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Unformatted text preview: Brief Calculus Supplement Notes 4.1 4.4 Applications of Derivatives Section OneExtrema On an Interval Review Dot/Cross from Unit I Ex 1R: (x 3)(x + 2) < 0 Ex 2R: (2x 3)(x + 5) 2 > D Extrema Let f be defined on an interval I containing c . 1. f(c) is the minimum value of f on I if f(c) < f(x) for all x in I. Also, (c,f(c)) is a minimum point. 2. f(c) is the maximum value of f on I if f(c) > f(x) for all x in I. Also, (c,f(c)) is a maximum point. The min and max of a function on an interval are the extreme values , or extrema , of the function on the interval. The min. and max. of a function on an interval are also called the absolute min and absolute max on the interval. T The Extreme Value Theorem If f is continuous on [a, b], then f has both an absolute minimum and an absolute maximum on the interval. D Relative Extrema 1. If there is an open interval containing c on which f(c) is a maximum value, then (c, f(c)) is a relative maximum point. 2. If there is an open interval containing c on which f(c) is a minimum value, then (c, f(c)) is a relative minimum point. D Critical Number Let f be defined at c. If ( 29 c f = 0 or if f is not differentiable at c, then c is a critical number of f. T Relative Extrema Occur Only at Critical Numbers If f has a relative minimum or relative maximum at x = c, then c is a critical number of f, and c is a zero of ( 29 x f . Guidelines for Finding Extrema on a Closed Interval: 1. Find the critical numbers of f in (a,b) 2. Evaluate f at each critical number in (a,b) 3. The least of these output values is the minimum. The greatest of these output values is the maximum. 4. The critical number (the x value) together with the minimum or maximum value(the y value) is the minimum or maximum point. Brief Calculus Notes Increasing and Decreasing Functions/1 st Derivative Test ***Dot/Cross used here*** D Increasing Function f is increasing on I if for any two inputs, 2 1 x and x in I such that 2 1 x x < implies ( 29 ( 29 2 1 x f x f < . D Decreasing Function f is increasing on I if for any two inputs, 2 1 x and x in I such that 2 1 x x < implies ( 29 ( 29 2 1 x f x f . T Test for Increasing/Decreasing Function 1. If ( 29 x f , 2200 ) , ( b a x f is increasing on [a, b] 2. If ( 29 < x f , 2200 ) , ( b a x f is decreasing on [a, b] 3. If ( 29 = x f , 2200 ) , ( b a x f is constant on [a, b] Guidelines for Finding Intervals on Which a Function is Increasing/Decreasing Let f be continuous on the interval (a,b). To find the open intervals on which f is increasing or decreasing, use the following steps. 1. Locate the critical number(s) of f in (a,b), and use these numbers to determine test intervals using dot/cross....
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This note was uploaded on 02/28/2012 for the course MAT MAT 212 taught by Professor Michaeldereck during the Spring '12 term at Mesa CC.
- Spring '12