Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
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" 2<br />
x sin ( 1 ) Fx<br />
x<br />
, x ! 0,<br />
( )= #<br />
$ 0, x = 0.<br />
F' ( x)= 2x sin( 1<br />
x)! cos ( 1 ) x<br />
, x " 0.<br />
F h F<br />
F'<br />
lim<br />
h h<br />
h<br />
lim<br />
h<br />
h h<br />
h<br />
h<br />
sin<br />
( )! ( 0)<br />
( 0)=<br />
" 0<br />
2 ( 1 ) x<br />
=<br />
F<br />
" 0<br />
= lim sin ( 1<br />
)= 0 .<br />
h"<br />
0<br />
lim F'x ! 0<br />
'<br />
( 0)<br />
( )<br />
does not exist, but<br />
does exist (and equals 0).