Pure A1&2 Level Maths OCR tiny little things to remember

set notation and interval notation for -4

set notation: {x: x>4} U {x: x<9}

interval notation: (-4,9)

the discriminant

graph reflections

reflection in x axis: y = -f(x)

reflection in y axis: y = f(-x)

if you times the gradients of two perpendicular lines, what do you get

-1

graph of y = ln(x)

graph of y = a^x

x-axis is an asymptote, y-intercept is 1

what is sinx in terms of cosx and vice versa (A1 formula)

sinx == cos(90-x)

cosx == sin(90-x)

cosine rule both ways

what do we know when dy/dx = 0

and when d2y/dx2 = 0

when dy/dx = 0, we know there is a stationary point

when d2y/dx2 = 0, we know that no conclusion can be drawn

what do you do for integration if the curve goes below the x-axis

you need to find the areas of the parts below and above seperately

what to look out for when criticising proofs

flaws in logic, mistakes in algebra, mistakes in arithmatic

proof that root 2 is irrational

one-one, many-one , one-many

a functions has to be either:

- one-one (every y value has one x value)

- many - one (one y value that comes from more than one x value)

a mapping is not a function if it is:

- one-many (a single x value corresponds to more than one y value)

what happens if you input the inverse function to its functions and vice versa

f(f-1(x)) == f-1(f(x)) == x

what types of functions have inverse functinos

only one-one functions

similarities between y = f(x) and y = f^-1(x)

- reflection on the line y = x

- the domain of f^-1(x) == range of f(x)

- range of f^-1(x) == domain of f(x)

when you are doing two vertical transformations or two horizontal transformations, what order do you do them in

- for y = pf(x) + c:

- stretch is performed before translation

- for y = f(qx + d):

- translation is performed before stretch

*remember the order doesn't matter if its one horizontal and one vertical and vice versa

for graph of y = I mx + c I where is it reflected and where is the vertex

-part of graph below the x-axis is reflected over the x-axis

- vertex at (-c/m , 0)

how do you express this inequality without the modulus symbol

I x-a I < b

I x-a I < b is the same as:

a-b < x < a+b

formula for a arithmetic sequence and geometric sequence

arithmetic: Un = a + (n-1)d where a = first term, d = difference

geometric: Un = ar^(n-1)

*remember for geometric, if mod(r) < 1, the series converges and it has a sum to infinity

polynomial division layout and requirements of the degrees of the powers

P(x)/ax+b == Q(x) + r/ax+b

-where Q(x) is the quotient and r is the remainder

- you can only do polynomial division if degree of numerator is same or greater than degree of denominator

* if degree of denominator is greater, than you do partial fractions

how to rewrite (a + bx)^n so that you can do general binomial expansion, and what is it valid for

(a + bx)^n == a^n( 1 + bx/a)^n

this expansion is valid for mod(bx/a) < 1

for graph y = a sinb(x+c) + d what does each letter represent

for y = a sinb(x+c) + d

mod(a) = amplitude

d = central value

d- mod(a) = minimum value

d + mod(a) = maximum value

2pi/mod(b) = period

domain and range of inverse trig functions

arcsinx: domain [ -1, 1 ] range [-90 , 90]

arcosx: domain [ -1, 1 ] range [ 0 , 180 ]

arctanx: domain R range [ -90 , 90 ]

*remember these are just the opposites of domain/range of normal trig functions

equation for length of an arc and area of a sector

l = rx

A = 1/2 r^2 x

where r is radius and x is angle in radians

trig identity for 2cos^2(x) and 2sin^2(x) to remove the 'squared' in integration

2cos^2(x) == 1 + cos2x

2sin^2(x) == 1 - cos2x

the two Pythagorean identities for reciprocal trigonomic functions

sec^2(x) == 1 + tan^2(x)

cosec^2(x) == 1 + cot^2(x)

how to differentiate a ln(bx + c)

and integrate a/bx+c

f'(a ln(bx+c)) = ab/bx+c

integral = a/b (lnbx+c)

chain rule

used to differentiate composite functions

derivative of a^(kx)

a^(kx) ln(a^k)

how to differentiate y in respect to x (implicit differentiation)

d/dx (f(y)) == f'(y) x dy/dx

e.g. d/dx(y) = 1 dy/dx

steps for integration by subsitution

1. differentiate substitution and express dx in terms of du

2. simplify resulting expression

3. find the limits for u

inverse chain rule

integral of f(x)(ax+b)

is 1/a g(x) (ax+b) where g(x) is the integral of f(x)

integral of f'(x)/f(x)

lnf(x) + c

what substitution should you use to find the integral of sqrt(a^2 - x^2)

x = asinu

difference of the second derivative for a convex curve vs a concave curve and at a point of inflection

- a convex curve has f''(x) > 0

- a concave curve has f''(x) < 0

- at inflection f''(x) = 0 and the curve changes from convex to concave or vice cersa

how to integrate a parametric equation with respect to t to find the area bound by the x axis

A = y x (dx/dt) dt

where A = the area

*make sure the parameters are in terms of t

how to find the area between the y-axis and the lines y =a and y = b and the curve g(y)

A = integral of g(y) .dy

where the parameters are a and b

how to solve most differential equations

separate the variables

* initial conditions can be used to find the constant of integration

limitations of the sign change method (when may it not work)

- if the graph of f(x) has a vertical asymptote

- if it has a break

- if it has a tangent to the x-axis

when does an iteration converge and when does it diverge

- converges if mod(g'(x)) < 1 near the root

- diverges if mod(g'(x)) > 1 near the root

*if the equation has several roots, different rearrangements might converge to different roots

rectangle method vs trapezium

- rectangle rule will have an upper bound and lower bound for the area, as the rectangles get smaller the bounds will get closer together. The actual area is the limit of the sum of the rectangles

- for the trapezium rule, you need fewer trapezia than rectangles to achieve the same level of accuracy

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