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Unformatted text preview: 3.1 Extrema on an Interval Calculus Home Page Problems for 3.1 Title: intro (1 of 13) Extrema on an Interval Calculus Home Page Describe what happens to the graph on [1, 4] Problems for 3.1 Title: consider y values (2 of 13) Extrema on an Interval
Describe what happens to the graph on [1, 4] Beyond [1, 4], for all x, tendency is: We need a way to describe the points where the graph of f turns: from up to down, from down to up. Calculus Home Page Problems for 3.1 Title: student response (3 of 13) 3.1 Extrema on an Interval Definition of Extrema: Let f be defined on and interval I containing c. Note that I can be open or closed.) ( 1. f(c) is a minimum of fon I if f(c) < f(x) for all x in . I 2. f(c) is a maximum of fon I if f(c) > f(x) for all x in . I The maximum and minimum values are extreme values. Note that these are values of f (or y), not values of x. The minimum and maximum on the interval are called the absolute minimum and absolute maximum. Extreme Value Theorem
If f is continuous on a losed interval [a, b], then f has both a c minimum and a maximum on the interval. Definition of Relative Extrema: 1. If there is an pen interval contain c on which f(c) is a maximum o , then f(c) is called a relative maximum of f. Or f has a relative max at (c,f(c)). 2. If there is an pen interval contain c on which f(c) is a minimum o , then f(c) is called a relative minimum of f. Or f has a relative min at (c,f(c)). Calculus Home Page Problems for 3.1 Title: def: extrema (4 of 13) Problem 1. At each labeled point, determine if there is a relative min or max, or an absolute min or max. As drawn in book, neither, as drawn here it appears to be a rel. max. Calculus Home Page Problems for 3.1 Title: overview example (5 of 13) Or CV, critical value Calculus Home Page Problems for 3.1 Title: Critical Number (6 of 13) Calculus Home Page Problems for 3.1 Title: f' not exist at x=0 (7 of 13) There are 3 Critical Numbers Calculus Home Page Problems for 3.1 Title: Finding CNs (8 of 13) Problems for 3.1 Calculus Home Page Title: Oct 176:09 PM (9 of 13) Calculus Home Page Problems for 3.1 Title: Oct 176:18 PM (10 of 13) Calculus Home Page Problems for 3.1 Title: Oct 176:29 PM (11 of 13) Calculus Home Page
Problems for 3.1 Title: Oct 176:36 PM (12 of 13) Calculus Home Page
Problems for 3.1 Title: Oct 176:40 PM (13 of 13) ...
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