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2022-2023學(xué)年度第一學(xué)期蕪湖市中學(xué)教學(xué)質(zhì)量統(tǒng)測(cè)高一年級(jí)數(shù)學(xué)試題卷注意事項(xiàng):1.本試卷滿分為100分,考試時(shí)間為120分鐘.2.本試卷包括“試題卷”和“答題卷”兩部分.“試題卷”共4頁(yè),“答題卷”共6頁(yè).3.請(qǐng)務(wù)必在“答題卷”上答題,在“試題卷”上答題是無(wú)效的.4.考試結(jié)束后,請(qǐng)將“答題卷”交回.一、單選題(本題共8小題,每小題3分,共24分.在每小題給出的四個(gè)選項(xiàng)中,只有一個(gè)是符合題目要求的)1.設(shè)集合SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】A【解析】【分析】根據(jù)集合的交集,可得答案.【詳解】由題意,SKIPIF1<0.故選:A.2.不等式SKIPIF1<0的解集是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0或SKIPIF1<0 D.SKIPIF1<0或SKIPIF1<0【答案】B【解析】【分析】根據(jù)一元二次不等式的解法計(jì)算可得;【詳解】由SKIPIF1<0,解得SKIPIF1<0,即原不等式的解集為SKIPIF1<0;故選:B.3.SKIPIF1<0()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【答案】C【解析】【分析】由誘導(dǎo)公式化簡(jiǎn)后得結(jié)論.【詳解】SKIPIF1<0.故選:C.4.已知命題SKIPIF1<0,則命題SKIPIF1<0為()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】C【解析】【分析】給定命題是全稱(chēng)量詞命題,由全稱(chēng)量詞命題的否定的意義即可得解.【詳解】因SKIPIF1<0是全稱(chēng)量詞命題,則命題SKIPIF1<0為存在量詞命題,由全稱(chēng)量詞命題的否定意義得,命題SKIPIF1<0:SKIPIF1<0.故選:C5.若SKIPIF1<0,SKIPIF1<0,則下列不等式成立是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】A【解析】【分析】根據(jù)不等式的性質(zhì)可判斷A,取特值可判斷B,C,D.【詳解】對(duì)于A,因?yàn)镾KIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,故A正確;對(duì)于B,若SKIPIF1<0,則SKIPIF1<0,故B正確;對(duì)于C,若SKIPIF1<0,則SKIPIF1<0,故C不正確;對(duì)于D,若SKIPIF1<0,則SKIPIF1<0,故D不正確.故選:A.6.折扇是一種用竹木或象牙做扇骨,?紙或綾絹?zhàn)錾让娴哪苷郫B的扇子,如圖1,其平面圖如圖2的扇形SKIPIF1<0,其中SKIPIF1<0,SKIPIF1<0SKIPIF1<0,則扇面(曲邊四邊形SKIPIF1<0)的面積是()A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【答案】B【解析】【分析】由扇形面積公式計(jì)算(大扇形面積減去小扇形面積).【詳解】由已知SKIPIF1<0,SKIPIF1<0,扇面面積為SKIPIF1<0故選:B.7.下列說(shuō)法正確的是()A.“SKIPIF1<0”是“SKIPIF1<0”的既不充分也不必要條件B.“SKIPIF1<0”是“SKIPIF1<0”的充分不必要條件C.若SKIPIF1<0,則“SKIPIF1<0”是“SKIPIF1<0”的必要不充分條件D.在SKIPIF1<0中,角SKIPIF1<0,SKIPIF1<0均為銳角,則“SKIPIF1<0”是“SKIPIF1<0是鈍角三角形”的充要條件【答案】D【解析】【分析】利用充分不必要條件,必要不充分條件,充要條件定義進(jìn)行逐項(xiàng)判定.【詳解】對(duì)于A,因?yàn)镾KIPIF1<0能夠得到SKIPIF1<0,反之不成立,所以“SKIPIF1<0”是“SKIPIF1<0”的必要不充分條件,A錯(cuò)誤;對(duì)于B,因?yàn)镾KIPIF1<0時(shí),SKIPIF1<0,而當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,所以“SKIPIF1<0”是“SKIPIF1<0”的必要不充分條件,B錯(cuò)誤;對(duì)于C,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,無(wú)法得出SKIPIF1<0;當(dāng)SKIPIF1<0,SKIPIF1<0,所以SKIPIF1<0,C錯(cuò)誤;對(duì)于D,因?yàn)榻荢KIPIF1<0,SKIPIF1<0均為銳角,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,由于SKIPIF1<0所以SKIPIF1<0,即SKIPIF1<0,所以SKIPIF1<0是鈍角三角形;反之依然成立,D正確.