数学实验习题集及部分答案

2012-2013学年第二学期数学实验综合实验题目一、Matlab 基本操作 参看上课ppt二、Matlab 绘图操作 1. 绘制一个正方体；2. 在同一个坐标系下绘制以下两个函数的图形，要求1y 用红色的实线点标记为星号，2y 用蓝色的虚线点标记为小方格，步长为0.01，x 轴和y 轴分别给出标识，图中有网格表示，其中：212sin(2),;cos (),22y x x y x x πππ=-≤≤=-≤≤.x1=-pi:0.01:pi; y1=sin(2*x1);x2=-2*pi:0.01:2*pi; y2=cos(x2).^2;plot(x1,y1,'r-*',x2,y2,'b--s') xlabel('X');ylabel('Y') grid on83.在绘图窗口1绘出1y 曲线, 在绘图窗口2绘出2y 曲线, 1y 为红色实线, 2y 为蓝色虚线,步长为0.01.x 轴和y 轴分别给出标识，其中参数方程为233222,()11at at x x y y t t a x===++-，直角方程：()sin 2sin23sin302y x x x x π=++≤≤.t=0:0.01:2*pi;a=2;x1=a*t.^2./(1+t.^2);y1=a*t.^3./(1+t.^2); y2=sin(t)+2*sin(2*t)+3*sin(3*t); plot(x1,y1,'r-',t,y2,'b--') xlabel('X');ylabel('Y')0.4cos )cos ,(30.4cos )sin ,0.4sin u v y u v z v+=+=.5.绘制椭球面3cos sin ,2cos cos ,sin x u v y u v z u ===的图形.clear;uu=linspace(0,2*pi,50);vv=uu; [u,v]=meshgrid(uu,vv);x=3*cos(u).*sin(v);y=2*cos(u).*cos(v);z=sin(u); mesh(x,y,z)6.绘制单叶双曲面3sec sin,2sec cos,4tanx u v y u v z u===的图形. clear;uu=linspace(0,2*pi,50);vv=uu;[u,v]=meshgrid(uu,vv);x=3*sec(u).*sin(v);y=2*sec(u).*cos(v);z=4*tan( u);mesh(x,y,z)7.绘制双曲抛物面22,,3u vx u y v z-===的图形.clear;uu=linspace(0,2*pi,50);vv=uu; [u,v]=meshgrid(uu,vv);x=u;y=v;z=(u.^2-v.^2)/3;mesh(x,y,z)8.绘制旋转面ln sin,ln cos,x u v y u v z u===的图形.clear;uu=linspace(0,2*pi,50);vv=uu;[u,v]=meshgrid(uu,vv);x=log(u).*sin(v);y=log(u).*cos(v);z=u;mesh(x,y,z)9.绘制圆锥面sin,cos,x u v y u v z u===的图形. clear;uu=linspace(0,2*pi,50);vv=uu;[u,v]=meshgrid(uu,vv);x=u.*sin(v);y=u.*cos(v);z=u;mesh(x,y,z)10.绘制正螺面sin,cos,4x u v y u v z v===的图形.clear;uu=linspace(0,2*pi,50);vv=uu;[u,v]=meshgrid(uu,vv);x=u.*sin(v);y=u.*cos(v);z=4*v; mesh(x,y,z)11.绘制贝壳曲面cos sin ,sin sin ,cos ln tan 0.22vx u v y u v z v u ===++的图形.clear;uu=linspace(0,2*pi,50);vv=uu;[u,v]=meshgrid(uu,vv);x=cos(u).*sin(v);y=sin(u).*cos(v);z=cos(v)+log (abs(tan(v/2)+eps))+0.2*u; mesh(x,y,z)三、计算极限 1.1)1232(lim +∞→++x x x x ； syms x;limit(((2*x+3)/(2*x+1))^(x+1),x,inf)ans =exp(1) 2.xx xx sin 2cos 1lim0-→； syms x;limit((1-cos(2*x))/(x*sin(x)),x,0) ans =23. 求极限)1(lim 2x x x x -++∞→； symsx;limit(x*(sqrt(x^2+1)-x),x,inf,'left') ans =1/24.4586lim 224+-+-→x x x x x ； syms x;limit((x^2-6*x+8)/(x^2-5*x+4),x,4) ans =2/3 5.)25sin(2lim nnn ∞→； syms n;limit(2^n*sin(5/2^n),n,inf) ans =5clear;limit(2^n*sin(5/2^n),'n',inf) ??? Reference to a cleared variable n.6. 