AntidifferentiationReversing DifferentiationSlide 3Slope FieldsSlide 5Finding an Antiderivative Using a Slope FieldAntiderivative NotationBasic Integration RulesPracticeApplicationsSlide 11AssignmentAntidifferentiationLesson 5.1Reversing Differentiation•An antiderivative of function f is a function Fwhich satisfies F’ = f•Consider the following:•We note that two antiderivatives of the same function differ by a constant43( ) 2'( ) 8F x xF x x==43( ) 2 17'( ) 8G x xG x x= +=Reversing Differentiation•General antiderivativesf(x) = 6x2 F(x) = 2x3 + Cbecause F’(x) = 6x2k(x) = sec2(x) K(x) = tan(x) + Cbecause K’(x) = k(x)Slope Fields•Slope of a function f(x)at a point agiven by f ‘(a)•Suppose we know f ‘(x)substitute different values for a draw short slope lines for successive values of y•Example'( ) 2f x x=Slope Fields•For a large portion of the graph, when•We can trace the line for a specific F(x)specifically when the C = -3'( ) 2f x x=Finding an Antiderivative Using a Slope Field•Given•We can trace the version of the original F(x) which goes through the origin. 2'( )xf x e=Antiderivative Notation•Notation•Where …C is an arbitrary constantF is an antiderivative of fcalled the indefinite integral of fsatisfies condition F’(x) = f(x) for all x in domain of f( ) ( )f x dx F x C= +�Basic Integration Rules•Note the list of rules on pg 321,2•Notably the power rule•Otherwise, it is just the “reverse” of the derivative rules1; 11ln ; 1nnuC nu dunu C n+�+ �-�=+��+ =-��Practice•Find the indefinite integral3 5( 8 15 )t t dt- +�2 31 2 3dtt t t� �-� �� ��Applications•Given slope 3x2 – 6 for a function F(x) which contains the point (2,3), what is F(x)?•Given marginal revenue R’(x) = 180 + 0.15x What is the the demand function p(x)Note that R(x) = x*p(x) and R(0) = 0Applications•Consider an area function A(t)•Can be shown thaty=f(x)( ) ( )A t C f t dt+ =�f(x)Assignment•Section 5.1•Page 281•3, 5, 9, 13, 17, 21, 23, 27, 31, 35, 39, 41, 43,
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