PURE YEAR 2 Flashcards
how can a rational number be expressed
a/b
a and b are integers
in an arithmetic sequence what is constant
the difference between consecutive terms
formula for nth term of arithmetic sequence
un= a + (n-1) d
formula for first n terms of an arithmetic sequence
Sn= n/2 (2a + (n-1) d)
a geometric sequence has what
a common ratio between consecutive terms
formula for nth term of geometric sequence
un = a r^(n-1)
formula for sum of first n terms of geometric sequence
Sn= (a (1-r^n) / 1-r)
OR
Sn= (a (r^n - 1)/ r-1)
when is a geometric sequence convergent
|r|<1
sum to infinity of a geometric sequence
a / 1-r
when can you sum to infinity
when a geometric sequence is convergent
what is a recurrence relation
u (n+1) = f (u(n))
defines each term of a sequence as a function of the previous term
when is a sequence increasing
u(n+1) > u(n) for all n
when is a sequence decreasing
u(n+1) < u(n) for all n
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
radian to degrees
x 180/pi
degrees to radians
x pi/180
arc length equation
l= θr
radius x angle (in radians)
sector area equation
1/2 r^2 θ
(in radians)
segment area equation
1/2 r^2 (θ - sinθ)
when θ is small and measured in radians:
approximation for sinθ
θ
when θ is small and measured in radians:
approximation for tanθ
θ
when θ is small and measured in radians:
approximation for cosθ
1 - θ^2 / 2
sec x =
1 / cos x
cosec x =
1 / sin x
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<x<b
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<x<1
range -90<arcsinx<90 (pi/2)
y=arccos x graph:
domain
range
domain -1<x<1
range 0<arccosx< 180 (pi)
y=arctan x graph:
domain
range
domain x is all real numbers
range -90<arctan<90 (pi/2)
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<k<90)
Rcosk= b, Rsink=a and R= sqrt (a^2 +b^2)
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