故選:D.8.已知實(shí)數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,那么實(shí)數(shù)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的大小關(guān)系是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】A【解析】【分析】利用作差法,結(jié)合對(duì)數(shù)的運(yùn)算,以及對(duì)數(shù)函數(shù)的性質(zhì),可得答案.【詳解】SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,即SKIPIF1<0,可得SKIPIF1<0;SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,即SKIPIF1<0,可得SKIPIF1<0;SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,即SKIPIF1<0,可得SKIPIF1<0;綜上,SKIPIF1<0.故選:A.二、多選題(本題共4小題,每小題4分,共16分.在每小題給出的選項(xiàng)中,有多個(gè)選項(xiàng)符合題目要求,全部選對(duì)的得4分,部分選對(duì)的得2分,有選錯(cuò)的得0分)9.下圖為冪函數(shù)SKIPIF1<0的大致圖象,則SKIPIF1<0的解析式可能為()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】AC【解析】【分析】根據(jù)奇函數(shù)的性質(zhì),以及冪函數(shù)的性質(zhì),可得答案.【詳解】對(duì)于A、C,SKIPIF1<0,SKIPIF1<0,顯然為奇函數(shù),且指數(shù)在0到1之間,在第一象限是越增越慢的,故A、C正確;對(duì)于B、D,SKIPIF1<0,SKIPIF1<0,顯然為偶函數(shù),故B、D錯(cuò)誤.故選:AC.10.下列說(shuō)法中正確的是()A.SKIPIF1<0在SKIPIF1<0上單調(diào)遞增B.SKIPIF1<0與SKIPIF1<0的圖象相同C.不等式SKIPIF1<0的解集為SKIPIF1<0D.SKIPIF1<0的圖象對(duì)稱(chēng)中心為SKIPIF1<0【答案】ABC【解析】【分析】根據(jù)正弦函數(shù)的性質(zhì)可判斷A,根據(jù)誘導(dǎo)公式及余弦函數(shù)的性質(zhì)可判斷B,根據(jù)輔助角公式及正弦函數(shù)的圖象函數(shù)性質(zhì)可判斷C,根據(jù)正切函數(shù)的性質(zhì)可判斷D.【詳解】對(duì)于A:因?yàn)镾KIPIF1<0的單調(diào)增區(qū)間為SKIPIF1<0,所以函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增,故A正確;對(duì)于B:因?yàn)镾KIPIF1<0,SKIPIF1<0,所以SKIPIF1<0與SKIPIF1<0的圖象相同,故B正確;對(duì)于C:由SKIPIF1<0,可得SKIPIF1<0,則SKIPIF1<0,即SKIPIF1<0,所以不等式的解集為SKIPIF1<0,故C正確;對(duì)于D:對(duì)于函數(shù)SKIPIF1<0的圖象對(duì)稱(chēng)中心為SKIPIF1<0,故D錯(cuò)誤.故選:ABC.11.已知SKIPIF1<0,SKIPIF1<0,則下列選項(xiàng)一定正確的是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】ABD【解析】【分析】對(duì)于A,利用等量代換整理函數(shù)解析式,利用二次函數(shù)的性質(zhì),可得答案;對(duì)于B,利用基本不等式,可得答案;對(duì)于C,利用反例,可得答案;對(duì)于D,利用等量代換整理函數(shù)解析式,利用導(dǎo)數(shù)研究其最值,可得答案.