微积分英文课件PPT (5).ppt

The best way to learn mathematics is to do mathematics Chapter 4 Applications of differentiation 4.1 Maximum and minimum values 4.2 The Mean value Theorem 4.3 How Derivatives Affect the shape of a Graph 4.4 Indeterminate Forms and L’ Hospital’s Rule 4.5 Summary of Curve Sketching 4.6 Graphing with Calculus and Calculators 4.7 Optimization Problems 4.8 Applications to business and Economics 4.9 Newton’s Method 4.10 Antiderivatives 4.1 Maximum and minimum values Example : Example: Example: Example: The maximum and minimum values of f are called the extreme values of f. c d (c,f(c)) (d,f(d)) We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. Example This function has no maximum or minimum 机动 目录 上页 下页 返回 结束 Example This function has no maximum or minimum Caution: The conditions cannot be weakened. Example Caution: The conditions is sufficient, but not necessary. This function has minimum value f(-1)=-4 This function has maximum value f(-4) 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. c Proof: Without loss of generality, we consider local Maximum: We have if if Therefore Because exists, Thus 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 cr