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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

Increasing and Decreasing IntervalsWhere Does the Fun End?

Anne Dudley

Michael Holtfrerich

Joshua Whitney

Glendale Community College

Glendale, Arizona


Increasing decreasing activity 1
Increasing & DecreasingActivity 1

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

Once you have determined your answers, talk to neighbors about their answers.


Graph 1 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?

  • (–1, 1]

  • (–1, 1)

  • [–1, 1]

  • Other


Graph 2 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?

  • Other


Graph 2 DecreasingUse your clicker to indicate your answer. What is the largest interval on which the function is decreasing?

  • Other


Discussion
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?

Will it confuse your students?


Graph 3 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?

  • Other


Graph 4 increasing use your clicker to indicate your answer where is the function increasing
Graph 4 IncreasingUse your clicker to indicate your answer.Where is the function increasing?

  • Other


What d o textbooks say
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 Iif whenever 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 d o textbooks say1
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 d o textbooks say2
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 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 d o textbooks say3
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 d o textbooks say4
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 Differing Ideas

Two Point Idea of Increasing

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


Apply the definitions
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.


Graph 5 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?

  • Other


Graph 6 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?


Graph 7 aka final exam use your clicker to indicate your answer where is the function increasing
Graph 7 (AKA Final Exam) Use your clicker to indicate your answer.Where is the function increasing?

  • Other


Dudfreney intervals
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
Historical Ideas

Introduction to

Infinitesimal Analysis,

O. Veblen, 1907


Historical ideas1
Historical Ideas

Theory of Functions

of Real Variables,

J. Pierpont, 1905


Historical ideas2
Historical Ideas

Differential and

Integral Calculus,

G. Osborne, 1907


Historical ideas3
Historical Ideas

An Elementary

Treatise on the Calculus,

G. Gibson, 1901



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