2006年三峡大学第一届“求索杯”数学竞赛
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-SIZE: 16pt;"><font face="Times New Roman"><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font size="3"><p></p></font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"></p></font></span></b><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman"> <p></p></font></span><p></p><span lang="EN-US" style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';"><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-SIZE: 16pt;"><font face="Times New Roman">2006</font></span></b><b style="mso-bidi-font-weight: normal;"><span style="FONT-SIZE: 16pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">年三峡大学第一届“求索杯”数学竞赛</span></b><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="FONT-SIZE: 16pt;"><p></p></span></b></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">时间</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">120</font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分钟</span><span style="FONT-SIZE: 14pt;"><font face="Times New Roman"> </font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">总分</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">100</font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><p></p></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US" style="FONT-SIZE: 14pt;"><span style="mso-spacerun: yes;"><font face="Times New Roman"> </font></span></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">姓名：</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">_____<span style="mso-spacerun: yes;"> </span></font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">学号：</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">______ </font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">专业：</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">______<span style="mso-spacerun: yes;"> </span></font></span><span style="FONT-SIZE: 14pt; FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">成绩：</span><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman">_______<p></p></font></span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US" style="FONT-SIZE: 14pt;"><font face="Times New Roman"> <p></p></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; mso-list: l3 level1 lfo1; tab-stops: list 21.0pt;"><span lang="EN-US" style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">一、</span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">填空题</span><span lang="EN-US"><font face="Times New Roman">(15</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分</span><span lang="EN-US"><font face="Times New Roman">)</font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l2 level1 lfo2; tab-stops: list 18.0pt;"><span lang="EN-US"><font face="Times New Roman">1.<span style="FONT: 7pt 'Times New Roman';"> </span></font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shapetype id="_x0000_t75" stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" opreferrelative="t" ospt="75" coordsize="21600,21600"><font face="Times New Roman"> <stroke joinstyle="miter"></stroke><formulas><f eqn="if lineDrawn pixelLineWidth 0"></f><f eqn="sum @0 1 0"></f><f eqn="sum 0 0 @1"></f><f eqn="prod @2 1 2"></f><f eqn="prod @3 21600 pixelWidth"></f><f eqn="prod @3 21600 pixelHeight"></f><f eqn="sum @0 0 1"></f><f eqn="prod @6 1 2"></f><f eqn="prod @7 21600 pixelWidth"></f><f eqn="sum @8 21600 0"></f><f eqn="prod @7 21600 pixelHeight"></f><f eqn="sum @10 21600 0"></f></formulas><path gradientshapeok="t"></path><lock aspectratio="t" vext="edit"></lock></font></shapetype><shape id="_x0000_i1025" oole="" type="#_x0000_t75" style="WIDTH: 84.75pt; HEIGHT: 18pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image001.wmz"></imagedata></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在实数轴上有</span><span lang="EN-US"><font face="Times New Roman">__________</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">个零点。</span><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US"><p></p></span></i></b></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l2 level1 lfo2; tab-stops: list 18.0pt;"><span lang="EN-US"><font face="Times New Roman">2.<span style="FONT: 7pt 'Times New Roman';"> </span></font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1026" oole="" type="#_x0000_t75" style="WIDTH: 45pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image003.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">由方程</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1027" oole="" type="#_x0000_t75" style="WIDTH: 116.25pt; HEIGHT: 18pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image005.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">确定，则</span><span lang="EN-US"><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1028" oole="" type="#_x0000_t75" style="WIDTH: 49.5pt; HEIGHT: 42pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image007.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">＝＿＿＿＿</span><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US"><font face="Times New Roman"> <p></p></font></span></i></b></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">3.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">若</span><span lang="EN-US"><span style="mso-text-raise: -33.0pt;"><shape id="_x0000_i1029" oole="" type="#_x0000_t75" style="WIDTH: 180pt; HEIGHT: 63pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image009.wmz"></imagedata></font></shape></span><span style="mso-spacerun: yes;"><font face="Times New Roman"> </font></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，则常数</span><span lang="EN-US"><font face="Times New Roman">a=_____,b=______,c=______</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">。</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">二、选择题：（</span><span lang="EN-US"><font face="Times New Roman">15</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">１</span><span lang="EN-US"><font face="Times New Roman">.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1030" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image011.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">与</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1031" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image013.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">均可导，且</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1032" oole="" type="#_x0000_t75" style="WIDTH: 60.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image015.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，则必有＿＿＿＿．</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">A</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">）</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1033" oole="" type="#_x0000_t75" style="WIDTH: 66pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image017.wmz"></imagedata></font></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(B) </font><span style="mso-text-raise: -15.0pt;"><shape id="_x0000_i1034" oole="" type="#_x0000_t75" style="WIDTH: 2in; HEIGHT: 45pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image019.