Increasing and Decreasing Intervals Where Does the Fun End ?
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Increasing and Decreasing IntervalsWhere Does the Fun End?
Anne DudleyMichael Holtfrerich
Joshua WhitneyGlendale Community College
Glendale, Arizona
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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.
Increasing & DecreasingActivity 1
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1. (–1, 1]2. (–1, 1)3. [–1, 1]4. 5. Other
Graph 1 IncreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4 5
0%
50%
0%
50%
0%
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
x
y
f(x) = -(1/3) x3 + x ,11,
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1. 2. 3. 4. Other
Graph 2 IncreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
0% 0%
100%
0%
-1 1 2 3
1
2
3
4
5
x
y
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1. 2. 3. 4. Other
Graph 2 DecreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is decreasing?
1 2 3 4
25% 25%25%25%
-1 1 2 3
1
2
3
4
5
x
y
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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?
Discussion
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Graph 3 IncreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
25% 25%25%25%
-1 1
1
2
3
4
5
x
y
1. 2. 3. 4. Other
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Graph 4 IncreasingUse your clicker to indicate your answer.
Where is the function increasing?
1 2 3 4
50% 50%
0%0%
1. 2. 3. 4. Other
-2 -1 1 2
-3
-2
-1
1
2
3
4
5
6
7
x
y
f(x) = x3+3
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Ostebee & Zorn Calculus, 2nd Edition (p.56)Definition: Let I denote the interval (a, b). A function f is increasing on I if
whenever a < x1 < x2 < b. f is decreasing on I if 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.
What Do Textbooks Say?
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Graph 1 Increasing (Ostebee & Zorn )Use your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
0% 0%0%
100%
1. 2. 3. 4. Other
1,1 1,1 1,1
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
x
y
f(x) = -(1/3) x3 + x
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Hughes-Hallett Calculus, 5th edition (p. 165) and Swokowski Calculus, 2nd edition (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 .
What Do Textbooks Say?
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Graph 1 Increasing (Hughes-Hallett)Use your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
0% 0%
100%
0%
1. 2. 3. 4. Other
1,1 1,1 1,1
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
x
y
f(x) = -(1/3) x3 + x
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Larson Edwards Calculus, 5th edition (p. 219) A function f is increasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).A function f is decreasing on an interval if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).
What Do Textbooks Say?
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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].
x
y
a bx1 x2
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Larson Edwards Calculus, 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?
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Cynthia Young Precalculus, 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 interval I if for any x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).
What Do Textbooks Say?
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Two Point Idea of Increasing
Slope of the Tangent Line Idea of Increasing(one point idea)
Two Differing Ideas
-2 -1 1 2
-3
-2
-1
1
2
3
4
5
6
7
x
y
f(x) = x3+3
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Now determine the largest interval on which the function is increasing for the three new examples.
Be prepared to clicker your choice.
Apply the Definitions
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Graph 5 IncreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
50% 50%
0%0%
1. 2. 3. 4. Other
1 2 3
-1
1
2
3
x
y
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1. 2. 3. 4.
Graph 6 IncreasingUse your clicker to indicate your answer.
What is the largest interval on which the function is increasing?
1 2 3 4
0% 0%
100%
0%
1 2 3
1
2
3
4
5
x
y
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Graph 7 (AKA Final Exam) Use your clicker to indicate your answer.
Where is the function increasing?
1 2 3 4 5
20% 20% 20%20%20%
-1 1 2 3 4 5 6
-1
1
2
3
4
5
6
x
y1. 2. 3. 4. 5. Other
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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 x1 and x2 in I, where x1 < x2, then f(x1) < f(x2).A function f is decreasing on an open interval I if for all x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).
Dudfreney Intervals
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Historical Ideas
Introduction to Infinitesimal Analysis,O. Veblen, 1907
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Historical Ideas
Theory of Functionsof Real Variables,J. Pierpont, 1905
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Historical Ideas
Differential and Integral Calculus,G. Osborne, 1907
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Historical Ideas
An Elementary Treatise on the Calculus,G. Gibson, 1901
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Anne – [email protected]
Michael – [email protected]
Josh – [email protected]
Thanks for Your Participation