Understanding Extrema on Closed Intervals in Calculus: A Comprehensive Guide
Explore the concept of extrema on closed intervals in calculus, including definitions, critical numbers, relative and absolute extrema, and how to find extrema using step-by-step methods. Learn how to determine minimum and maximum values of functions on specified intervals effectively.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Chapter Chapter 3 3.1: Extrema On An Interval Extrema On An Interval .1: HONORS CALCULUS/CALCULUS HONORS CALCULUS/CALCULUS
Definition of Extrema 1. ? ? ?? ??? ??????? ?? ? ?? ? ?? ? ? ? ? ? ?? ?. 2. ? ? ?? ??? ??????? ?? ? ?? ? ?? ? ? ? ? ? ?? ?. If f is continuous on a closed interval [a, b], then f has both a minimum and a maximum on the interval. If f is defined at c, then c is called a CRITICAL NUMBER of f if and ? ? = ? or if ? ? ?? ????????? ?? ?. RELATIVE EXTREMA occur only at CRITICAL NUMBERS. If f has a relative min or a relative max at ? = ?, then c is a critical number of f.
Finding Extrema On A Closed Interval To find the extrema of a continuous function f on a Closed Interval [a, b], use the following steps: 1. Find the critical numbers of f in (a, b) 2. Evaluate f at each critical number in (a, b) 3. Evaluate f at each endpoint of [a, b] 4. The Least of These Values is the Minimum/The Greatest is the Maximum
Ex. 1) Find the Absolute Extrema of ? ? = ??? ??? on the interval [ ?,?].
Ex. 2) Find the Absolute Extrema of ? ? = ?? ???/? on the interval [ ?,?].
Ex. 3) Find the Absolute Extrema of ? ? = ????? ????? on the interval [?,??].
