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Math Bridging Course Tutorial 3Chris TC Wong30/8/20121/9/2012
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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
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Review on Maximum and Minimum Concept• Existence of Maximum and Minimum
• For this function, does global Maximum exists on…• [-5,5]• (-5,5)
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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)
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Strange things• Closed?• Bounded?• Continuous function?
• V.s.• Open (Supremum and Infimum)• Unbounded (what is infinity ?)• Not continuous function (Where is the “break point”?)
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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
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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 +
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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.
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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}
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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 + + +
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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
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Exercises
• g• h• p• q
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Q&A