1 . ①在微积分中,求极限有一种重要的数学工具——洛必达法则,法则中有一结论:若函数
,
的导函数分别为
,
,且
,则
;
②设
,k是大于1的正整数,若函数
满足:对任意
,均有
成立,且
,则称函数
为区间
上的k阶无穷递降函数.
结合以上两个信息,回答下列问题:
(1)证明
不是区间
上的2阶无穷递降函数;
(2)计算:
;
(3)记
,
;求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22add663bd26e87d972a10dc5fd9ada1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62ceac3910b9f134bab0b92e8d9a9eb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74acc4d2f565d7088e8d737718e89602.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e0c1abf0378a7f5d79672f622b275e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e54d86850a733707433da2e423a5c81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/580f20b900b6d8c9e90c84a0588ae74d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c3e441923ed3c1a32720d6aeac2f599.png)
结合以上两个信息,回答下列问题:
(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64d1f6f459292de1002f863203ce91a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fab11f38ab8593932082ec4d9c8c91f.png)
(2)计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8063898825e02107b7e04f6eba28cb8c.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/602d05de8ada4a6f4d53bab28430f684.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d40b0c4fd043d372c463db08659e779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caea9a696f22c76f8f4563ac45d124b1.png)
您最近一年使用:0次
2024-04-18更新
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454次组卷
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6卷引用:广东省广州市天河中学高中部2023-2024学年高二下学期基础测试数学试题
广东省广州市天河中学高中部2023-2024学年高二下学期基础测试数学试题(已下线)模块五 专题5 全真拔高模拟5(人教B版高二期中研习)四川省广安市华蓥中学2023-2024学年高二下学期4月月考数学试题广东省广州市天河中学2023-2024学年高二下学期第二次月考数学试题黑龙江省哈尔滨市双城区兆麟中学2023-2024学年高二下学期5月期中考试数学试题(已下线)专题14 洛必达法则的应用【练】
名校
2 . 已知函数
,
,
.
(1)当
时,判断函数
的奇偶性并证明;
(2)当
且
时,利用函数单调性的定义证明函数
在
上单调递增;
(3)求证:当
且
时,方程
在
内有实数解.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0cc9b1b321520eae2bf944a9c85c9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19339e3904e9541ff26b30ae5f1242b2.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/657435e1fda84118e7f63c97505c8b75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5d6243e93c41978871cb23d8e66148d.png)
(3)求证:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4d795709b0abcf47bceec2250f2f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91a871ef7bf13de3e15489d65b57a3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99caed81bfb141d6e7dac8f6fe9db069.png)
您最近一年使用:0次
名校
3 . 已知函数
,
为函数
的反函数
(1)讨论
在
上的单调性,并用定义证明;
(2)设
,求证:
有且仅有一个零点
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4ccec118032fd96e0713b04c3a27a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028517e8bebe634441e0a5c79828e88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12be206d66e65eb92ef08bad8cd8f71d.png)
(1)讨论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae4a2b3998705e51dbade9ada0873b2b.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041f581f277a2de1ef41c354b6e6991e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18c49ca8562b98657ca9c499093f7233.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7d126a2ae5babaf18b9082a975cdc52.png)
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解题方法
4 . 已知函数
.
(1)用单调性定义证明:
在
上单调递增;
(2)若函数
有3个零点
,满足
,且
.
①求证:
;
②求
的值(
表示不超过
的最大整数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4247d7790d83be16bc74aa5e5d12dd63.png)
(1)用单调性定义证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6d804ef44bfc64f824b0ccef71765e.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f8994d83bf4a688c0ab897a5a40fa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d995c5d2e1e0305d805032e18997986a.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28cbe8f17c4472d8663f9ccbe3b98f6.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59077d1948911b13d68a572eadbca3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
您最近一年使用:0次
名校
5 . 设函数
定义域为
.若整数
满足
,则称
与
“相关”于
.
