第十二次课 2 0土木工程1、2班 75（101）人
① 罗尔中值定理
② 拉格朗日中值定理及其应用 ③ 柯西中值定理：
20土木工程1、2班 75（101）人
---------------------------------------------+ 罗尔定理
1.?f?x??on??a,b??c.t.;??2.?f?x??in??a,b??diff.;??????a,b???f?????0.
?3.?f?a??f?b??拉格朗日中值定理
1.?f?x??on??a,b??c.t.;?f?b??f?a???????a,b??f??.?????b?a2.?f?x??in??a,b??diff.;?? ?????????????????f?b??f?a??f?????b?a?1.?f?x??on??a,b??c.t.;?f?b??f?a???.??????a,b???f????b?a2.?f?x??in??a,b??diff.;?????????????????f?b??f?a?H?x??f?x??xb?a1.?H?x???on??a,b??c.t.;??2.?H?x??in??a,b??diff.??????a,b?,H?????0.?3.?H?a???H?b?.?????????????f?b??f?a??0?.b?af?b??f?a?f?b??f?a?H?a???H?b??f?a??a?f?b??bb?ab?af?b??f?a??f?a??f?b???b?a??0
b?aH?????f?????应用之一
函数在某区间I上导数恒等于零，则函数为常数函数.
f?x??on??I??f??x??0.?f?x??C.????????????????proof:?x1,x2?x1?x2??I????f?x??on???x1,x2???c.t, ---------------+
????????x1,x2?,?f?x2??f?x1??f?????x2?x1??02.f?x??in??x1,x2??is?diff.,??f?b??f?a???a???b.b?a②?f?b??f?a??f?????b?a????a???b.①?f?????③?f?b??f?a??f??a???b?a???b?a????有限增量公式0??????a???b?a?b?ab?a??????????????????1,???????y?f?????x,??a???b.应用之一： 证明：恒等式
??a??af?x??arcsinx?arccosx?proof:??f??x??12?2121?x1?x??f?x??arcsinx?arccosx?C,??0.
?2?22????f??arcsin?arccos???2?222??拉格朗日中值定理证明不等式的基本原理：
1.?f?x??on??a,b??c.t.;????????a,b???f?b??f?a??f?????b?a?.2.?f?x??in??a,b??diff.;?? ????????????????????if??f??x??,?f??a??b?a??f?b??f?a??f??b??b?a?;if??f??x??,?f??a??b?a??f?b??f?a??f??b??b?a?;
nbn?1?a?b??an?bn?nan?1?a?b??????a?b?0,n?1?.???????????proof:sup?f?x??xn,????b,a??c.t.???in?b,a??is?diff.?????b,a????b???a?? an?bn?n?n?1?a?b?f??x??nxn?1,nbn?1?a?b??an?bn?n?n?1?a?b??nan?1?a?b?ln?1?x??ln?1?0?sup?f?x??ln?1?x???on??0,x??c.t,???in??0,x??is?diff..?????0,x????0???x?ln?1?x??ln?1?0??f??x??1?is1?x1x1??111,x?ln?1?x??ln?1?0??x?x.1?x1??1?0
1.?f?x??,F?x??on??a,b??c.t.??f????f?b??f?a?2.?f?x?,F?x??in??a,b??diff.??????a,b?,???F?F?b??F?a????3.F??x??0??????????????????f?b??f?a??f?????b?a??f????f?b??f?a??????F???F?b??F?a?F?b??F?a??F?????b?a???
1.?f?x??,F?x??on??a,b??c.t.??f????f?b??f?a?2.?f?x?,F?x??in??a,b??diff.??????a,b?,???F???F?b??F?a??3.F??x??0?????????????????proof:sup?H?x??f?x??F?x?f?b??f?a?F?b??F?a?1.?H?x???on??a,b??c.t.??f?b??f?a?2.?H?x??in??a,b??diff.?????a,b?,?H?????0?f?????F?????0F?b??F?a??3.?H?a??H?b?.?f????f?b??f?a????F???F?b??F?a??????????????????????????????H?a??H?b??f?a??F?a??f?a??f?b??f?b??f?a?f?b??f?a??f?b??F?b?F?b??F?a?F?b??F?a?f?b??f?a??F?b??F?a???0F?b??F?a?
f????f?b??f?a?f?b??f?a???0?f?????F?????0F????F?b??F?a?F?b??F?a?
2024年高等数学课件-第十二次课 20土木工程1、2班 75(101)人
