asymptotes? range?

period 180 or pi vertical asymptotes at all values of x for which tanx=0 (0,180,360) range y is all real numbers

identities derived from sin^2+ cos^2 = 1

1 + tan^2 = sec^2 1 + cot^2 = cosec^2

y=arcsin x graph: domain range

domain -1

y=arccos x graph: domain range

domain -1

y=arctan x graph: domain range

domain x is all real numbers range -90

addition formula: sin (A+B)

sinAcosB + cosAsinB

addition formula: sin(A-B)

sinAcosB - cosAsinB

addition formula: cos(A+B)

cosAcosB - sinAsinB

addition formula: cos(A-B)

cosAcosB + sinAsinB

addition formula: tan(A+B)

tanA + tanB / 1-tanAtanB

addition formula: tan(A-B)

tanA - tanB / 1 + tanAtanB

double angle formulae: sin2A

2sinAcosA

double angle formula: cos2A

cos^2 A - sin^2 A 2cos^2 A -1 1- 2sin^2 A

double angle formula: tan2A

2tanA/1-tan^2 A

how can a sin x +- b cos x be expressed

Rsin (x +- k) or Rcos (x +- k) (R>0 and 0

how to convert between cartesian and parametric equations

using substitution to eliminate parameter t

for parametric equations x=p(t) and y=q(t) with cartesian equation y=f(x), what is domain and range of cartesian

domain of f(x) is range of p(t) range of f(x) if range of q(t)

when does newton raphson not work

stationary point at point x=n tangent to curve at point would not meet x-axis tangent to curve at point is horizontal