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Day-4-filled - Math 1314 Day 4 From last time: We learned...

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Math 1314 Day 4 From last time: We learned to find the intervals on which a function is increasing and intervals on which a function is decreasing. Example 6.5: Find the intervals on which f is increasing and the intervals on which f is decreasing: 4 3 ( ) 3 4 9 f x x x = + +
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Finding Relative Extrema The first derivative can also help up find the x coordinate(s) of any high points or low points on the graph of a function. This will allow us to find the x coordinates of the “peaks and valleys” in the graphs. These high points and/or low points are called relative extrema of a function. An extremum is called a relative (local) maximum if it is higher than the points located nearby. An extremum is called a relative (local) minimum if it is lower than the points located nearby. Definition : A function f has a relative maximum at c x = if there exists an open interval ( 29 b a , containing c such that ) ( ) ( c f x f for all x in ( 29 b a , . A function f has a relative minimum at c x = if there exists an open interval ( 29 b a , containing c such that ) ( ) ( c f x f for all x in ( 29 b a , . To find the relative extrema, we must first find the critical points of the function. Definition : A critical point of a function f is any point x in the domain of f such that 0 ) ( ' = x f or ) ( ' x f does not exist. Once we find the critical points, we can use a line test to determine whether the critical point gives us a maximum, a minimum or neither. The First Derivative Test To find the relative extrema of a function, 1. Determine the critical points of f . 2. Determine the sign of f’ ( x ) to the left and to the right of each critical point. (a) if f’ ( x ) changes sign from positive to negative as we move across a critical point c x = from left to right, then ) ( c f is a relative maximum. (b) if f’ ( x ) changes sign from negative to positive as we move across a critical point c x = from left to right, then ) ( c f is a relative minimum. (c) if f’ ( x ) does not change sign as we move across a critical point c x = from left to right, then ) ( c f is not a relative extremum.
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Example 7 : Find the relative extrema if 32 24 3 ) ( 2 3 + - - = x x x x f .
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Example 8 : Find the relative extrema if 4 3 ( ) 5 f x x x = -
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Example 9: Find the relative extrema if 3 ( ) x f x x e =
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Example 10 : After birth, an infant normally will lose weight for a few days and then start gaining. A model for the average W (in pounds) of infants over the first two week following birth is , 14 0 , 3032 . 7 3974 . 033 . ) ( 2 + - = t t t t W where t is measured in days. Find the interval(s) on which weight is expected to increase and the interval(s) on which weight is expected to decrease based on this model.
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Section 12.2 Applications of the Second Derivative
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Day-4-filled - Math 1314 Day 4 From last time: We learned...

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