pure Flashcards by Unknown Unknown

new formula of a line

y-y₁ = m(x-x₁)

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gradient of a straight line through 2 point

change in y / change in x

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midpoint of a line

(x₁+x₂/2 , y₁+y₂/2)

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quadratic formula

-b ∓ √b²-4ac / 2a

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set notation for and inequality

{x: x> -3} ∩ {x: x<2}

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set notation for or inequality

{x: x<-5} ∪ {x: x>1}

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discriminant

b² - 4ac

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discriminant when 2 different real roots

b² - 4ac > 0

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discriminant when no real roots

b² - 4ac < 0

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discriminant when 2 equal/repeated roots

b² - 4ac = 0

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general form of a circle with centre (a, b)

(x-a)² + (y-b)² = r²

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gradient of normal to a point on a circle

gradient from centre to radius

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gradient of tangent to a point on a circle

negative reciprocal of normal gradient

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the angle in a semicircle is…

a right angle

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a line from the centre of a circle to a chord, at a right angle to that chord will…

bisect the chord and is the shortest distance from the chord to the centre

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tangents from the circle to the same point outside the circle…

have the same length

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(x+3) is a factor of 6x³-bx² + 18, find the value of b.

substitute x=-3

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f(x) → -f(x)

reflection in x axis

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f(x) → f(-x)

reflection in y axis

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sine rule

a/sinA = b/sinB = c/sinC

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cosine rule

a² = b² + c² - 2bc cosA

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area of a triangle trig

0.5 × a × b × sinC

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sin graph repetitions

add 360
180 minus x

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cos graph repetitions

add 360
minus x plus 360

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tan graph repetitions

- plus 180 - minus 180

cos²x + sin²x

tan(x)

sin(x) / cos(x)

logₐa

logₐ1

logₐ(1/a)

-1

logₐx + logₐy

logₐ(xy)

logₐx - logₐy

logₐ(x/y)

logₐ(xⁿ)

nlogₐx

logₐ(aˣ)

general equation for exponential growth

V = Akᵗ

y = axⁿ

log(y) = log(a) + nlog(x)

y = kbˣ

log(y) = log(k) + xlog(b)

when to use (nCr)

(a+b)ⁿ where n is a positive whole number

when to use n(n-1)/2!

(1+x)ⁿ where n is a rational number

differentiate eⁿˣ

neⁿˣ

gradient of perpendicular

negative reciprocal

when is a function increasing

gradient is positive

when is a function decreasing

gradient is negative

show that the square of any number is positive in symbols

x² ≥ 0

gradient at stationary point

second derivative at minimum point

> 0

second derivative at maximum point

< 0

integration steps

- add 1 to power - divide by power

method to integrate 3x² if limit is unknown

factorise 3

vertical line test

If no vertical line will ever cross the graph more than once, it is a function.

horizontal line test

If no horizontal line will ever cross the graph more than once it is a one-to-one function, else many-to-one.

ℕ

natural numbers: all positive whole numbers

ℤ

integers: all whole numbers

ℚ

rational numbers: all numbers that can be written as an integer fraction

ℝ

real numbers: all possible decimals

transformation from f(x) to f⁻¹(x)

reflection in line y=x

domain of inverse is

range of original

ff⁻¹(x)

how many radians is 360º

2π

1ᶜ

angle subtended at the centre of the circle by an arc equal in length to the radius

length of an arc of a circle

l = rθ

area of a sector of a circle with radians

A = ½r²θ

tan²x + 1

sec²x

cot²x + 1

cosec²x

what does tan(arcsin¾) mean

find tanθ if sinθ=¾

sin2θ

2sinθcosθ

cos2θ

cos²θ - sin²θ

tan2θ

2tanθ / 1-tan²θ

asinθ - bcosθ

sin(θ - α)

acosθ - bsinθ

cos(θ + α)

if both transformations are in the x direction...

do translation first

how to describe transformations for a quadratic

complete the square

differentiate sin(x)

cos(x)

differentiate cos(x)

-sin(x)

steps of chain rule

- bring down exponent - multiply derivative of bracket - reduce exponent by 1 - keep bracket the same

differentiate ln(x)

1/x

how to differentiate ln(...)

derivative of bracket / bracket

differentiate aᵏˣ

k(ln(a))aᵏˣ

when to use product rule

when there is more than 1 unknown

differentiate y

dy/dx

how to determine whether concave

2nd derivative < 0

how to determine whether convex

2nd derivative > 0

point of inflection

between concave and convex

2nd derivative at point of inflection

how to make an equation equal to zero

make numerator 0

how to make an equation undefined

make denominator 0

how to solve an equation with sin ∓ cos

harmonic form

what type of function does not have an inverse

many to one

when to integrate using partial fractions

when the denominator has more than one x

order to choose for u when integrating by parts

Ln Algebra Trig Exponentials

how to integrate lnx

u=lnx v=1

integral of ln(x)

xln(x) - x + c

increasing sequence

If each term of the sequence is greater than the one immediately preceding it.

decreasing sequence

If each term of the sequence is less than the one immediately preceding it.

periodic sequence

A sequence that repeats itself at regular intervals.

order of a sequence

the number of terms before the sequence repeats

inductive sequence

Defines the term in a sequence based on the previous term.

convergent sequence

tends towards a limit

general rule for an arithmetic sequence

Uₙ = a + (n-1)d

general rule for a geometric sequence

Uₙ = arⁿ⁻¹

formula to differentiate parametric equations

(dy/dt) / (dx/dt)

formula to integrate parametric equations

∫y (dx/dt)