(1)设
,
,写出所有与
“相关”于
的整数;
(2)设
满足:任取不同的整数
,
与
均“相关”于
.求证:存在整数
,使得
都与
“相关”于
;
(3)是否存在实数
,使得函数
,
满足:存在
,能使所有与
“相关”于
的非零整数组成一个非空有限集?若这样的
存在,指出
和
的大小关系(无需证明),并求出
的取值范围;若这样的
不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cae7b35b6dbdeafb82810fb8239121c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b056a90a2751f04ba5fff3dc5c1d0674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d53f0be9e922c54b74dc21ef147d81c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f713f92ce74aa961b391fe544e609a68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e196c4f6545854d1fee5fea9609d01d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0afb80007983e5b99dcdeebf87d18ff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cec48f249cb8141bad725728996fbf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0becd5c2342ac2cef8d24b6e7e8a0abc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4b983d54b0b2d085456306ef564bda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8e8936c9fe1e81726455908657a29fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
(3)是否存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7489d1555eaffa5d7d26d37df5d4355a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e196c4f6545854d1fee5fea9609d01d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8380733ca1aaccc36e3b0c658bd6011b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cee175df20c19745745059464e643079.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
您最近一年使用:0次
2024高三·全国·专题练习
6 . 已知
定义在
上且
,
,当a,
,
时,有
.
(1)试判断函数
在
上是增函数还是减函数,并证明该结论.
(2)设
,求证:
.
(3)若
,求x的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd1db6c94b94afc372212a81cc1f4dd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc4c79d578d4b0d74b84c3f6579e8806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96bc2eeaca8a8ce4bcce2bff011a11bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71b84952d33957e5b90d8cd3b3bcc127.png)
(1)试判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d188ec2580e273ce87e51653a2177ee.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e966083a8c143204a48b8406f62dcc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d65ffb2191a2d5182e671b73f30598c7.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a60168f2f9ae5baa0c753bc8154d46d6.png)
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名校
7 . 给出集合
对任意
,都有
成立
.
(1)若
,求证:函数
;
(2)由于(1)中函数
既是周期函数又是偶函数,于是张同学猜想了两个结论:
命题甲:集合
中的元素都是周期为6的函数;
命题乙:集合
中的元素都是偶函数;
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305c81b6a05c983ef0dd04962d546bd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b005e1e4b8e41c0028cd464835c464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ca8bdc812627d925f00ed7c145d696.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5314a9d2205a2beba0dcffb8fd943b18.png)
(2)由于(1)中函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
命题甲:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
命题乙:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
您最近一年使用:0次
名校
8 . 已知函数
.
(1)求证:函数
是定义域为
的奇函数;
(2)判断函数
的单调性,并用单调性的定义证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbd9e52b79fb84c320dc522e13d4f0b.png)
(1)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
(2)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
您最近一年使用:0次
2024-01-24更新
|
654次组卷
|
4卷引用:河南省洛阳市强基联盟2023-2024学年高一上学期12月联考数学试题
9 . 已知函数
,
.
(1)求证:
为偶函数;
(2)设
,判断
的单调性,并用单调性定义加以证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf688908975687a9bead59e017acacc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4741b2cc342a055aefb2d825e45ce77e.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aae37cac299cbe3ccac181b2175287f.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa662f0273f0921c1fa4727f632395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a813b5adbf5c7082561237894ba6d599.png)
您最近一年使用:0次
名校
10 . 已知函数
.
(1)判断并证明
的零点个数
(2)记
在
上的零点为
,求证;
(i)
是一个递减数列
(ii)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58ae99833cb675e5e36c58f345eb03e7.png)
(1)判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d64af919a56a107e0fc0a417e481648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
(i)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1fd18a909cecbaee7115d6b15631d83.png)
(ii)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167f4f8fd4d3de714b87f05e57a3ba3b.png)
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2024-06-04更新
|
558次组卷
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2卷引用:湖南省长沙市第一中学2024届高三下学期模拟考试数学试卷(一)