1'2 Day 2: Functions and Their Properties

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1.2 Day 2 Functions and Their Properties

• p. 81-101

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Continuity

• A graph is continuous at a point if it does not
  come apart or break at that point.

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Calculus Phobe on Continuity

• http//www.calculus-help.com/funstuff/phobe.html

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This graph is Continuous!
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This graph has a removable discontinuity or hole!
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This graph has a non-removable discontinuity or
vertical asymptote!
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This graph has a non-removable discontinuity or
Jump Discontinuity!
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Example 1

• Determine whether there is a discontinuity at x
  0. Is it removable or non-removable?

The function does have a non-removable
discontinuity at x 0. There is a Vertical
Asymptote!
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Example 2

• Determine whether there is a discontinuity at x
  0. Is it removable or non-removable?

The function does have a removable discontinuity
at x 0. There is a hole in the graph!
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You Try!

• Does the function have a discontinuity? Is it
  removable or non-removable? Is it a vertical
  asymptote, hole, or jump?
• a) c)
• b)

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Intervals of Increase and Decrease
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The Function is Increasing

• The Slope of the Tangent Line is Positive

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The Function is Decreasing

• The Slope of the Tangent Line is Negative

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The Function is Constant

• The Slope of the Tangent Line is Zero

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• The function is decreasing on (-8, -2
• The function is constant on -2, 2
• The function is increasing on 2, 8)

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-2
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Maximum and Minimum Points
Maximum
Minimum
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Local and Absolute Extrema
Local Max
Local Max
Local Min
Local Min
Absolute Min Local Min
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Absolute and Local Extrema

• Local Maximum
• A Point at the Top of a Graph Mountain
• Local Minimum
• A Point at the Bottom of a Graph Valley
• Absolute Maximum (Also a Local Max)
• The Absolute Highest Point on the Graph
• Absolute Minimum (Also a Local Min)
• The Absolute Lowest Point on the Graph

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Boundedness

• A function can either be Bounded Above, Bounded
  Below, Unbounded, or Bounded (Bounded above and
  below)

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Bounded Below
The Function does not go down to negative
infinity!
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Bounded Above
The function does not go up to positive infinity!
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Unbounded
The Function goes up to positive infinity
The Function goes down to negative infinity
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Bounded
The Function does not go up to positive infinity
or down to negative infinity