Unit Four Notes--Application of Change

Unit Four Notes--Application of Change - Brief Calculus...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 20

Unit Four Notes--Application of Change - Brief Calculus...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online