7dot7FindingK - Finding K Derrick Smith Introduction The...

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Finding K Derrick Smith Introduction The error bounds for the integration approximation techniques you learned in class need you to find a value of K such that ( ) f x K , ( ) f x K ′′ or ( ) 4 ( ) f x K for all x in [ , ] a b . After finishing this supplement, you will know the two ways to find K : the High Road and the Quick and Dirty Method. The High Road All of the approximation methods require finding an upper bound on some function. For now, forget that the function is some derivative and just concentrate on how to find the upper bond of a function. The High Road is based on the following theorem from Calculus 1: THE MAXIMUM VALUE THEOREM If f is continuous on the closed interval[ , ] a b , then f attains is absolute maximum value ( ) f c at some number c in the interval. Moreover, either c is one of the endpoints of the interval, or c is a critical number of f . (Recall that c is a critical number of f iff ( ) 0 f c = or ( ) f c is undefined.) So, to find K for an arbitrary function f on a closed interval, you need to do two things: 1) Find all places in the interval where the derivative of f is 0. 2) Look at values of f for all the points you found in 1) and at the endpoints. 3) Let K be the absolute value of f that has the largest magnitude. Let’s do a few examples. Example 1:
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7dot7FindingK - Finding K Derrick Smith Introduction The...

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