Math Bridging Course Tutorial 3Chris TC Wong30/8/20121/9/2012

Review on Maximum and Minimum Concept• Do you know what does it mean to be bigger/smaller?• Introduction to Metric:

• is a function satisfying some conditions:• d(x,y)=0 x=y

• The distance between two elements is zero iff they are the same thing• d(x,y)>=0 for any x,y

• distance suppose to be greater than 0• d(x,y)=d(y,x)

• symmetric• d(x,y)+d(y,z)>=d(x,z)

• Triangle inequality

Review on Maximum and Minimum Concept• Existence of Maximum and Minimum

• For this function, does global Maximum exists on…• [-5,5]• (-5,5)

Extreme value theorem• If a real-valued function f is continuous in

the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in [a,b] such that:

• f ( c ) <= f ( x ) < f ( d ) for any x in [a,b]• (http://en.wikipedia.org/wiki/Extreme_value_theorem)

Strange things• Closed?• Bounded?• Continuous function?

• V.s.• Open (Supremum and Infimum)• Unbounded (what is infinity ?)• Not continuous function (Where is the “break point”?)

Assume things are nice• The function is differentiable. (Hence also continuous)

• i.e. first derivative exists.• First derivative test.

• Nicer : the function is twice differentiable• i.e. second derivative exists.• Second derivative test.

• Very Nice : the function is “smooth”• i.e. Derivative of any order exists

First derivative test• Compute by hand?

• Make use of a table can speed things up• Examples:

X<-1 X=-1 -1<X<1 X=1 X>1

f(x) ↗ 2/3 ↘ -2/3 ↗

f’(x) + 0 - 0 +

Caution :• What if the first derivative does not exist on certain point?

• E.g. • Ignore the point.• (What if the first derivative does not exists on the whole interval?)• (http://en.wikipedia.org/wiki/File:WeierstrassFunction.svg)

• How about boundary cases?• E.g.

Algorithm• Read carefully about the function• Differentiate the function• Finding local max/min• Compute function value on Boundary points• Compute function value on non-differentiable points• Return max{f(BoundaryPts),f(non-d-able-pts),localMaxs} and

min{f(BoundaryPts),f(non-d-able-pts),localMins}

Second Derivative test• It is just first derivative test with extra thing done but require

much more.• Same example

X<-1 x=-1 -1<x<0 X=0 0<X<1 X=1 X>1

f(x) ↗ 2/3 ↘ 0 ↘ -2/3 ↗

f’(x) + 0 - - - 0 +

f‘’(x) - - - 0 + + +

Who cares about point of inflexion?

• Second derivative only provide some clues on it.• Point of inflexion does not necessarily appears at points where

f’’(x)=0• Remember the case which f’(x) does not exists?

• Consider this function :

• Why brother using second derivative test?• Hint : Sometimes the modeled world just isn’t perfect.

• Let us face something like this : for function

Exercises

• g• h• p• q

Q&A