Math1011_Week11_4_456 - Math1011 Section 4.4 Given f(x) =...

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Math1011 Section 4.4 11/11/2009 Given f(x) = x 3 - 12x + 1 Find the intervals of concavity and the inflection points. Given the above information plus f(x) is increasing on (- , -2) and (2, ), decreasing on (-2, 2)when f’(x) < 0 local maximum at (-2, 17) local minimum at (2, -15) Graph f(x)
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Math1011 Section 4.4 1 Given f(x) = x 3 - 12x + 1 Find the intervals of concavity and the inflection points. ''( ) 6 0 roots at 0 fx x == = 0 f’’(x) - + Concave down on (- , 0) when f’’(x) < 0 Concave up on (0, + ) when f’’(x) > 0 Inflection point at (0, 1) when f(x) changes concavity Given the above information plus f(x) is increasing on (- , -2) and (2, ), decreasing on (-2, 2)when f’(x) < 0 local maximum at (-2, 17) local minimum at (2, -15) Graph f(x) -20 -15 -10 -5 0 5 10 15 20 -5 -4 -3 -2 -1 0 1 2 3 4 5 Local Max (2,17) Inflection Point (0,1) Local Min (-2,-15) (Doc #011w.43.01A)
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Math1011 Section 4.4 2 Find the intervals on which is increasing or decreasing. () 1 0 x fx xe = Find the local maximum and minimum values of ƒ. Find the intervals of concavity and the inflection points. From the information above, and f(0)=0, f(-1)=-3.6 and f(-2)=-2.7 Sketch a graph of f(x).
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Math1011 Section 4.4 2 Find the intervals on which is increasing or decreasing. () 1 0 x fx xe = ' ( )1 0 ( ) ( 1 0 0 1 0 1 0 ( 1 ) roots at 1 xx dd dx e ex =+= + = =− + -1 f’(x) - + Increasing on (-1, ) when f’(x) > 0 Decreasing on (- , -1) when f’(x) < 0 Find the local maximum and minimum values of ƒ. f’(x)=0 and changes from negative to positive at x = -1, so local minimum at (-1, -10/e) Find the intervals of concavity and the inflection points. ''( ) ( 1) (10 ) 10 ( 1) '( ) ( 1)10 10 10 ( 2) roots at 2 =+ + + + = + = -2 f’’(x) - + Concave down on (- , -2) when f’’(x) < 0 Concave up on (-2, + ) when f’’(x) > 0 Inflection point at (-2, -20/e 2 ) when f(x) changes concavity From the information above, and f(0)=0, f(-1)=-3.6 and f(-2)=-2.7 Sketch a graph of f(x). (Doc #011w.44.02t)(Stewart Section 4.3 #11) -6 -4 -2 0 2 4 6 8 10 -10 -8 0 2
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Math1011 Section 4.4 3 2 2 For ( ) , find vertical and horizontal asymptotes, 1 intervals of increase and decrease, local max and min values, intervals of concavity, inflection points, and sketch the graph. x fx =
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Math1011 Section 4.4 3 2 2 For ( ) , find vertical and horizontal asymptotes, 1 intervals of increase and decrease, local max and min values, intervals of concavity, inflection points, and sketch the graph. x fx = 2 2 22 2 2 2 x 1 x (/) 1 10 ( 1 /) x x 1 x Horizontal Asymptotes: lim ( ) lim divide by highest order in denominator lim 1 Horizontal Asymptote at y = 1 Vertical Asymptotes: lim ( ) lim xx c C →∞ = == =∞ =±∞ 2 when 0.
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This note was uploaded on 11/22/2009 for the course CHEM 2070 at Cornell University (Engineering School).

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Math1011_Week11_4_456 - Math1011 Section 4.4 Given f(x) =...

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