145lim 1---→x xx x ；syms x;limit((sqrt(5*x-4)-sqrt(x))/(x-1),x,1) ans =2 7.)(lim 22x x x x x --++∞→syms x;limit(sqrt(x^2+x)-sqrt(x^2-x),x,inf,'left') ans =18.)2cos 2ln(lim 6x x π→syms x;limit(log(2*cos(2*x)),x,pi/6) ans =0 9.)12)(11(lim 2xx x -+∞→ syms x;limit((1+1/x)*(2-1/x^2),x,inf)ans =2 10．xx x x x x -++-+→2sin 1sin 1tan 1limsyms x;limit((sqrt(1+tan(x))-sqrt(1+sin(x)))/(x*sqrt(1+sin(x)^2)-x),x,0) ans =1/2 11.)112(lim 2xx x +-∞→ syms x;limit(2-1/x+1/x^2,x,inf) ans =2 12.xx x tan 2)(sin lim π→syms x;limit(sin(x)^tan(x),x,pi/2) ans =113.121lim 22---∞→x x x xsyms x;limit((x^2-1)/(2*x^2-x-1),x,inf) ans =1/214.52lim 21+-→x x xsyms x;limit(sqrt(x^-2*x+5),x,1) ans =6^(1/2) 15.35)3)(2)(1(limnn n n n +++∞→ syms n;limit(((n+1)*(n+2)*(n+3))/(5*n^3),n,inf)ans =1/5 16. xx x sin 30)21(lim +→ symsx;limit((1+2*x)^(3/sin(x)),x,0) ans =exp(6) 17.xxx arctan lim∞→ symsx;limit(atan(x)/x,x,inf) ans =018. xx x 11lim 0-+→syms x;limit((sqrt(x+1)-1)/x,x,0) ans =1/2四、计算偏导数(x y z∂∂∂2和22x z ∂∂)1.13323+--=xy xy y x zsyms x y;diff(diff(x^3*y^2-3*x*y^2-x*y+1,y,1),x,1), diff(x^3*y^2-3*x*y^2-x*y+1,x,2)ans =6*y*x^2 - 6*y - 1 ans =6*x*y^2clear;syms x y;diff(x^3*y^2-3*x*y^2-x*y+1,x,2) ans =6*x*y^2sin()0,xy dy e x y dx+=, syms xy;dx=diff(exp(x*y)+x*sin(y),x),dy=diff(exp(x*y )+x*sin(y),y),dx =sin(y) + y*exp(x*y) dy =x*exp(x*y) + x*cos(y) dy_dx=-1*dx/dydy_dx =-(sin(y) + y*exp(x*y))/(x*exp(x*y) + x*cos(y))2.22444y x y x z -+=syms x y;diff(diff(x^4+y^4-4*x^2*y^2,y,1),x,1), diff(x^4+y^4-4*x^2*y^2,x,2) ans =(-16)*x*yans =12*x^2 - 8*y^2 3.222ax ya xy z ++=syms x ya;diff(diff(x*y^2+y*a^2+a*x^2,y,1),x,1), diff(x*y^2+y*a^2+a*x^2,x,2) ans =2*y ans =2*a 4.yx z tanln = syms x y a;diff(diff(log(tan(x/y)),y,1),x,1), diff(log(tan(x/y)),x,2)ans =(x*(tan(x/y)^2 + 1)^2)/(y^3*tan(x/y)^2) - (tan(x/y)^2 + 1)/(y^2*tan(x/y)) - (2*x*(tan(x/y)^2 + 1))/y^3ans =(2*(tan(x/y)^2 + 1))/y^2 - (tan(x/y)^2 + 1)^2/(y^2*tan(x/y)^2)5.)