tailieunhanh - Calculus and its applications: 2.2

"Calculus and its applications: " - Using Second derivatives to find maximum and minimum values and sketch graphs have objective: find the relative extrema of a function using the second-derivative test, sketch the graph of a continuous function. | 2012 Pearson Education, Inc. All rights reserved Slide Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVE Find the relative extrema of a function using the Second-Derivative Test. Sketch the graph of a continuous function. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: Suppose that f is a function whose derivative f exists at every point in an open interval I. Then f is concave up on I if f is concave down on I f is increasing over I. if f is decreasing over I. Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 4: A Test for Concavity 1. If f (x) > 0 on an interval I, then the graph of f is concave up. ( f is increasing, so f is turning up on I.) 2. If f (x) Slide THEOREM 5: The Second Derivative Test for Relative Extrema Suppose that f is differentiable for every x in an open interval (a, b) and that there is a critical value c in (a, b) for which f (c) = 0. Then: 1. f (c) is a relative minimum if f (c) > 0. 2. f (c) is a relative maximum if f (c) p. 216, the concluding sentence of Theorem 5 says “f(x)” instead of f(c). 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Graph the function f given by and find the relative extrema. 1st find f (x) and f (x). Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs 2012 Pearson Education, Inc. All rights reserved Slide Example 1 | 2012 Pearson Education, Inc. All rights reserved Slide Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVE Find the relative extrema of a function using the Second-Derivative Test. Sketch the graph of a continuous function. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: Suppose that f is a function whose derivative f exists at every point in an open interval I. Then f is concave up on I if f is concave down on I f is increasing over I. if f is decreasing over I. Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 4: A Test for Concavity 1. If f (x) > 0 on an interval I, then the graph of f is concave up. ( f is increasing, so f is turning up on I.) 2. If f (x) < 0 on an interval I, then the graph of f is concave down. ( f is decreasing, so f is turning down on I.) Using Second Derivatives to .