pure normal

fixed point behaviour based of derivative

• converges if modulus of derivative is less than one near the root

• diverges if modulus of derivative is more than one near the root

at point where tangent of parametric curve parallel to x axis?

dy/dt = 0

at point which tangent is parallel to y axis ?

dx/dt = 0

integrating parametric

convex curve?

• a curve that curves UPwards is conVEX

• second derivative greater than 0

concave curve?

• a curve that curves DOWNWARDS is conCAVE

• has second derivative less than zero

how to prove a point of inflections is actually a point of inflection

• be careful not to assume a stationary point is a POI when second derivative is zero

• it could be local min, local max, or POI

• POI is when changes from convex to concave, (or vice versa) so second derivative changes sign!!!

how to find limit of a sequence

order for vertical translations

order for two horizontal transformations

modulus inequality

use proof by contradiction to prove there is an infinite number of primes

Assume that there is a largest prime number, P.

Construct another number N that is the product of all the prime numbers up to and including P.

Consider N + 1. This is one greater than a number divisible by all the primes up to and including P, so it cannot be divisible by any of the primes up to and including P.

Therefore N + 1 is either itself prime, or is divisible by primes larger than P.

Either way, a prime larger than P has been discovered which contradicts the premise that there is a largest prime number.

Therefore there are an infinite number of prime numbers.

prove root 2 is irrational

assume root 2 equals p/q, where p and q are two integers with no common factors

then 2 = psquared / q squared

2 q squared = p squared

means p squared is even so p must also be even

p = 2k p squared = 4 k squared

so 4 k squared = 2 q squared

so 2k squared = q squared

means q squared is even so q must also be even

both p and q are even so share a factor of 2 - CONTRADICTION

when is a mapping a function?

if every x value maps to a single y value!!

types of functions

• one-to-one - if every y-value corresponds to only one x-value eg straight line

• many-to-one - if there are some y-values that come from more than one x-value eg quadratic

vertical line test?

• if a mapping is a function, any vertical line will cross its graph at most once.

horizontal line test?

• if a function is one-to-one, any horizontal line will cross the graph at most once.

which functions have inverse functions

ONLY ONE TO ONE!!!

• horizontal line crosses graph at most once!!!!

why does fixed point iterations sometimes not work?

more than one root - if a function has more than one root it can sometimes give wrong root

or diverge away from root