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 5.25pt; mso-char-indent-count: .5; mso-char-indent-size: 10.5pt;"><span lang="EN-US"><font face="Times New Roman">(C) </font><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1035" oole="" type="#_x0000_t75" style="WIDTH: 75pt; HEIGHT: 15.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image021.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(D) </font><span style="mso-text-raise: -11.0pt;"><shape id="_x0000_i1036" oole="" type="#_x0000_t75" style="WIDTH: 123.75pt; HEIGHT: 27.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image023.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 5.25pt; mso-char-indent-count: .5; mso-char-indent-size: 10.5pt;"><span lang="EN-US"><font face="Times New Roman">2.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1037" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image025.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">满足：</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1038" oole="" type="#_x0000_t75" style="WIDTH: 123pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image027.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，又在</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1039" oole="" type="#_x0000_t75" style="WIDTH: 32.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image029.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">有</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1040" oole="" type="#_x0000_t75" style="WIDTH: 53.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image031.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，则</span><span lang="EN-US"><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1041" oole="" type="#_x0000_t75" style="WIDTH: 35.25pt; HEIGHT: 30.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image033.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">＝</span><span lang="EN-US"><font face="Times New Roman">________.<b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><p></p></i></b></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 5.25pt; mso-char-indent-count: .5; mso-char-indent-size: 10.5pt;"><span lang="EN-US"><font face="Times New Roman">(A) </font><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1042" oole="" type="#_x0000_t75" style="WIDTH: 18.75pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image035.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(B) </font><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1043" oole="" type="#_x0000_t75" style="WIDTH: 11.25pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image037.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(C) </font><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1044" oole="" type="#_x0000_t75" style="WIDTH: 20.25pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image039.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(D) </font><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1045" oole="" type="#_x0000_t75" style="WIDTH: 12pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image041.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 5.25pt; mso-char-indent-count: .5; mso-char-indent-size: 10.5pt;"><span lang="EN-US"><font face="Times New Roman">3.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1046" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image043.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">连续，则</span><span lang="EN-US"><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1047" oole="" type="#_x0000_t75" style="WIDTH: 98.25pt; HEIGHT: 30.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image045.wmz"></imagedata></font></shape></span><font face="Times New Roman">________.<b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><p></p></i></b></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 5.25pt; mso-char-indent-count: .5; mso-char-indent-size: 10.5pt;"><span lang="EN-US"><font face="Times New Roman">(A) </font><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1048" oole="" type="#_x0000_t75" style="WIDTH: 30pt; HEIGHT: 15.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image047.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(B) </font><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1049" oole="" type="#_x0000_t75" style="WIDTH: 41.25pt; HEIGHT: 30.75pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image049.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>(C) </font><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1050" oole="" type="#_x0000_t75" style="WIDTH: 35.25pt; HEIGHT: 18pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image051.wmz"><font face="Times New Roman"></font></imagedata></shape></span><font face="Times New Roman"><span style="mso-spacerun: yes;"> </span>(D) </font><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1051" oole="" type="#_x0000_t75" style="WIDTH: 51.75pt; HEIGHT: 18pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image053.wmz"><font face="Times New Roman"></font></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">二、解答题（</span><span lang="EN-US"><font face="Times New Roman">70</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l1 level1 lfo3; tab-stops: list 18.0pt;"><span lang="EN-US"><font face="Times New Roman">1．<span style="FONT: 7pt 'Times New Roman';"> </span></font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1052" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image055.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">除</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1053" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 14.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image057.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">与</span><span lang="EN-US"><font face="Times New Roman">1</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">两点外，对全体实数都有定义，并满足等式：</span><span lang="EN-US"><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1054" oole="" type="#_x0000_t75" style="WIDTH: 110.25pt; HEIGHT: 30.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image059.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，求符合这一条件的所有函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1055" oole="" type="#_x0000_t75" style="WIDTH: 27pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image055.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">。（</span><span lang="EN-US"><font face="Times New Roman">5</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">2.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1056" oole="" type="#_x0000_t75" style="WIDTH: 38.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image062.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在域</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1057" oole="" type="#_x0000_t75" style="WIDTH: 149.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image064.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上连续，对任意</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1058" oole="" type="#_x0000_t75" style="WIDTH: 47.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image066.