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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

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.

- (–1, 1]
- (–1, 1)
- [–1, 1]
- Other

- Other

- Other

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?

- Other

- Other

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

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

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 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].

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.

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 Point Idea of Increasing

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

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

Be prepared to clicker your choice.

- Other

- Other

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).

Introduction to

Infinitesimal Analysis,

O. Veblen, 1907

Theory of Functions

of Real Variables,

J. Pierpont, 1905

Differential and

Integral Calculus,

G. Osborne, 1907

An Elementary

Treatise on the Calculus,

G. Gibson, 1901

Anne – [email protected]

Michael – [email protected]

Josh – [email protected]