calc notes lecture 15

calc notes lecture 15 - Example 2 Find all relative extrema...

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Lecture Section Objectives Assignment 15 3.2 Extrema and the First-Derivative Test 3.2: 1, 5-11 odd, 19-29 odd, 35, 39 Understanding Goals: 1. Understand the difference between relative extrema and absolute extrema. 2. Understand how to use the First-Derivative Test to find the relative extrema of functions. 3. Understand how to find absolute extrema of continuous functions on a closed interval. 4. Understand how to find minimum and maximum values of real-life models and interpret the results in context. A function has a relative extremum at points where the function changes from increasing to decreasing or vice versa. 1
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For a continuous function, the relative extrema must occur at critical numbers of the function. 2
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3
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Example 1 : Find all relative extrema of the function 3 2 ( ) 2 3 36 14 f x x x x = - - + . Interval Test Value Sign of f '( x ) Conclusion
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Unformatted text preview: Example 2 : Find all relative extrema of the function 2 3 ( ) 2 3 f x x x =-. 4 5 Example 3 : Find the minimum and maximum values of 2 2 ( ) 3 t g t t = + on the interval [ ] 1,1-. 6 Example 4 : Coughing force the trachea (windpipe) to contract, which affects the velocity v of the air passing through the trachea. The velocity of the air during coughing is 2 ( ) , v k R r r r R =-< where k is constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity? Example 5 : Poiseuille’s Law asserts that the speed of blood that is r centimeters from the central axis of an artery of radius R is ( 29 2 2 ( ) S r c R r =-, where c is a positive constant. Where is the speed of the blood greatest? 7...
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calc notes lecture 15 - Example 2 Find all relative extrema...

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