# Increasing and Decreasing Intervals Where Does the Fun End ? - PowerPoint PPT Presentation

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Increasing and Decreasing Intervals Where Does the Fun End ?. Anne Dudley Michael Holtfrerich Joshua Whitney Glendale C ommunity College Glendale, Arizona. Increasing & Decreasing Activity 1. Determine the largest interval on which each function (on the handout) is increasing.

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Increasing and Decreasing Intervals Where Does the Fun End ?

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## Increasing and Decreasing IntervalsWhere Does the Fun End?

Anne Dudley

Michael Holtfrerich

Joshua Whitney

Glendale Community College

Glendale, Arizona

### Increasing & DecreasingActivity 1

Determine the largest interval on which each function (on the handout) is increasing.

• (–1, 1]

• (–1, 1)

• [–1, 1]

• Other

• Other

• Other

### Discussion

If you used closed intervals for your two previous answers, are you OK with the graph being both increasing and decreasing at x = 1?

• Other

• Other

### What Do Textbooks Say?

Ostebee & ZornCalculus, 2ndEdition (p.56)

Definition: Let I denote the interval (a, b).

A function fis increasing on Iif

whenever a < x1 < x2 < b.

fis decreasing on Iifwhenever a < x1 < x2 < b.

Fact: If for all x in I, then f increases on I.

If for all x in I, then f decreases on I.

Graph 1 Increasing (Ostebee & Zorn )Use your clicker to indicate your answer.What is the largest interval on which the function is increasing?

• Other

### What Do Textbooks Say?

Hughes-HallettCalculus, 5th edition (p. 165) and

Swokowski Calculus, 2ndedition (p.147)

Let f(x) be continuous on [a, b], and differentiable on (a, b).

If f’(x) > 0 on a < x < b, then f is increasing on .

If f’(x)≥ 0 on a < x < b, then f is non-decreasing on .

Graph 1 Increasing (Hughes-Hallett)Use your clicker to indicate your answer.What is the largest interval on which the function is increasing?

• Other

### What Do Textbooks Say?

Larson EdwardsCalculus, 5th edition (p. 219)

A function f is increasing on an interval if for any two numbers x1 and x2in the interval, x1 < x2 implies

f(x1) < f(x2).

A function f is decreasing on an interval if for any two numbers x1 and x2in the interval, x1 < x2 implies

f(x1) > f(x2).

### The Problem with “Any”

The following function isnot increasing on [a,b].

But it does fit the precedingdefinition.

For the two numbers

x1< x2, f(x1) < f(x2)

should imply that f(x) is

increasing on [a,b].

### What Do Textbooks Say?

Larson EdwardsCalculus, 5th edition (p. 219) (Theorem)

Let f (x) be continuous on [a, b], and differentiable on (a, b).

If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b].

If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b].

If f’(x) = 0 for all x in (a, b), then f is constant on [a, b].

All textbook examples and the answers in homework are open intervals.

### What Do Textbooks Say?

Cynthia YoungPrecalculus, 1st edition (p. 128)

A function f is increasing on an open interval I if for any x1 and x2 in I, where x1 < x2, then f(x1) < f(x2).

A function f is decreasing on an open intervalI if for any x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).

### Two Differing Ideas

Two Point Idea of Increasing

Slope of the Tangent Line Idea of Increasing(one point idea)

### Apply the Definitions

Now determine the largest interval on which the function is increasing for the three new examples.

Be prepared to clicker your choice.

• Other

• Other

### Dudfreney Intervals

We propose this definition for all textbooks at the College Algebra (Pre-Calculus) level and below:

A function f is increasing on an open interval I if for all x1and x2 in I, where x1 < x2, then

f(x1) < f(x2).

A function f is decreasing on an open intervalI if for all x1and x2 in I, where x1 < x2, then

f(x1) > f(x2).

### Historical Ideas

Introduction to

Infinitesimal Analysis,

O. Veblen, 1907

### Historical Ideas

Theory of Functions

of Real Variables,

J. Pierpont, 1905

### Historical Ideas

Differential and

Integral Calculus,

G. Osborne, 1907

### Historical Ideas

An Elementary

Treatise on the Calculus,

G. Gibson, 1901