=>
It follows that
<=>
If and only if
N
Natural numbers
POSITIVE integers
Z
Integer
Q
quotients / rational numbers
Expressed as a FRACTION
R
Real numbers
not IMAGINARY
a ^ m/n
n rt(a^ m)
b^2 - 4ac > 0
2 disctint roots
b^2 - 4ac = 0
1 repeated root
b^2 - 4ac < 0
No roots
Turning point (a,b)
(x - a) + b
x = (quadratic formula)
-b +- sqrt(b^2 -4ac) / 2a
=< when region drawing
SOLID line
< when region drawing
DOTTED line
=< on number line
FILLED dot
< on number line
OPEN dot
=< in interval notation
SQUARE bracket [
< in interval notation
CURVED bracket (
Interval bracket for infinity
CURVED ( as <
define polynomial
Finite expression with N indices
state factor theorem
If f(a) = 0 then (x-a) is a factor
When to use identity symbol
Simplifying expressions
y = k / x
top right, bottom left
Gets flatter when k increases
fg(x) meaning
f of g(x)
Inverse function notation
f ^-1 (x)
Inverse function of e ^x
ln x
midpoint
( x1 + x2 / 2 , y1 + y2 / 2)
Collinear definition
Of the SAME LINE
tangent definition
Gradient at one point on a curve
When b^2 - 4ac = 0
Point which only touches at one point
Normal defintion
Line 90 degrees to tangent at one point
r^2 =
(x-a) + (y-b) with centre (a,b)
Define circumcircle
unique circle around 3 vertices of a triangle
Why is a = 1 not an exponential graph
Because 1^x is always 1
a^x = b
loga b = x
loga (a^x) =
x
a ^ (loga x) =
x
e =
2.71819
differentiate e^kx
ke^kx
Exponential decay
Ae^kx
Exponential decay
Ae^-kx
Differentiate e^-kx
-ke^kx
loga 1 =
0
log a (1/x) =
- loga x
log (x + y) =
CAN’T BE SIMPLIFIED
y = (exponential graph)
A (coefficient of e)
Make linear graph with logs
Logs of both sides
Separate RHS
constant is gradient
Variables are axis
loga y = kx + loga b (what on axis)
y = loga y
x = loga b
gradient = k
Derivation from first principles
lim h-> 0 f(x + h) - f(x) / h
As h -> 0 …
nh -> 0 and nh^2 -> 0
Find equation on normal at point (x,y)
dy/dx to find tangent
x -1/grad is gradient
Solve y=mx+c
is interval () strict or not?
NOT
Is interval[] strict or not
STRICT
find if maximum or minimum
d2y / dx2 using value for x
If n > 0 then MINIMUM
If n < 0 then MAXIMUM
If n = 0 then point of INFLECTION/ MAX/ MIN (use table)
Define dV / dr
rate of change in volume WRT radius
|a| =
sqrt( Δx^2 + Δy^2 + Δz^2)
a hat = (unit vector)
a / |a|
Find vector angle with y axis
cosθ = y / |a|
Find vector angle with x axis
cosθ = x / |a|
Find vector angle with z axis
cosθ = z / |a|
