pure core AS Flashcards
1
Q
A^m/n
A
(n√a)^m
2
Q
discriminant
A
b^2 - 4ac
3
Q
b^2 - 4ac < 0
A
no real roots
4
Q
b^2 - 4ac = 0
A
1 real root
5
Q
b^2 - 4ac > 0
A
2 real roots
6
Q
dotted line
A
<>
7
Q
solid line
A
<= >=
8
Q
[]
A
included in the range
9
Q
()
A
not included in the range
10
Q
midpoint
A
( (x1+x2)/2 , (y1+y2)/2)
11
Q
length of line
A
√(x1-x2)^2 + (y1-y2)^2
12
Q
the angle in a semi circle is always
A
a right angle
13
Q
the perpendicular line from the centre of a circle to a chord
A
perpendicularly bisects the chord
14
Q
the tangent to a circle at a point
A
is perpendicular to the radius through that point
15
Q
tanx =
A
sinx/cosx
16
Q
sin^2 x =
A
1 - cos^2 x
17
Q
y = f(x) + 3
A
translation 3 in the y direction
18
Q
y = f(x+3)
A
translation -3 in the x direction
19
Q
y = 3f(x)
A
stretch of scale factor 3 in the y direction
20
Q
y = f(3x)
A
stretch of scale factor 1/3 in the x direction
21
Q
y = -f(x)
A
reflection in the x axis
22
Q
y = f(-x)
A
reflection in the y axis
23
Q
nCr
A
n! / r!(n-r)!
24
Q
normal to curve
A
perpendicular to tangent at particular point
25
vector polar form
(r, θ)
26
vector/component form
(X, Y)
| Xi + Yj
27
position vector
starts at the origin
28
unit vector
magnitude of 1
| divide a direction by its magnitude to get its unit vector
29
vectors with common multiples
are parallel
30
vectors are perpendicular if
their dot/scalar product equals zero
31
Inverse of
| "A to the power of X equals B"
log to the base a of b equals x
32
log(xy) =
log(x) + log(y)
33
log(x/y)
log(x)-log(y)
34
log to the base a of a =
1
35
log1/y =
-logy
36
log1=
0
37
degrees to radians
a*π/180
38
radians to degrees
a*180/π
39
arc length formula degrees
ϴ/360 * 2πr
40
arc length formula radians
rϴ
41
sector area formula degrees
ϴ/360 * πr^2
42
sector area formula radians
1/2r^2ϴ
43
Small angle approximation sine
ϴ = sinϴ = tanϴ
44
small angle approximation cos
cosϴ = 1 - 1/2ϴ^2
45
inputs of a function
domain
46
outputs of a function
range
47
composite functions
work from inside out
48
a composite function has a domain...
of the first function
49
for an inverse function to exist
the function must be one-to-one
50
geometric relationship between inverse function and function
reflected in line y=x (domain and range swap)
51
period of a sequence
how often the sequence repeats
52
increasing sequence
every term is greater than the previous term
53
decreasing sequence
every term is less than the previous term
54
diverging sequence
the difference between each term gets greater away from a point
55
converging sequence
the difference between each term gets less towards a point
56
sum of an arithmetic sequence
s = 1/2n (a+l)
57
term in an arithmetic sequence (last term)
l = a + (n-1)d
58
example of a geometric sequence
a, ar, ar^2, ar^3, ar^4...
59
sum of terms in a geometric sequence
s = (a(1-r^n))/(1-r)
60
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
61
sketching y=|f(x)|
swap all the negative y values to positive y values
62
sketching y =f(|x|)
get rid of -x graph and mirror positive x in the y axis
63
Convex curve
Curve underneath chord
| F''(X)>0
64
Concave curve
Curve above chord
| F''(X)<0
65
chain rule
dy/dx = dy/du * du/dx
66
Product rule
Vdu+Udv
67
Quotient rule
(Vdu- udv )/v^2
68
Sin cos identity
Sin^2x+cos^2x = 1
69
Tan sin cos identity
Tanx = sinx/cosx
70
Tan sec identity
Tan^x + 1 = sec^2x
71
Cot cosec identity
1+cot^2x = cosec^2x
72
compound angle formulae sin
sin(θ+ϕ) = sinθcosϕ + sinϕcosθ
73
compound angle formulae cos
cos(θ+ϕ) = cosθcosϕ - sinθsinϕ
74
compound angle formulae tan
tan(θ+ϕ) = (tanθ+tanϕ)/(1-tanθtanϕ)
75
Double angle formulae sin
sin(2θ) = 2sinθcosθ
76
Double angle formulae cos
cos(2θ)= cos^2(θ) - sin^2(θ)
77
Double Angle formulae tan
tan(2θ) = 2tanθ / 1- tan^2(θ)
78
compound angle formulae proof sin
triangle with obtuse angle at C
| area of ABC = area of ADC + area of DBC
79
compound angle formulae proof cos
let θ = 90-1 in sin compound formula
80
d/dx lnx
1/x
81
d/dx a^x
a^x*lna
82
d/dx e^x
e^x
83
d/dx cosx
-sinx
84
d/dx sinx
cosx
85
d/dx tanx
sec^2x
86
d/dx cotx
-cosec^x
87
d/dx secx
sec xtanx
88
d/dx cosecx
-cosecxcotx
89
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
90
turning point from implicit differentiation
solve simultaneously with the equation of the curve
91
∫f'(x)/f''(x) =
ln|f(x)| + c
92
∫sinkx dx
-1/k coskx +c
93
∫coskx dx
1/k sinkx + c
94
∫ sec^2 kx dx
1/k tan kx + c
95
∫e^kx dx
1/ke^kx + c
96
∫1/x dx
ln|x| + c
97
parametric -> cartesian method 1
- rearrange one equation in terms of the parameter
| - substitute into the second equation
98
parametric -> cartesian method 2
- add the equations
- rearrange this for the parameter
- substitute into either equation
99
parametric equations for a circle (centre (a,b))
```
x = a + rcosθ
y = b + rsinθ
```
100
parametric differentiation (parameter t)
dy/dx = (dy/dt) / (dx/dt)
101
parametric integration
∫ydx = ∫y* (dx/dt) dt
102
change of sign methods
find a solution to f(x) by finding the range where the function changes sign
103
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
104
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
105
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
106
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
107
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
108
Trapezium rule
A = 1/2h [(y0 + yn) + 2(y1 + y2 + .... + yn-1)
109
trapezium rule overestimate
convex curve
110
trapezium rule underestimate
concave curve
111
upper/lower bounds for trapezium rule
use square with different corners touching the curve
