微积分 修改版SECTION 3.1.ppt

SECTION 3.1 Increasing and Decreasing Functions; Relative Extrema 极点 ?. Increasing and Decreasing Functions CHAPTER 3 Additional Applications of the Derivative Assume that f (x) is a function defined on the interval (a , b), then f (x) is increasing (递增的) on (a , b) if for any x1, x2 in (a , b), f (x1) f (x2) whenever x1 x2 . Assume that f (x) is a function defined on the interval (a , b), then f (x) is decreasing (递减的) on (a , b) if for any x1, x2 in (a , b), f (x1) f (x2) whenever x1 x2 . Def (定义) 几何意义： 切线斜率0 切线斜率0 Theorem (定理) Assume that f (x) is differentiable (可导的) on (a , b), then 注意：1. 上述结论反之不成立。 2. 函数增减区间可能的端点： （如：y = x3） Analysis： Procedure 1 for Determining Intervals of Increase and decrease for f (x) Step1. Find all values of x for which f′ (x) = 0 or f′ (x) is not defined to get some open intervals. Step2. Decide whether f ′(x) is positive or negative on each above intervals.（注：当f ′(x)可分解为一次因式乘积时，常用 “穿针引线”法） Example 3.1.2 (P201) Find the intervals of increase and decrease for the function Solution 1 Step1 We find that which is not defined at x = 2 and has f ′(x) = 0 at x = 0 and x = 4. Thus , we get 4 open intervals on which f ′(x) ≠ 0 : (-∞,0), (0,2), (2,4), (4,+ ∞). Step2 It’s easy to see that It follows that 先对x(x-4)用“穿针引线”法判定符号 We conclude that f (x) is increasing for x 0 and for x 4 and that it is decreasing for 0 x 2 and for 2 x 4. Procedure 2 for Determining Intervals of Increase and decrease for f (x) Step1. Find all values of x for which f ′(x) = 0 or f ′(x) is not continuous to get some open intervals. Step2. Choose a test number c from each interval a x b determined in step 1 and evaluate f ′(c). Then, 取数检验法 Analysis： Solution 2 Find the intervals of increase and decrease for the function Step1 We find that which is discontinuous at x = 2 and has f ′(x) = 0