(cos )sin(2xy xy z +=syms x ya;diff(diff(sin(x*y)+cos(x*y)^2,y,1),x,1), diff(sin(x*y)+cos(x*y)^2,x,2)ans =- 2*x*y*cos(x*y)^2 - 2*cos(x*y)*sin(x*y) + cos(x*y) + 2*x*y*sin(x*y)^2 - x*y*sin(x*y) ans =- 2*y^2*cos(x*y)^2 + 2*y^2*sin(x*y)^2 - y^2*sin(x*y) 6.22y x z +=syms x y a;diff(diff(sqrt(x^2+y^2),y,1),x,1), diff(sqrt(x^2+y^2),x,2) ans =-(x*y)/(x^2 + y^2)^(3/2)ans =1/(x^2 + y^2)^(1/2) - x^2/(x^2 + y^2)^(3/2) 7.xye z =syms x y a;diff(diff(exp(x*y),y,1),x,1), diff(exp(x*y),x,2)ans =exp(x*y) + x*y*exp(x*y) ans =y^2*exp(x*y) 8.y x z =syms x y a;diff(diff(x^y,y,1),x,1), diff(x^y,x,2)ans =x^y/x + x^(y - 1)*y*log(x) ans =x^(y - 2)*y*(y - 1) 五、计算级数1.将函数x y arctan =在x=0处展开成5阶taylor 级数；taylor(atan(x),6,0) ans =x^5/5 - x^3/3 + x taylor(atan(x))ans =x^5/5 - x^3/3 + x2.将函数2x e y =在x=0处展开成6阶taylor 级数； taylor(exp(x^2),7,0)ans =x^6/6 + x^4/2 + x^2 + 1 taylor(exp(x^2),7)ans =x^6/6 + x^4/2 + x^2 + 13.将函数y=arccosx 在x=0处展开成7阶taylor 级数；taylor(acos(x),8,0)ans =- (5*x^7)/112 - (3*x^5)/40 - x^3/6 - x + pi/2 4.将函数xy 1=在x=-1处处展开成8阶taylor 级数；taylor(1/x,9,1)ans =(x - 1)^2 - x - (x - 1)^3 + (x - 1)^4 - (x - 1)^5 + (x - 1)^6 - (x - 1)^7 + (x - 1)^8 + 25.将函数xxe y =在x=0处展开成3阶taylor 级数； taylor(x*exp(x),4,0) ans =x^3/2 + x^2 + x6.将函数x y tan =在x=0处展开成4阶taylor 级数；taylor(tan(x),5,0) ans =x^3/3 + x7.将函数x y 2cos =在x=0处展开成5阶taylor 级数；taylor(cos(x)^2,6,0) ans =x^4/3 - x^2 + 18.将函数)1ln(2x x y ++=在x=0处展开成5阶taylor 级数；taylor(log(x+sqrt(1+x^2)),6,0) ans =(3*x^5)/40 - x^3/6 + x9.将函数xe y x-=1在x=0处展开成3阶taylor 级数；taylor(exp(x)/(1-x),4,0)ans =(8*x^3)/3 + (5*x^2)/2 + 2*x + 1 六、计算不定积分1. ⎰-+dx x x )1)(1(3;syms x;int((sqrt(x)+1)*(sqrt(x^3)-1))Warning: Explicit integral could not be found. ans =int((x^(1/2) + 1)*((x^3)^(1/2) - 1), x) 2.⎰x x xdx22sin cos 2cos ； syms x;int(cos(2*x)/(cos(x)^2*sin(x)^2)) ans =(t*cos(2*x))/(cos(x)^2*sin(x)^2)3. ⎰∙-∙dx xxx 32532；syms x;int((2*3^x-5*2^x)/3^x)ans =2*x - (5*2^x)/(3^x*log(2/3)) 4.⎰+dx x x x2)ln (ln 1 symsx;int((1+log(x))/(x*log(x))^2) ans =-1/(x*log(x))423x t a ++⎰syms x t a;int(4*x+2*t+3*a) ans =(3*a + 2*t + 4*x)^2/8 syms x t a;int(4*x+2*t+3*a,x) syms x t a;int(4*x+2*t+3*a,t) syms x t a;int(4*x+2*t+3*a,a) syms x t a;int(2*t+3*a) syms x t a;int(3*a,a,1,2) ans =9/2ans =(3*a + 2*t)^2/4 ans =(3*a + 2*t + 4*x)^2/8 ans =(3*a + 2*t + 4*x)^2/4 ans =(3*a + 2*t + 4*x)^2/65.