×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

1 review

10

0

2

# MUSIC THEORY Music 16

UCI
GPA 3.54

Staff

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
2
WORDS
KARMA
25 ?

1 review
Kristin Honowitz

## Popular in Music

This 2 page Class Notes was uploaded by Shad Aufderhar on Saturday September 12, 2015. The Class Notes belongs to Music 16 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/201837/music-16-university-of-california-irvine in Music at University of California - Irvine.

×

## Reviews for MUSIC THEORY

-Kristin Honowitz

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/12/15
LECTURE 16 21 Critical Points Properties and Applications 211 Absolute Maxima and Minima De nition De nition Suppose the domain of a function f includes a closed interval lab A point c is said to be 0 an absolute maximum of f on ab if fc Z for all z in lab 0 an absolute minimum of f on ab if fc g for all z in lab Note Absolute maxima and minima of a function on an open interval and on an arbitrary set of points are de ned analogously eg see absolute mazima in SampM Examples 0 on the interval 047r the function sinz has two absolute maxima and two absolute minima nd them 0 on the interval 12 the function fltzgti 171 for 0 zlt1 7 1 for 1 g I g 2 has absolute maxima at the point z 0 and at all points in the interval 12 on the zaxis but no absolute minima 212 Finite Differences De nition Suppose a function f is de ned in a neighborhood of a point 6 Exercise This exercise is an ef cient way to clarify the limit formulas for the rightsided and leftsided deriva tives Let h be a small positive number such that the closed interval 6 7 hc h is contained in the domain of f Write down the slope of the line passing through 0 the points 6 and C h fc o the points 6 and c 7 h fc 7 De nition For a small It gt 0 the expressions fC h HQ 11 and f C 7 f C i h h are called respectively the right7sia ea and left7sia ea nite ali e39rences of f at c with step Another term for these is a 7ight7sia ea left7sia ea fc h7a i e39rence Note Finite differences have the following geometric interpretation A rightsided nite f c hdifference is the slope of the line passing through the points CfC 6hfch Similarly a leftsided fc hdifference is the slope of the line passing through the points CfC 6 hfC 0 Note that for both differences It is positive 213 The Derivative at an Extremum Analysis by the lst Derivative Theorem Maxima of Di 39erentiable Functions Suppose a function is differentiable on the open interval lab and suppose c is a point in lab at which f attains a a maximum on lab ie fc 2 for all z in lab 24 THEN f c 0 Veri cation To see this consider the rightsided nite difference f6 h 0 f lt25 where h is positive and small enough that C h lies in la b By 24 we have fc 1117 m S 0 26 Letting h tend to 0 from the right and taking the limit we obtain by Nonpositivity Lecture 5 section 462 that NC S 0 27 Carrying out a similar procedure for the leftsided nite difference fc ee my b0 11 we obtain NC 2 0 28 The inequalities 27 and 28 must hold simultaneously and this is possible only if f C 0 Corollary Minima of Di 39erentiable Functions Suppose a function is differentiable on the open interval lab and suppose c is a point in lab at which f attains a minimum on lab ie fc g for all z in lab 29 THEN f c 0 Veri cation To see this introduce the function Then f has a maximum at 6 hence satis es f c 0 as we veri ed above It follows that f c 0 Terminology Extrema Absolute maxima and minima are collectively called absolute extrema How to Picture the Results in this Section Drop a horizontal bar onto a 2D hill77 The bar will hit the highest point of the hill How to Remember the Results in this Section At an absolute extremum the tangent is horizontal

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Amaris Trozzo George Washington University

#### "I made \$350 in just two days after posting my first study guide."

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com