高等数学-一元微分学-导数的应用-单调性与极值

高等数学-一元微分学-导数的应用-单调性与极值

单调性与极值

取得极值求参数

例1
image-20200421003426239
image-20200421003408417

求函数极值最值拐点

例1
image-20200419195708739
image-20200419195723861
例2
image-20200419200119412
image-20200419200408718
例3
image-20200421215552152
image-20200421215539424
例4
image-20200421223721753
image-20200421223647790
image-20200421223707139
例5
image-20200421234610269
image-20200421234549317

不等式的证明

例1
image-20200419105526487
image-20200419105612557
例2
image-20200419105817774
image-20200419110336084
image-20200419110432991
例3
image-20200419111217580
image-20200419111349386
image-20200419111417677
例4
image-20200419111515417
image-20200419111621331
image-20200419111738948
image-20200419111823189
image-20200419111854517
例5
image-20200420133940016
image-20200420134002071
例6
image-20200421211229118
image-20200421211213122
例7
image-20200421212306546
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例8
image-20200421212720592
image-20200421212704141
例9
image-20200421230538382
image-20200421230518064
例10
image-20200421232406926
image-20200421232344310
例11
image-20200421233509457
image-20200421233445561

方程根的讨论

零点定理证明方程有根

例1
image-20200419112411767

罗尔定理加原函数证明方程有根

image-20200419112651426
例1
image-20200419112806674
image-20200419112837164
image-20200419112910603

单调法求方程根的数目

  • 设立函数，关注定义域
  • 找出驻点和不可导点，考察这些点处的值
  • 关注区间两侧，做草图
例1
image-20200419114101538
image-20200419114230396
image-20200419114257690
例2
image-20200419114323890
image-20200419114404601
image-20200419114626282
例3
image-20200422004431575
image-20200422004417498
例4
image-20200422004808801
image-20200422004739575
image-20200422004756135
例5
image-20200422105544816
image-20200422105531412

已知方程根数目求参数的范围

例1
image-20200420121227949
image-20200420121202318
例2
image-20200420223346656
image-20200420223327541

函数极值点的判定

  • 设立函数，关注定义域
  • 找出驻点和不可导点
  • 极值点的判别法
    • 第一判别法：一阶导数左边大于0右边小于0，极大值
    • 第二判别法：二阶导数小于0，极大值
例1
image-20200419120525676
image-20200419120600695
image-20200419120619583
例2
image-20200419120740848
image-20200419120833477
image-20200419120906304
image-20200419121043544
image-20200419121155925
例3
image-20200419121249527
image-20200419121336818
例4
image-20200420210515185
image-20200420210441704
例5
image-20200420222151454
image-20200420222110963
image-20200420222133567
例6
image-20200420224050066
image-20200420224032845
例7
image-20200420224820402
image-20200420224803435
例8
image-20200421000705445
image-20200421000637031
例9
image-20200421002732769
image-20200421002713453

凹凸性或拐点的判定

二阶导数与0的比较判断凹凸性

例1
image-20200419132232287
image-20200419132359015
例2
image-20200419132543609
image-20200419132619855
image-20200419132646963
image-20200419132736921
例3
image-20200419133037795
image-20200419134810917

同理，x1，x2也是拐点

例4
image-20200420211117238
image-20200420211103328
例5
image-20200421003114473
image-20200421003056157
例6
image-20200422120254671
image-20200422120242960

求函数渐近线

例1
image-20200419142633183
例2
image-20200419144107200
image-20200419144232633
例3
image-20200419222130464
image-20200419222110214
例4
image-20200420122142599
image-20200420122128178
例5
image-20200420222837617
image-20200420222805112
例6
image-20200421115714521
image-20200421115659052
例7
image-20200421224917390
image-20200421225234846
例8
image-20200422104544611
image-20200422104605915