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Math 140. 3.1 – Increasing and Decreasing Functions; Relative Extrema. Let be defined on the interval . Suppose and are in that interval . is ___________________ on that interval if when is ___________________ on that interval if when .
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Math 140 3.1 – Increasing and Decreasing Functions; Relative Extrema
Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when
Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing
Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing decreasing
Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)
Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)
Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?
Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?
Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?
Ex 1. Find the intervals of increase and decrease for
Ex 1. Find the intervals of increase and decrease for
If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur.
If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number
If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number
If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point
If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point
First Derivative Test for Relative Extrema Let be a critical number. Relative max at
First Derivative Test for Relative Extrema Let be a critical number. Relative max at Relative min at
First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum
First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum Not a relative extremum
Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.
Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.
Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.
Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.