【詳解】對(duì)于A,由SKIPIF1<0,則SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,故A正確;對(duì)于B,由SKIPIF1<0,即SKIPIF1<0,則SKIPIF1<0,當(dāng)且僅當(dāng)SKIPIF1<0時(shí)等號(hào)成立,故B正確;對(duì)于C,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,而SKIPIF1<0,故C錯(cuò)誤;對(duì)于D,由SKIPIF1<0,則SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,令SKIPIF1<0,SKIPIF1<0,令SKIPIF1<0,解得SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0;當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,所以函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增,在SKIPIF1<0上單調(diào)遞減,故SKIPIF1<0,故D正確;故選:ABD.12.已知函數(shù)SKIPIF1<0圖象關(guān)于SKIPIF1<0軸對(duì)稱(chēng),且SKIPIF1<0,SKIPIF1<0都有SKIPIF1<0.若不等式SKIPIF1<0,對(duì)SKIPIF1<0恒成立,則SKIPIF1<0的取值可以為()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0【答案】BC【解析】【分析】由題可得SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱(chēng),且在SKIPIF1<0上單調(diào)遞增,進(jìn)而將不等式轉(zhuǎn)化為SKIPIF1<0,對(duì)SKIPIF1<0恒成立,然后利用換元法結(jié)合二次函數(shù)的性質(zhì)可得SKIPIF1<0的取值范圍,即得.【詳解】因函數(shù)SKIPIF1<0圖象關(guān)于SKIPIF1<0軸對(duì)稱(chēng),所以SKIPIF1<0的圖象關(guān)于SKIPIF1<0對(duì)稱(chēng),又SKIPIF1<0,SKIPIF1<0都有SKIPIF1<0,所以函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增,因?yàn)椴坏仁絊KIPIF1<0,對(duì)SKIPIF1<0恒成立,所以SKIPIF1<0,對(duì)SKIPIF1<0恒成立,令SKIPIF1<0,則SKIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0,對(duì)SKIPIF1<0恒成立,因?yàn)镾KIPIF1<0,SKIPIF1<0,SKIPIF1<0,故SKIPIF1<0,所以BC正確,AD錯(cuò)誤.故選;BC.【點(diǎn)睛】方法點(diǎn)睛:恒(能)成立問(wèn)題的解法:若SKIPIF1<0在區(qū)間SKIPIF1<0上有最值,則(1)恒成立:SKIPIF1<0;SKIPIF1<0;(2)能成立:SKIPIF1<0;SKIPIF1<0.若能分離常數(shù),即將問(wèn)題轉(zhuǎn)化為:SKIPIF1<0(或SKIPIF1<0),則(1)恒成立:SKIPIF1<0;SKIPIF1<0;(2)能成立:SKIPIF1<0;SKIPIF1<0.三、填空題(本題共4小題,每題4分,共16分)13.已知SKIPIF1<0為第三象限角,SKIPIF1<0,則SKIPIF1<0___________.【答案】SKIPIF1<0【解析】【分析】根據(jù)同角三角函數(shù)商式關(guān)系以及平方和關(guān)系,可得答案.【詳解】由SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,由SKIPIF1<0,則SKIPIF1<0,由SKIPIF1<0為第三象限角,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0.故答案為:SKIPIF1<0.14.函數(shù)SKIPIF1<0為偶函數(shù),當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,則SKIPIF1<0時(shí),SKIPIF1<0___________.【答案】SKIPIF1<0【解析】【分析】由偶函數(shù)的定義求解.【詳解】SKIPIF1<0時(shí),SKIPIF1<0,SKIPIF1<0是偶函數(shù),∴SKIPIF1<0,故答案為:SKIPIF1<0.15.科學(xué)家通過(guò)生物標(biāo)本中某種放射性元素的存量來(lái)估算該生物的年代,已知某放射性元素的半衰期約為1620年(即:每經(jīng)過(guò)1620年,該元素的存量為原來(lái)的一半),某生物標(biāo)本中該元素的初始存量為SKIPIF1<0,經(jīng)檢測(cè)生物中該元素現(xiàn)在的存量為SKIPIF1<0,(參考數(shù)據(jù):SKIPIF1<0)請(qǐng)推算該生物距今大約___________年.