⎰xdx e x 2sinsyms x; int(exp(x)*sin(x)^2) ans =1/5*(sin(x)-2*cos(x))*exp(x)*sin(x)+2/5*exp(x) 6.⎰+2142)1(dx x x syms x; int(x^2+x^(-4),1,2) ans =21/8 7.⎰-++02222x x dxsyms x; int((x^2+2*x+2)^2,-2,0) ans =56/158.⎰-+dx x x )1)(1(3syms x; int((sqrt(x)+1)*(sqrt(x^3)-1)) ans=-2/3*x^(3/2)+1/3*(x^3)^(1/2)*x^(3/2) +2/5*(x^3)^(1/2)*x-x七、计算二重积分 1.⎰⎰++Ddxdy y y x x)3(323，D={}10,10|),(≤≤≤≤y x y x ；syms x y;int(int('x^3+3*x^2*y+y^3','x',0,1),'y',1,2) ans =11/2 2.⎰⎰++Ddxdy y x y 221，其中D 是直线y=x 、x=-1和y=1所围成的闭区域； (-1<x<1,-1<y<1)syms x y; int(int('y*sqrt(1+x^2+y^2)','x',-1,1),'y',-1,1) ans =0 3.⎰⎰+Ddxdy y x )23(，其中D 是由两坐标轴及直线x+y=2所围成的闭区域. (0<x<2,0<y<2)syms x y; int(int('3*x+2*y','x',0,2),'y',0,2) ans =20 4.⎰⎰Dxydxdy ，其中D 是由直线y=1、x=2及y=x 所围成的闭区域； (1<x<2,1<x<2)syms x y; int(int('x*y','x',1,2),'y',1,2) ans =9/4 5.⎰⎰≤+++yy x dxdy y x 222)1(syms x y;double(int （int （'1+x+y','x','-sqrt(2*y-y.^2)','sqrt(2*y-y.^2)'),'y',0,1）) 6.⎰⎰+Dydxdy x sin )1(，其中D 是顶点（0,0），（1,0），（1,2）和（0,1）的梯形闭区域； syms x y;int(int('(1+x)*sin(y)','x',0,1), 'y',0,2)ans =-3/2*cos(2)+3/2 7.⎰⎰-Ddxdy y x )(22，其中D={}x y x y x sin 0,0|),(≤≤≤≤π；syms x y; int(int('x^2+y^2','y',0,'sin(x)'), 0,pi)ans =pi^2-32/98.⎰⎰+-+Ddxdy y x y)963(2，其中{}4|),(22≤+=y x y x D ；syms x y; int(int('y^2+3*x-6*y+9','x','-sqrt(4-y^2)','sqrt(4-y^2)'),0,2) ans =20*pi-32七、计算行列式1.1100121001310014a a a a ---.2.222233331111a b c d a b c d a b c d3.1100121001310014a a a a ---4.x yx yx yxy xy yxsyms xy;A=diag(x*ones(6,1),0)+diag(y*ones(5,1),1);A(6,1)=y;det(A) ans =x^6 - y^65.1112131415xa x a x a x a x a a -----6.1111111111a b c def7.cos 112cos 112cos 112cos 112cos 112cos t t t t t t8.111111111211111113111111141111111511111116b b b b b b ++++++9.12345623456134561245612356123461234510.11111a b a b a b a b a b a b a b a b a b a b a b +⨯+⨯+⨯+⨯+⨯+11.1234512345123451234512345x a x x x x x x b x x x x x x b x x x x x x b x x x x x x b----- 八、方程的解1. 4312251160x x x -++= roots([1 -12 0 25 116]) ans =11.7473 2.7028 -1.2251 + 1.4672i -1.2251 - 1.4672i2. 322.50.33320x x x -+-= roots([1 -2.5 0.33 -32]) ans =4.2193 -0.8596 + 2.6163i -0.8596 - 2.6163i 3. 322sin(4)0x x ex x x ---+=在[-1,4]内的根。