pure core AS Flashcards
A^m/n
(n√a)^m
discriminant
b^2 - 4ac
b^2 - 4ac < 0
no real roots
b^2 - 4ac = 0
1 real root
b^2 - 4ac > 0
2 real roots
dotted line
<>
solid line
<= >=
[]
included in the range
()
not included in the range
midpoint
( (x1+x2)/2 , (y1+y2)/2)
length of line
√(x1-x2)^2 + (y1-y2)^2
the angle in a semi circle is always
a right angle
the perpendicular line from the centre of a circle to a chord
perpendicularly bisects the chord
the tangent to a circle at a point
is perpendicular to the radius through that point
tanx =
sinx/cosx
sin^2 x =
1 - cos^2 x
y = f(x) + 3
translation 3 in the y direction
y = f(x+3)
translation -3 in the x direction
y = 3f(x)
stretch of scale factor 3 in the y direction
y = f(3x)
stretch of scale factor 1/3 in the x direction
y = -f(x)
reflection in the x axis
y = f(-x)
reflection in the y axis
nCr
n! / r!(n-r)!
normal to curve
perpendicular to tangent at particular point
vector polar form
(r, θ)
vector/component form
(X, Y)
Xi + Yj
position vector
starts at the origin
unit vector
magnitude of 1
divide a direction by its magnitude to get its unit vector
vectors with common multiples
are parallel
vectors are perpendicular if
their dot/scalar product equals zero
Inverse of
“A to the power of X equals B”
log to the base a of b equals x
log(xy) =
log(x) + log(y)
log(x/y)
log(x)-log(y)
log to the base a of a =
1
log1/y =
-logy
log1=
0
degrees to radians
a*π/180
radians to degrees
a*180/π
arc length formula degrees
ϴ/360 * 2πr
arc length formula radians
rϴ
sector area formula degrees
ϴ/360 * πr^2
sector area formula radians
1/2r^2ϴ
Small angle approximation sine
ϴ = sinϴ = tanϴ
small angle approximation cos
cosϴ = 1 - 1/2ϴ^2
inputs of a function
domain
outputs of a function
range
composite functions
work from inside out
a composite function has a domain…
of the first function
for an inverse function to exist
the function must be one-to-one
geometric relationship between inverse function and function
reflected in line y=x (domain and range swap)
period of a sequence
how often the sequence repeats
increasing sequence
every term is greater than the previous term
decreasing sequence
every term is less than the previous term
diverging sequence
the difference between each term gets greater away from a point
converging sequence
the difference between each term gets less towards a point
sum of an arithmetic sequence
s = 1/2n (a+l)
term in an arithmetic sequence (last term)
l = a + (n-1)d
example of a geometric sequence
a, ar, ar^2, ar^3, ar^4…
sum of terms in a geometric sequence
s = (a(1-r^n))/(1-r)
what happens if the common ratio is between -1 and 1
r^n tends to 0 and n tend to infinity
so s = a/(1-r)
series converges and has a sum to infinity
sketching y=|f(x)|
swap all the negative y values to positive y values
sketching y =f(|x|)
get rid of -x graph and mirror positive x in the y axis
Convex curve
Curve underneath chord
F’‘(X)>0
Concave curve
Curve above chord
F’‘(X)<0
chain rule
dy/dx = dy/du * du/dx
Product rule
Vdu+Udv
Quotient rule
(Vdu- udv )/v^2
Sin cos identity
Sin^2x+cos^2x = 1
Tan sin cos identity
Tanx = sinx/cosx
Tan sec identity
Tan^x + 1 = sec^2x
Cot cosec identity
1+cot^2x = cosec^2x
compound angle formulae sin
sin(θ+ϕ) = sinθcosϕ + sinϕcosθ
compound angle formulae cos
cos(θ+ϕ) = cosθcosϕ - sinθsinϕ
compound angle formulae tan
tan(θ+ϕ) = (tanθ+tanϕ)/(1-tanθtanϕ)
Double angle formulae sin
sin(2θ) = 2sinθcosθ
Double angle formulae cos
cos(2θ)= cos^2(θ) - sin^2(θ)
Double Angle formulae tan
tan(2θ) = 2tanθ / 1- tan^2(θ)
compound angle formulae proof sin
triangle with obtuse angle at C
area of ABC = area of ADC + area of DBC
compound angle formulae proof cos
let θ = 90-1 in sin compound formula
d/dx lnx
1/x
d/dx a^x
a^x*lna
d/dx e^x
e^x
d/dx cosx
-sinx
d/dx sinx
cosx
d/dx tanx
sec^2x
d/dx cotx
-cosec^x
d/dx secx
sec xtanx
d/dx cosecx
-cosecxcotx
differentiating implicitly
- differentiate with respect to x
- if y term differentiate and multiple by dy/dx
- if xy product use product rule and multiply by dy/dx
- factor out the dy/dx
turning point from implicit differentiation
solve simultaneously with the equation of the curve
∫f’(x)/f’‘(x) =
ln|f(x)| + c
∫sinkx dx
-1/k coskx +c
∫coskx dx
1/k sinkx + c
∫ sec^2 kx dx
1/k tan kx + c
∫e^kx dx
1/ke^kx + c
∫1/x dx
ln|x| + c
parametric -> cartesian method 1
- rearrange one equation in terms of the parameter
- substitute into the second equation
parametric -> cartesian method 2
- add the equations
- rearrange this for the parameter
- substitute into either equation
parametric equations for a circle (centre (a,b))
x = a + rcosθ y = b + rsinθ
parametric differentiation (parameter t)
dy/dx = (dy/dt) / (dx/dt)
parametric integration
∫ydx = ∫y* (dx/dt) dt
change of sign methods
find a solution to f(x) by finding the range where the function changes sign
drawbacks of change of sign methods
- if the curve touches the axis it won’t work
- if roots are close together it is easy to miss changes in sign
- if there is discontinuity in the function the method may say there are false roots
fixed point iteration
rearrange the function into the form x = g(x)
input values of x into g(x) to find the next value of x until the outputted value is repeating
what does the iterative formula do?
instead of looking for where y=f(x) crosses the x axis look for where y=g(x) intersects with the line y=x
drawing spirals / staircases
- draw y = g(x) and y=x
- mark this initial estimate
- draw a vertical line to the curve
- draw a horizontal line to y=x, mark the x value
- repeat atleast 5 times
Newton Raphson Method
- draw a tangent to the curve
- use a triangle to find the gradient of the tangent, this can be written as f’(x)
f(x1)/ x1 - x2 - rearrange for the estimated root value (x2)
- iterate over to find the expected root
Trapezium rule
A = 1/2h [(y0 + yn) + 2(y1 + y2 + …. + yn-1)
trapezium rule overestimate
convex curve
trapezium rule underestimate
concave curve
upper/lower bounds for trapezium rule
use square with different corners touching the curve
