微积分英文课件：Chapter3 Applications of Derivatives.pptVIP

Chapter3 Applications of Derivatives 3.1 Extreme Values of Functions The maximum and minimum values of f are called the extreme values of f. Example Problem: The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and minimum value. But it does not tell us how to find these extreme values. We start by looking for local extreme values. Caution: 1) The condition is just sufficient. We cannot expect to locate extreme values simply by setting and solving for x. For example: 2)There may be an extreme value even when does not exist. For example: Fermat’s Theorem does suggest that we should at least start looking for extreme values of f at the numbers c where or does not exist. Definition: A critical number of a function f is a number c in the domain of f such that either or does not exist. Conclusion: If f has a local maximum or minimum at c, then c is a critical number of f. To find an absolute maximum or minimum of a continuous function on a closed interval , we note that either it is local {in which case it occur at a critical number by the above conclusion} or it occurs at an endpoint of the interval. The Closed Interval Method To find the absolute maximum and minimum values of a continous function f on a closed interval [a, b]: 1) Find the values of f at the critical numbers of f in (a,b). 2) Find the values of f at the endpoints of the interval. 3) The largest of the values from step 1) and step 2) is the absolute maximum value; the smallest of these values is the absolute minimum value. Example Find the absolute maximum and minimum values of the function Solution: Since f is continuous on the given closed interval, we can use the Closed Interval Method: 4.2 The Mean Value Theorem Rolle’ s Theorem: Let f be a function that satisfy the following three hyp