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Math1011_Week10_4_123

Math1011_Week10_4_123 - Math1011 Section 4.1 Use the graph...

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Math1011 Section 4.1 11/4/2009 Use the graph to state the absolute and local maximum and minimum values of the function on [ 0 , 7 ].
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Math1011 Section 4.1 1 Use the graph to state the absolute and local maximum and minimum values of the function on [ 0 , 7 ]. Absolute Min Absolute Max Local max Local min The function is continuous! Critical points at x = 1.1, 2.8, 5.1, and 6.0 End points at x = 0.0, 7.0 Local maximum at (0, 3), (2.8, 4.8) and (6, 5.4) Local minimum at (1.1, 1.3) and (5.1, 1.9), (7, 0.4) Absolute max at (6, 5.4) Absolute minimum at (7, 0.4) (Doc #011w.41.01t)
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Math1011 Section 4.1 2 Classify all local extreme values and determine whether any of these extreme values are absolute. Sketch the graph of the function. 4 1 ( ) x x f x + =
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Math1011 Section 4.1 2 Classify all local extreme values and determine whether any of these extreme values are absolute. Sketch the graph of the function. 4 1 ( ) x x f x + = 4 4 3 4 2 1 4 4 4 4 2 1 4 4 4 2 4 4 4 4 1 4 1 1 1 1 1 ( 1) 4 2 1 4 2 1 4 4 Find critical points: '( ) '( ) '( ) Critical Points: 1 0 1 (Add term to other side and Cross Multiply) 1 2 x d d x x dx dx x x x x x x x x x x x x x x x f x f x f x x x x x 1 + + + − + − + ⋅ + + + + + + = = = = = + = + = + = 2 4 4 4 4 4 4 -1 1 2 2 / 1/ 1 / 1/ 1 1/ 1 1 ______ 1_______1________ ' - + - local minimum at -1, , local max at x = 1 lim lim lim 0 Divide x x x x x x x x x x x x x x f x y y + + + →∞ →∞ →∞ = = ± = = = = = = 2 4 by highest order in denominator x 1 is an absolute min and x = 1 is an abs max. x x = = − -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 -40 -30 -20 -10 0 10 20 30 40 (Doc #011w.41.02t)
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Math1011 Section 4.1 3 Find the absolute maximum and absolute minimum values of ƒ on the given interval. 2/3 ( ) (20 ), [ 1,4] f x x x on =
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Math1011 Section 4.1 3 Find the absolute maximum and absolute minimum values of ƒ on the given interval. 2/3 ( ) (20 ), [ 1,4] f x x x on = 2/3 2/3 2/3 5/3 1/3 2/3 5 2 3 3 4 Closed Interval Method ( ) is continuous (x and (20-x) are continuous) on closed interval Find critical points: ( ) (20 ) 20 '( ) 20( ) 0 Note: critical point at x=0 '( ) f x f x x x x x f x x x f x = = = = = 2/3 1/3 2/3 1/3 0 5 3 3 40 5 3 3 2/3 2/3 2/3 3 0 15 120 8
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