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

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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.
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 = -
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.
Section 12.2 Applications of the Second Derivative

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## This note was uploaded on 02/21/2012 for the course MATH 1314 taught by Professor Marks during the Fall '08 term at University of Houston.

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

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