tailieunhanh - Calculus and its applications: 2.4

"Calculus and its applications: " - Using derivatives to find absolute maximum and minimum values have objective: find absolute extrema using maximum-minimum principle 1, find absolute extrema using maximum-minimum principle 2. | 2012 Pearson Education, Inc. All rights reserved Slide Using Derivatives to Find Absolute Maximum and Minimum Values OBJECTIVE Find absolute extrema using Maximum-Minimum Principle 1. Find absolute extrema using Maximum-Minimum Principle 2. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: Suppose that f is a function with domain I. f (c) is an absolute minimum if f (c) ≤ f (x) for all x in I. f (c) is an absolute maximum if f (c) ≥ f (x) for all x in I. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 7: The Extreme Value Theorem A continuous function f defined over a closed interval [a, b] must have an absolute maximum value and an absolute minimum value over [a, b]. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 8: Maximum-Minimum Principle 1 Suppose that f is a continuous function defined over a closed interval [a, b]. To find the absolute maximum and minimum values over [a, b]: a) First find f (x). b) Then determine all critical values in [a, b]. That is, find all c in [a, b] for which f (c) = 0 or f (c) does not exist. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 8: Maximum-Minimum Principle 1 (continued) c) List the values from step (b) and the endpoints of the interval: a, c1, c2, ., cn, b. d) Evaluate f (x) for each value in step (c): f (a), f (c1), f (c2), ., f (cn), f (b). The largest of these is the absolute maximum of f over [a, b]. The smallest of these is the absolute minimum of f over [a, b]. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Find the absolute maximum and minimum values of over the interval [–2, ]. a) b) Note that f (x) exists for . | 2012 Pearson Education, Inc. All rights reserved Slide Using Derivatives to Find Absolute Maximum and Minimum Values OBJECTIVE Find absolute extrema using Maximum-Minimum Principle 1. Find absolute extrema using Maximum-Minimum Principle 2. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: Suppose that f is a function with domain I. f (c) is an absolute minimum if f (c) ≤ f (x) for all x in I. f (c) is an absolute maximum if f (c) ≥ f (x) for all x in I. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 7: The Extreme Value Theorem A continuous function f defined over a closed interval [a, b] must have an absolute maximum value and an absolute minimum value over [a, b]. Using Derivatives to Find Absolute Maximum and Minimum Values 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 8: Maximum-Minimum Principle 1 Suppose that f is a .