微积分基础实验报告

15曲阜师范大学《数学实验课程》实验报告实验名称:微积分基础（一）班级专业：2013级姓名：学号：日期：成绩：一、实验目的及要求1.了解mathematics的绘图，建表，求和，累乘等命令；2.掌握借助mathematics解决数学问题的方法；二、实验准备软件命令：Plot，Sum，Product，Table；理论知识：函数单调性，近似公式等三、实验内容与步骤练习一Graphics`Legend`Plot[{Sin[x],0.8x,x,1.2x},{x,-Pi,Pi},PlotStyle{RGBColor[0,1,0],RGBColor[1,0,0],RGBColor[0,0,1],RGBColor[1,1,0]},PlotLegend{Sinx,x,0.8x,1.2x},LegendPosition{1,.0}]-3-2-1123-3-2-11231.2x0.8xxSinx结论:由上图可知,y=x与y=sinx最接近,通过试验可知y=sinx在x=0处的导数为xg1PlotSinx,xPi332,x,Pi,Pig2GraphicsPointSize.02,PointPi3,3216-3-2-1123-3-2-1123g1PlotSinx,13xPi332,x,Pi,Pig2GraphicsPointSize.02,PointPi3,32-3-2-1123-1-0.50.511.5结论:由上图可知,k13时直线与正弦曲线在Pi3,32最接近.PlotSinx,xx36,xx36x5120,xx33x55x77,x,Pi,Pi,PlotStyleRGBColor0,1,0,RGBColor1,0,0,RGBColor0,0,1,RGBColor1,1,017-3-2-1123-2-112结论:由上图可知,yxx33x55x77最逼近ysinx.练习二Plotxx36,1x22,x,4,4,PlotStyleRGBColor0,1,0,RGBColor1,0,0-4-224-4-224结论:由上图可知,y0时,图像是上升的,y0时,图像是下降的.y0时,y有极大值;y上升时图像是凹的,下降时图像是凸的,y取极值时y的图像出现拐点.f[x_,n_]:=Sum[(-1)^k*x^(2*k+1)/((2*k+1)!),{k,0,n}];Do[Print[FindRoot[f[x,n],{x,3.0}]],{n,3,7}]{x3.07864}18{x3.14869}{x3.14115}{x3.14161}{x3.14159}练习三f[x_,n_]:=Sum[Sin[kx]/k,{k,1,n,2}];Plot[f[x,9],{x,-2Pi,2Pi}]-6-4-2246-0.75-0.5-0.250.250.50.75Graphicsf[x_,n_]:=Sum[Sin[kx]/k,{k,1,n,2}];Plot[f[x,566],{x,-2Pi,2Pi}]19-6-4-2246-0.75-0.5-0.250.250.50.75Graphics结论:由上图可知,当n值很大时,图像越来越接近于方形的波.改变函数f的定义f[x_,n_]:=Sum[Sin[kx]/k,{k,1,n}];Plot[f[x,9],{x,-2Pi,2Pi}]-6-4-2246-1.5-1-0.50.511.5Graphics练习四Plot[Sin[1/x],{x,-1,1}]20-1-0.50.51-1-0.50.51GraphicsPlot[Sin[1/x],{x,-0.1,0.1}]-0.1-0.050.050.1-1-0.50.51GraphicsPlot[Sin[1/x],{x,-0.001,0.001}]-0.001-0.00050.00050.001-1-0.50.51Graphics结论:由上图可知,区间越小,曲线震荡得越快.T=Table[{1/k,Sin[k]},{k,1,3000}];21P=ListPlot[T]0.0010.0020.0030.004-1-0.50.51Graphicsd=44T1=Table[{1/k,Sin[k]},{k,3,3000,d}];T2=Table[{1/k,Sin[k]},{k,6,3000,d}];P1=ListPlot[T1,PlotJoinedTrue,PlotStyleRGBColor[1,0,0]];P2=ListPlot[T2,PlotJoinedTrue,PlotStyleRGBColor[1,0,0]];Show[P,P1,P2]440.0010.0020.0030.004-0.8-0.6-0.4-0.2220.0010.0020.0030.004-0.20.20.40.60.80.0010.0020.0030.004-1-0.50.51Graphics练习五fgsin=Plot[Sin[x],{x,-4Pi,4Pi},PlotStyle{RGBColor[1,0,0]}];p[x_,n_]:=xProduct[1-x^2/((kPi)^2),{k,1,n}];fgproduct=Plot[p[x,5],{x,-4Pi,4Pi}];Show[fgsin,fgproduct]-10-5510-1-0.50.5123-10-5510-10-5510-10-5510-4-224Graphicsfgsin=Plot[Sin[x],{x,-4Pi,4Pi},PlotStyle{RGBColor[1,0,0]}];p[x_,n_]:=xProduct[1-x^2/((kPi)^2),{k,1,n}];fgproduct=Plot[p[x,50],{x,-4Pi,4Pi}];Show[fgsin,fgproduct]-10-5510-1-0.50.5124-10-5510-1-0.50.51-10-5510-1-0.50.51Graphics-10-5510-1-0.50.5125-10-5510-1-0.50.51-10-5510-1-0.50.51结论:由上图可知,当n增加时,Pnx逐渐向Sinx逼近.四、实验结果与分析本实验借助了mathematics软件对微积分学的极限，函数单调性，极值问题，傅里叶级数等问题进行验证，从函数图像和数值分析的角度，自己进行验证，实验，最终证明了理论的正确性。