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Math Bridging Course Tutorial 3

Math Bridging Course Tutorial 3. Chris TC Wong 30/8/2012 1/9/2012. 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

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Math Bridging Course Tutorial 3

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  1. Math Bridging Course Tutorial 3 Chris TC Wong 30/8/2012 1/9/2012

  2. 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

  3. Review on Maximum and Minimum Concept • Existence of Maximum and Minimum • For this function, does global Maximum exists on… • [-5,5] • (-5,5)

  4. Extreme value theorem • If a real-valued functionf 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)

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

  6. 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

  7. First derivative test • Compute by hand? • Make use of a table can speed things up • Examples:

  8. 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.

  9. 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}

  10. Second Derivative test • It is just first derivative test with extra thing done but require much more. • Same example

  11. 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

  12. Exercises • g • h • p • q

  13. Q&A

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