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，设</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1059" oole="" type="#_x0000_t75" style="WIDTH: 36pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image068.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是以</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1060" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image070.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为中心且全含于</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1061" oole="" type="#_x0000_t75" style="WIDTH: 11.25pt; HEIGHT: 14.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image072.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">内且各边与</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1062" oole="" type="#_x0000_t75" style="WIDTH: 11.25pt; HEIGHT: 14.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image074.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">的边平行的最大正方形，若总有</span><span lang="EN-US"><span style="mso-text-raise: -17.0pt;"><shape id="_x0000_i1063" oole="" type="#_x0000_t75" style="WIDTH: 101.25pt; HEIGHT: 30.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image076.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，问：</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1064" oole="" type="#_x0000_t75" style="WIDTH: 38.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image078.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1065" oole="" type="#_x0000_t75" style="WIDTH: 11.25pt; HEIGHT: 14.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image080.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上恒等于零吗？（过程）（</span><span lang="EN-US"><font face="Times New Roman">15</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman"> <p></p></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">3.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">证明：若级数</span><span lang="EN-US"><span style="mso-text-raise: -15.0pt;"><shape id="_x0000_i1066" oole="" type="#_x0000_t75" style="WIDTH: 32.25pt; HEIGHT: 35.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image082.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">收敛（</span><span lang="EN-US"><span style="mso-text-raise: -6.0pt;"><shape id="_x0000_i1067" oole="" type="#_x0000_t75" style="WIDTH: 71.25pt; HEIGHT: 18pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image084.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为正实数），则级数</span><span lang="EN-US"><span style="mso-text-raise: -15.0pt;"><shape id="_x0000_i1068" oole="" type="#_x0000_t75" style="WIDTH: 122.25pt; HEIGHT: 35.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image086.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">也收敛。（</span><span lang="EN-US"><font face="Times New Roman">15</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">4.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">计算</span><span lang="EN-US"><span style="mso-text-raise: -12.0pt;"><shape id="_x0000_i1069" oole="" type="#_x0000_t75" style="WIDTH: 119.25pt; HEIGHT: 30.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image088.wmz"></imagedata></font></shape></span><font face="Times New Roman">.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">（</span><span lang="EN-US"><font face="Times New Roman">5</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman">5.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">微分学中一个并不罕见的错误是将乘积的求导数的求导法则理解成</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1070" oole="" type="#_x0000_t75" style="WIDTH: 68.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image090.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，若</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1071" oole="" type="#_x0000_t75" style="WIDTH: 51.75pt; HEIGHT: 20.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image092.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，问是否存在一个开区间</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1072" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image094.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">以及定义于</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1073" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image096.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">上的一个非零函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1074" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image098.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">，使得这一错误的乘积求导法则对</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1075" oole="" type="#_x0000_t75" style="WIDTH: 27.75pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image100.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">中的</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1076" oole="" type="#_x0000_t75" style="WIDTH: 9.75pt; HEIGHT: 11.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image102.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">是成立的。（</span><span lang="EN-US"><font face="Times New Roman">10</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman"> <p></p></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">６</span><span lang="EN-US"><font face="Times New Roman">.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">设函数</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1077" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image104.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">在［０，１］上可导，并有</span></p><p class="MsoNormal" align="center" style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: center;"><span lang="EN-US"><span style="mso-text-raise: -9.0pt;"><shape id="_x0000_i1078" oole="" type="#_x0000_t75" style="WIDTH: 89.25pt; HEIGHT: 26.25pt;"><imagedata otitle="" src="file:///F:/msoclip1/03/clip_image106.wmz"></imagedata></shape></span></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">其中</span><span lang="EN-US"><span style="mso-text-raise: -3.0pt;"><shape id="_x0000_i1079" oole="" type="#_x0000_t75" style="WIDTH: 9.75pt; HEIGHT: 11.25pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image108.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">为实常数，试求</span><span lang="EN-US"><span style="mso-text-raise: -5.0pt;"><shape id="_x0000_i1080" oole="" type="#_x0000_t75" style="WIDTH: 26.25pt; HEIGHT: 15.75pt;"><font face="Times New Roman"> <imagedata otitle="" src="file:///F:/msoclip1/03/clip_image110.wmz"></imagedata></font></shape></span></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">。（</span><span lang="EN-US"><font face="Times New Roman">10</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">７</span><span lang="EN-US"><font face="Times New Roman">.</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">证明：任意正有理数必为调和级数中有限项之和。（</span><span lang="EN-US"><font face="Times New Roman">10</font></span><span style="FONT-FAMILY: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman';">分）</span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt;"><span lang="EN-US"><font face="Times New Roman"> <p></p></font></span></p><p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt 21pt; TEXT-INDENT: -21pt; mso-list: l3 level1 lfo1; tab-stops: list 21.0pt;"> </p></span>
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