微积分SECTION 3.2.pptVIP

SECTION 3.2 Concavity and Points of Inflection 1. Concavity 凹向 Def (定义) If the function f (x) is differentiable on the interval a x b , then the graph of f (x) is concave upward (凹上的) on a x b if f ′(x) is increasing on a x b Slope=f ′(x) concave downward (凹下的) ？ Informally, a concave upward portion of graph “holds water”, while a concave downward portion “spills water”. f ′(x) is increasing on a x b Theorem (定理) Suppose f (x) has the second derivative on a x b , then 注意： 1. 上述结论反之不成立。 如：y = x4 2. Do not confuse the concavity of a graph with its “direction” (rising or falling). y〞=12 x2≥0 A function f may be increasing or decreasing on an interval regardless of whether its graph is concave upward or concave downward on that interval. Procedure for Determining Intervals of Concavity for f (x) Step1. Find all values of x for which f ″(x) = 0 or f ″(x) does not exist to get some open intervals. (定分点) Step2. Check the sign of f ″(x) on each above intervals. Determine intervals of concavity for the function f (x) = 2x6 - 5x4 + 7x - 3 Example 3.2.1 (P217) Solution (板书) П. Inflection Points 拐点 Def. A point P(c , f (c)) is called an inflection point of f (x) if f (x) is continuous at x = c and the concavity changes at P. (函数在x = c 连续，在P点凹向改变。) Theorem If a point (c , f (c)) is an inflection point of f (x), then f ″(c) = 0 or f ″(c) does not exist. (拐点的可能范围) 注： f (x)的拐点是f ′(x)的极点。（？） 如：y = x4 Procedure for Finding the Inflection Points for Continuous f (x) （求拐点的步骤） Step1 Find all numbers c in the domain of f (x) where f ″(c) = 0 or f ″(c) does not exist. Step2 For each number c, Check the sign of f ″(x) on each side of c. Example 3.2.2 (P219) In each case, find all inflection points of the given function. a. f (x) = 3x5 - 5x4 - 1 b. g (x) = x1/3 Ш. Curve Sketching with the Second Derivative 利用二阶导数作图 Geometrically, inflection points occur at “twists” on a g