高等數(shù)學(xué)知識點歸納

1、1第一講第一講:極限與連續(xù)極限與連續(xù)一.數(shù)列函數(shù):1.類型:(1)數(shù)列:()nafn?1()nnafa??(2)初等函數(shù):(3)分段函數(shù):0102()()()xxfxFxxxfx??????00()()xxfxFxxxa??????(4)復(fù)合(含)函數(shù):f()()yfuux???(5)隱式(方程):()0Fxy?(6)參式(數(shù)一二):()()xxtyyt?????(7)變限積分函數(shù):()()xaFxfxtdt??(8)級數(shù)和函數(shù)(數(shù)一三

2、):0()nnnSxaxx??????2.特征(幾何):(1)單調(diào)性與有界性(判別)(單調(diào)定號)()fx000()(()())xxxfxfx????(2)奇偶性與周期性(應(yīng)用).3.反函數(shù)與直接函數(shù):11()()()yfxxfyyfx???????二.極限性質(zhì):1.類型:(含)(含)limnna??lim()xfx??x???0lim()xxfx?0xx??2.無窮小與無窮大(注:無窮量):3.未定型:0001000?????????4

3、.性質(zhì):有界性保號性歸并性三.常用結(jié)論:11nn?1(0)1naa??1()max()nnnnabcabc?????00!naan??1(0)xx???0lim1xxx???lim0nxxxe????lnlim0nxxx????0limln0nxxx???0xxex????????????3六.非常手段1.收斂準(zhǔn)則:(1)()lim()nxafnfx?????(2)雙邊夾:nnnbac??nnbca?(3)單邊擠:1()nnafa??2

4、1aa?naM?()0fx?2.導(dǎo)數(shù)定義(洛必達(dá)):00lim()xffxx??:::3.積分和:10112lim[()()()]()nnffffxdxnnnn????????4.中值定理:lim[()()]lim()xxfxafxaf??????????5.級數(shù)和(數(shù)一三):(1)收斂(如)(2)1nna???lim0nna????2!limnnnnn??121lim()nnnnaaaa??????????(3)與同斂散na11()n

5、nnaa?????七.常見應(yīng)用:1.無窮小比較(等價階):()(0)nfxkxx?:(1)(1)()(0)(0)(0)0(0)nnffffa????????()()!!nnnaafxxxxnn???:(2)00()xxnftdtktdt??:2.漸近線(含斜):(1)()limlim[()]xxfxabfxaxx???????()fxaxb????:(2)()()fxaxb????10x?3.連續(xù)性:(1)間斷點判別(個數(shù))(2)分段函