【答案】3780【解析】【分析】由指數(shù)函數(shù)模型求解.【詳解】設(shè)放射性元素的存量模型為SKIPIF1<0,由已知SKIPIF1<0,所以SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,設(shè)題中所求時(shí)間為SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,∴SKIPIF1<0,SKIPIF1<0.故答案為:3780.16.定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,則函數(shù)SKIPIF1<0的所有零點(diǎn)之和為_(kāi)__________.【答案】SKIPIF1<0【解析】【分析】畫(huà)出函數(shù)SKIPIF1<0與SKIPIF1<0圖象,根據(jù)對(duì)稱(chēng)性以及對(duì)數(shù)函數(shù)的運(yùn)算得出零點(diǎn)之和.【詳解】令SKIPIF1<0,即SKIPIF1<0,故函數(shù)SKIPIF1<0的零點(diǎn)就是函數(shù)SKIPIF1<0與SKIPIF1<0圖象交點(diǎn)的橫坐標(biāo),當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,函數(shù)SKIPIF1<0與SKIPIF1<0在SKIPIF1<0上圖象如圖所示:設(shè)SKIPIF1<0與SKIPIF1<0圖象交點(diǎn)的橫坐標(biāo)分別為SKIPIF1<0,由對(duì)稱(chēng)性可知,SKIPIF1<0,SKIPIF1<0.由SKIPIF1<0,結(jié)合奇偶性得出SKIPIF1<0,即SKIPIF1<0解得SKIPIF1<0,即SKIPIF1<0.故答案為:SKIPIF1<0四、解答題(本題共6小題,共44分)17.計(jì)算:(1)SKIPIF1<0;(2)SKIPIF1<0.【答案】(1)4(2)SKIPIF1<0【解析】【分析】根據(jù)對(duì)數(shù)運(yùn)算與指數(shù)運(yùn)算,可得答案.【小問(wèn)1詳解】原式SKIPIF1<0.【小問(wèn)2詳解】原式SKIPIF1<018.小明家院子中有塊不規(guī)則空地,如圖所示.小明測(cè)量并計(jì)算得出空地邊緣曲線擬合函數(shù)SKIPIF1<0,小明的爸爸打算改造空地,用家中現(xiàn)有的8米長(zhǎng)的柵欄如圖圍一面靠墻矩形空地SKIPIF1<0用來(lái)鋪設(shè)草皮,請(qǐng)問(wèn)小明的爸爸需要購(gòu)買(mǎi)多少平方米的草皮才能鋪滿矩形草地?(不考慮材料的損耗)【答案】SKIPIF1<0.【解析】【分析】設(shè)SKIPIF1<0,進(jìn)而可得SKIPIF1<0,根據(jù)條件可得方程,然后結(jié)合條件即得.【詳解】設(shè)SKIPIF1<0,因?yàn)镾KIPIF1<0,則SKIPIF1<0,所以SKIPIF1<0,解得SKIPIF1<0,即SKIPIF1<0,SKIPIF1<0,此時(shí)矩形SKIPIF1<0的面積為SKIPIF1<0,即小明的爸爸需要購(gòu)買(mǎi)6平方米的草皮才能鋪滿矩形草地.19.已知集合SKIPIF1<0,SKIPIF1<0.(1)當(dāng)SKIPIF1<0時(shí),求SKIPIF1<0;(2)若SKIPIF1<0,求實(shí)數(shù)SKIPIF1<0的取值范圍.【答案】(1)SKIPIF1<0;(2)SKIPIF1<0.【解析】【分析】(1)解分式不等式可得SKIPIF1<0集合,后根據(jù)并集的定義運(yùn)算即可;(2)由題可得SKIPIF1<0,然后分類(lèi)討論,結(jié)合子集的定義即得.【小問(wèn)1詳解】因?yàn)镾KIPIF1<0,SKIPIF1<0故SKIPIF1<0;【小問(wèn)2詳解】若SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,符合SKIPIF1<0;②SKIPIF1<0,SKIPIF1<0,不符合SKIPIF1<0,舍去;③SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0;綜上,實(shí)數(shù)SKIPIF1<0的取值范圍為SKIPIF1<0.20.已知函數(shù)SKIPIF1<0.(1)判斷函數(shù)奇偶性并證明;(2)設(shè)函數(shù)SKIPIF1<0,若函數(shù)SKIPIF1<0與SKIPIF1<0的圖象沒(méi)有公共點(diǎn),求實(shí)數(shù)SKIPIF1<0的取值范圍.