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[null,"       The Third Derivative"]
[]
[null,"This DIGMath module lets you investigate "]
[null,"     the properties of the third derivative"]
[null,"      and how it relates to the function "]
[null,null,"and the first two derivatives."]
[]
[]
["This exploration is based on the function f(x) =",{"$nulls":4},"x5-2x4-13x3+14x2+24x"]
[null,"but the results are typical for any function"]
[]
["Now enter the interval for x:  "]
[null,null,"xMin =",-3.5]
[null,null,"xMax =",4]
[]
["Select all the graphs you want to see  "]
[]
[null,true]
[{"$nulls":12},"The graphs of y = f(x) ",null,null,"= x5-2x4-13x3+14x2+24x"]
[null,true,{"$nulls":11},"and its first, second, and third derivatives"]
[]
[null,false,{"$nulls":10},"   Select the point on the curve at x0 =",{"$nulls":3},{"$f":"P24","v":-2}]
[]
[null,true,{"$nulls":12},20,{"$f":"D13+O24*(D14-D13)/100","v":-2}]
[]
[]
["Complete the following statements using your knowledge of derivatives",{"$nulls":11},"The value of the function f(x0) =",{"$nulls":3},{"$f":"LOOKUP(P$24,Sheet2!E$3:E$504,Sheet2!F$3:F$504)","v":48.13141266404287}]
[null,"and your investigations with this DIGMath graphing utility",{"$nulls":10},"The value of derivative f '( x0)  = ",{"$nulls":3},{"$f":"LOOKUP(P24,Sheet2!E3:E504,Sheet2!G3:G504)","v":-43.78327347917167}]
["1.  When f ''' > 0,  f '' is ",{"$nulls":6},"   Item 1",{"$nulls":4},"The value of second derivative f''( x0) = ",{"$nulls":3},{"$f":"LOOKUP(P$24,Sheet2!E$3:E$504,Sheet2!H$3:H$504)","v":-72.77374645810801}]
["2.  When f ''' < 0,  f '' is ",{"$nulls":6},"   Item 2",{"$nulls":4},"The value of third derivative f '''( x0) = ",{"$nulls":3},{"$f":"LOOKUP(P$24,Sheet2!E$3:E$504,Sheet2!I$3:I$504)","v":258.86281329556914}]
["3.  When f ''' > 0,  f ' is ",{"$nulls":6},"   Item 3"]
["4.  When f ''' < 0,  f ' is ",{"$nulls":6},"   Item 4"]
["5.  When f ''' = 0,  f '' may have a ",{"$nulls":6},"   Item 5"]
["6.  When f ''' = 0,  f ' is at an ",{"$nulls":6},"   Item 6"]
["7.  When f ''' > 0,  the concavity of f  is",{"$nulls":6},"   Item 7"]
["8.  When f ''' < 0,  the concavity of f  is ",{"$nulls":6},"   Item 8"]
["9.  When f ''' = 0,  the concavity of f  is either ",{"$nulls":6},"   Item 9"]
[]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["     Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

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