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This guide covers the concepts of monotonic functions, specifically increasing and decreasing intervals, and the application of the First Derivative Test for determining local extrema. It discusses the implications of a function being continuous and differentiable, and outlines conditions under which a function is monotonic on an interval. The guide provides methods to identify critical points, evaluate intervals of increase and decrease, and apply the First Derivative Test to find local maxima and minima. This information is crucial for understanding the behavior of functions in calculus.
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Math 180 4.3 – Monotonic Functions and the First Derivative Test
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.)
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.)
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing decreasing
Suppose that is continuous on and differentiable on . If __________ for all , then is ___________on . If __________ for all , then is __________on . Note: The above also works for infinite intervals. (ex: , , etc.) increasing decreasing
If is increasing or decreasing on an interval, it is said to be ___________on that interval.
If is increasing or decreasing on an interval, it is said to be ___________on that interval. monotonic
Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined
Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?
Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?
Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When is undefined In other words, where might change signs?
Ex 1. Identify the intervals of increase and decrease for
Ex 1. Identify the intervals of increase and decrease for
Critical points (where or undefined) give us a list of -values where might have a local max or local min. The First Derivative Test for Local Extrema tells us that if the sign of changes across a critical point, then we’ve got a local max or a local min.
Critical points (where or undefined) give us a list of -values where might have a local max or local min. The First Derivative Test for Local Extrema tells us that if the sign of changes across a critical point, then we’ve got a local max or a local min.
First Derivative Test for Local Extrema Let be a critical point. Local max at
First Derivative Test for Local Extrema Let be a critical point. Local max at Local min at
First Derivative Test for Local Extrema Let be a critical point. Not a local extremum
First Derivative Test for Local Extrema Let be a critical point. Not a local extremum Not a local extremum
Ex 2. Find all critical points for . Then find all local maxima and minima.
Ex 2. Find all critical points for . Then find all local maxima and minima.
Ex 3. Find the critical points of . Identify the intervals on which is increasing and decreasing. Find the function’s local and absolute extreme values.
Ex 3. Find the critical points of . Identify the intervals on which is increasing and decreasing. Find the function’s local and absolute extreme values.
