3.4 The Fundamental Theorem of Calculus (FTC)The Mean Value Theorem of IntegralsAnd Average Value* The Mean Value Theorem (p214):If Fis continuous on [ ],a band diff’able on the ( ),a bThen there exists a ( ),c a b�such that( )( ) ( )'F b F aF cb a-=- a c b* The Fundamental Theorem of Calculus (part 1):If fis continuous on [ ],a band Fis any antiderivative of fon [ ],a bThen( ) ( ) ( )baf x dx F b F a= -� f a b x0 xnProof:Let 1 2 1...na x x x b-< < < < <be any partition of [ ],a b(This partition divides the [ ],a binto nsubintervals: [ ] [ ] [ ] [ ] [ ]1 1 2 2 3 2 1 1, ; , ; , ;... , ; ,n n na x x x x x x x x b- - -)Whose widths are… ∆x1 ∆x2 ∆x3 ∆xn-1 ∆xn:By hypothesis ( ) ( )'F x f x=for all x in [ ],a bSo Fsatisfies the hypothesis of the MVT on each subintervalThus we can find numbers * * * *1 2 3, , ,...,nx x x xin the respective subintervals such that( )( ) ( )( )( ) ( )( )( ) ( )1*112 1*22 11*1'''n nnn nF x F aF xx aF x F xF xx xF x F xF xx x---=--=--=-M“OR”( )1F x( )F a- = ( )( )*1 1'F x x a� - = ( )*1 1f x x�D( )2F x( )1F x- = ( )( )*2 2 1'F x x x� - = ( )*2 2f x x�D( )3F x( )2F x- = ( )( )*3 3 2'F x x x� - = ( )*3 3f x x�D ADD M( )1nF x-( )2nF x--( )( )*1 1 2'n n nF x x x- - -= � - = ( )*1 1n nf x x- -�D( ) ( )1nF b F x-- = ( )( )*1'n nF x b x-� - = ( )*n nf x x�D( ) ( )F b F a- = ( )*1nk kkf x x=D�* Now increase n in such a way that the largest ∆xk gets smaller. ( ) ( )( )*1nk kkF b F a f x x=- = D�( ) ( )( )( )*10 0lim limnk kkF b F a f x xD � D=�- = D�( ) ( )( )( )*1lim limnk kn nkF b F a f x x�� ��=- = D�By definition of definite intergral… ( ) ( ) ( )baF b F a f x dx- =�Examples:]112 3001713x dx x= +� ( ) ( )3 31 11 71 0 713 3� � � �= + - +� � � �� � � � 1 171 0 713 3= + - - =* Note that we chose the constant to be 71, but we could have chosen C to be any number as it would have cancelled just like you see the 71’s cancelling in general we will choose c=0.3300sin cos 0xdx xpp�=- +���( )cos cos03p=- - -1 112 2=- + =Fundamental Theorem of Calculus (FTC) II* If fis continuous on an intervalI, then fhas an antiderivative onIIn particular, if ais any number in IThen the function Fdefined by( ) ( )xaF x f t dt=�Is an antiderivative of Fon I; that is ( ) ( )'F x f x=for allxin IAnother way to write is: ( ) ( )xadf t dt f xdx� �=� �� ��Example: 1) Find 5cos cosxdtdt xdx� �=� �� ��[ ]55cos sinxxd dtdt tdx dx� �=� �� ��[ ]sin sin 5dxdx= -cos 0 cosx x= - = 2) ( )223cos 2 cosxdtdt x xdx� �=� �� �� ��23sinxdtdx� �� �� �2sin sin 3dxdx� �= -� �2 22 cos 0 2 cosx x x x= + =*Note: If upper limit is more than just X then need a “chain rule” type piece. 3)( )11 2xF x t dt= - -�For what values of xis ( )0F x =( ) ( ) ( )1 5 31 1 11 0; 5 0; 3 0F F F-= = = = - = =� � �1 2y t= - -* Note that this type of function is a net sigh area accumulator. ( )2132 1 22F t dt= - - =-�( )0130 1 22F t dt= - - =�************************************************************************************************************MVT For Integrals:If fis continuous on [ ],a bthen there exists an *xin [ ],a bsuch that( )( )( )*baf x dx f x b a= � -� max min a x* b( ) ( )A b aa Mm b � � --Example:12013x dx =�( )*f xis called the “average value” of fon [ ],a b( )( )*1baf x f x dxa b=-� 1Work: ( )1* 201 1 11 0 1 3f x x dx= = �-�( )*13f x = 32 1( )2* *1 1 1 3.5773 3 33x x or= = = =
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