【答案】(1)偶函數(shù),證明見(jiàn)解析(2)SKIPIF1<0【解析】【分析】(1)根據(jù)偶函數(shù)的定義,可得答案;(2)根據(jù)函數(shù)與方程的關(guān)系,利用二次函數(shù)的性質(zhì),可得答案.【小問(wèn)1詳解】函數(shù)定義域?yàn)镾KIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0是偶函數(shù).【小問(wèn)2詳解】等價(jià)于方程SKIPIF1<0沒(méi)有實(shí)數(shù)根.令SKIPIF1<0,則SKIPIF1<0沒(méi)有大于SKIPIF1<0的根,令SKIPIF1<0.①SKIPIF1<0時(shí),SKIPIF1<0符合;②SKIPIF1<0時(shí),對(duì)稱(chēng)軸SKIPIF1<0,SKIPIF1<0,無(wú)正根符合;③SKIPIF1<0時(shí),對(duì)稱(chēng)軸SKIPIF1<0,SKIPIF1<0,有一根大于SKIPIF1<0,不符合.綜上,SKIPIF1<0.21.三角函數(shù)變形化簡(jiǎn)中常用“切割化弦”的技巧.其中“弦”指正弦函數(shù)與余弦函數(shù),“切”指正切函數(shù)與余切函數(shù),“割”指正割函數(shù)與余割函數(shù).設(shè)SKIPIF1<0是一個(gè)任意角,如圖所示它的終邊上任意一點(diǎn)SKIPIF1<0(不與原點(diǎn)重合)的坐標(biāo)為SKIPIF1<0,SKIPIF1<0與原點(diǎn)SKIPIF1<0的距離為SKIPIF1<0,則SKIPIF1<0的正割函數(shù)定義為SKIPIF1<0.(1)已知函數(shù)SKIPIF1<0,寫(xiě)出SKIPIF1<0的定義域和單調(diào)區(qū)間;(2)方程SKIPIF1<0在SKIPIF1<0所有根的和為SKIPIF1<0,求SKIPIF1<0的值.【答案】(1)詳見(jiàn)解析;(2)1.【解析】【分析】(1)利用正割函數(shù)的定義可得函數(shù)的定義域及函數(shù)的單調(diào)區(qū)間;或使用轉(zhuǎn)化思想,將對(duì)正割函數(shù)的研究轉(zhuǎn)化為已學(xué)的余弦函數(shù),進(jìn)而即得;(2)根據(jù)函數(shù)的奇偶性可得SKIPIF1<0,進(jìn)而即得.【小問(wèn)1詳解】解法一:根據(jù)正割函數(shù)定義,SKIPIF1<0是一個(gè)任意角,它的終邊上任意一點(diǎn)SKIPIF1<0(不與原點(diǎn)重合)的坐標(biāo)為SKIPIF1<0,因?yàn)镾KIPIF1<0,顯然SKIPIF1<0,因此角的終邊不能落在SKIPIF1<0軸上,結(jié)合終邊相同的角的表示,正割函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0,且因?yàn)镾KIPIF1<0,SKIPIF1<0是該函數(shù)的一個(gè)周期.SKIPIF1<0為大于0的定值,當(dāng)SKIPIF1<0時(shí),此時(shí)SKIPIF1<0越大即弧度制下的角越大,因此角終邊上的點(diǎn)的橫坐標(biāo)越小,SKIPIF1<0與橫坐標(biāo)的比值就越大,所以SKIPIF1<0為函數(shù)的一個(gè)單調(diào)增區(qū)間,結(jié)合該函數(shù)的周期,SKIPIF1<0為函數(shù)SKIPIF1<0的單調(diào)增區(qū)間,同理SKIPIF1<0為函數(shù)SKIPIF1<0的單調(diào)增區(qū)間,SKIPIF1<0和SKIPIF1<0為SKIPIF1<0的單調(diào)減區(qū)間;解法二:SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,定義域?yàn)镾KIPIF1<0,當(dāng)SKIPIF1<0時(shí),SKIPIF1<0在區(qū)間SKIPIF1<0和SKIPIF1<0單調(diào)遞減,所以SKIPIF1<0的單調(diào)增區(qū)間為SKIPIF1<0和SKIPIF1<0;SKIPIF1<0在區(qū)間SKIPIF1<0和SKIPIF1<0單調(diào)遞增,所以SKIPIF1<0的單調(diào)減區(qū)間為SKIPIF1<0和SKIPIF1<0;【小問(wèn)2詳解】因?yàn)镾KIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,故兩函數(shù)均為偶函數(shù),所以它們函數(shù)圖象的交點(diǎn)關(guān)于SKIPIF1<0軸對(duì)稱(chēng),因此方程SKIPIF1<0的根的和為0,也即SKIPIF1<0,所以SKIPIF1<0.22.已知函數(shù)SKIPIF1<0,在區(qū)間SKIPIF1<0
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