pure Flashcards
new formula of a line
y-y₁ = m(x-x₁)
gradient of a straight line through 2 point
change in y / change in x
midpoint of a line
(x₁+x₂/2 , y₁+y₂/2)
quadratic formula
-b ∓ √b²-4ac / 2a
set notation for and inequality
{x: x> -3} ∩ {x: x<2}
set notation for or inequality
{x: x<-5} ∪ {x: x>1}
discriminant
b² - 4ac
discriminant when 2 different real roots
b² - 4ac > 0
discriminant when no real roots
b² - 4ac < 0
discriminant when 2 equal/repeated roots
b² - 4ac = 0
general form of a circle with centre (a, b)
(x-a)² + (y-b)² = r²
gradient of normal to a point on a circle
gradient from centre to radius
gradient of tangent to a point on a circle
negative reciprocal of normal gradient
the angle in a semicircle is…
a right angle
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
tangents from the circle to the same point outside the circle…
have the same length
(x+3) is a factor of 6x³-bx² + 18, find the value of b.
substitute x=-3
f(x) → -f(x)
reflection in x axis
f(x) → f(-x)
reflection in y axis
sine rule
a/sinA = b/sinB = c/sinC
cosine rule
a² = b² + c² - 2bc cosA
area of a triangle trig
0.5 × a × b × sinC
sin graph repetitions
- add 360
- 180 minus x
cos graph repetitions
- add 360
- minus x plus 360
tan graph repetitions
- plus 180
- minus 180
cos²x + sin²x
1
tan(x)
sin(x) / cos(x)
logₐa
1
logₐ1
0
logₐ(1/a)
-1
logₐx + logₐy
logₐ(xy)
logₐx - logₐy
logₐ(x/y)
logₐ(xⁿ)
nlogₐx
logₐ(aˣ)
x
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
0
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)
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
0
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)
