微积分问题的计算机求解

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,第三章 微积分问题的计算机求解,微积分问题的解析解,函数的级数展开与级数求和问题求解,数值微分,数值积分问题,曲线积分与曲面积分的计算,3.1 微积分问题的解析解,3.1.1 极限问题的解析解,单变量函数的极限,格式1：,L=limit(fun,x,x0),格式2：,L=limit(fun,x,x0,left 或 right),例:试求解极限问题,syms x a b;,f=x*(1+a/x)x*sin(b/x);,L=limit(f,x,inf),L=,exp(a)*b,例:求解单边极限问题,syms x;,limit(exp(x3)-1)/(1-cos(sqrt(x-sin(x),x,0,right),ans=,12,在(-0.1,0.1)区间绘制出函数曲线：,x=-0.1:0.001:0.1;,y=(exp(x.3)-1)./(1-cos(sqrt(x-sin(x);,Warning:Divide by zero.,(Type warning off,MATLAB:,divideByZero to,suppress this warning.),plot(x,y,-,0,12,o),多变量函数的极限：,格式：,L,1,=limit(limit(f,x,x,0,),y,y,0,),或,L,1,=limit(limit(f,y,y,0,),x,x,0,),如果x,0,或y,0,不是确定的值，而是另一个变量的函数，如x-g(y)，则上述的极限求取顺序不能交换。,例：求出二元函数极限值,syms x y a;,f=exp(-1/(y2+x2)*sin(x)2/x2*(1+1/y2)(x+a2*y2);,L=limit(limit(f,x,1/sqrt(y),y,inf),L=,exp(a2),3.1.2 函数导数的解析解,函数的导数和高阶导数,格式：,y=diff(fun,x),%求导数(默认为1阶),y=diff(fun,x,n),%求n阶导数,例：,一阶导数：,syms x;f=sin(x)/(x2+4*x+3);,f1=diff(f);pretty(f1),cos(x)sin(x)(2 x+4),-,2 2 2,x +4 x+3 (x +4 x+3),原函数及一阶导数图：,x1=0:.01:5;,y=subs(f,x,x1);,y1=subs(f1,x,x1);,plot(x1,y,x1,y1,:),更高阶导数：,tic,diff(f,x,100);toc,elapsed_time=,4.6860,原函数4阶导数,f4=diff(f,x,4);pretty(f4),2,sin(x)cos(x)(2 x+4)sin(x)(2 x+4),-+4-12-,2 2 2 2 3,x +4 x+3 (x +4 x+3)(x +4 x+3),3,sin(x)cos(x)(2 x+4)cos(x)(2 x+4),+12-24-+48-,2 2 2 4 2 3,(x +4 x+3)(x +4 x+3)(x +4 x+3),4 2,sin(x)(2 x+4)sin(x)(2 x+4)sin(x),+24-72-+24-,2 5 2 4 2 3,(x +4 x+3)(x +4 x+3)(x +4 x+3),多元函数的偏导：,格式：,f=diff(diff(f,x,m),y,n),或,f=diff(diff(f,y,n),x,m),例：求其偏导数并用图表示。,syms x y z=(x2-2*x)*exp(-x2-y2-x*y);,zx=simple(diff(z,x),zx=,-exp(-x2-y2-x*y)*(-2*x+2+2*x3+x2*y-4*x2-2*x*y),zy=diff(z,y),zy=,(x2-2*x)*(-2*y-x)*exp(-x2-y2-x*y),直接绘制三维曲面,x,y=meshgrid(-3:.2:3,-2:.2:2);,z=(x.2-2*x).*exp(-x.2-y.2-x.*y);,surf(x,y,z),axis(-3 3-2 2-0.7 1.5),contour(x,y,z,30),hold on%绘制等值线,zx=-exp(-x.2-y.2-x.*y).*(-2*x+2+2*x.3+x.2.*y-4*x.2-2*x.*y);,zy=-x.*(x-2).*(2*y+x).*exp(-x.2-y.2-x.*y);%偏导的数值解,quiver(x,y,zx,zy)%绘制引力线,例,syms x y z;f=sin(x2*y)*exp(-x2*y-z2);,df=diff(diff(diff(f,x,2),y),z);df=simple(df);,pretty(df),2 2 2 2 2,-4 z exp(-x y-z)(cos(x y)-10 cos(x y)y x +4,2 4 2 2 4 2 2,sin(x y)x y+4 cos(x y)x y -sin(x y),多元函数的Jacobi矩阵:,格式：,J=jacobian(Y,X),其中，X是自变量构成的向量，Y是由各个函数构成的向量。,例：,试推导其 Jacobi 矩阵,syms r theta phi;,x=r*sin(theta)*cos(phi);,y=r*sin(theta)*sin(phi);,z=r*cos(theta);,J=jacobian(x;y;z,r theta phi),J=,sin(theta)*cos(phi),r*cos(theta)*cos(phi),-r*sin(theta)*sin(phi),sin(theta)*sin(phi),r*cos(theta)*sin(phi),r*sin(theta)*cos(phi),cos(theta),-r*sin(theta),0 ,隐函数的偏导数：,格式：,F=-diff(f,x,j,)/diff(f,x,i,),例：,syms x y;f=(x2-2*x)*exp(-x2-y2-x*y);,pretty(-simple(diff(f,x)/diff(f,y),3 2 2,-2 x+2+2 x +x y-4 x -2 x y,-,x(x-2)(2 y+x),3.1.3 积分问题的解析解,不定积分的推导：,格式：,F=int(fun,x),例：,用diff()函数求其一阶导数，再积分，,检验是否可以得出一致的结果。,syms x;y=sin(x)/(x2+4*x+3);y1=diff(y);,y0=int(y1);pretty(y0)%对导数积分,sin(x)sin(x),-1/2-+1/2-,x+3 x+1,对原函数求4 阶导数，再对结果进行4次积分,y4=diff(y,4);,y0=int(int(int(int(y4);,pretty(simple(y0),sin(x),-,2,x +4 x+3,例:,证明,syms a x;f=simple(int(x3*cos(a*x)2,x),f=1/16*(4*a3*x3*sin(2*a*x)+2*a4*x4+6*a2*x2*cos(2*a*x)-6*a*x*sin(2*a*x)-3*cos(2*a*x)-3)/a4,f1=x4/8+(x3/(4*a)-3*x/(8*a3)*sin(2*a*x)+.,(3*x2/(8*a2)-3/(16*a4)*cos(2*a*x);,simple(f-f1)%求两个结果的差,ans=,-3/16/a4,定积分与无穷积分计算:,格式：,I=int(f,x,a,b),格式：,I=int(f,x,a,inf),例：,syms x;I1=int(exp(-x2/2),x,0,1.5)无解,I1=,1/2*erf(3/4*2(1/2)*2(1/2)*pi(1/2),vpa(I1,70),ans=,I2=int(exp(-x2/2),x,0,inf),I2=,1/2*2(1/2)*pi(1/2),多重积分问题的MATLAB求解,例：,syms x y z;f0=-4*z*exp(-x2*y-z2)*(cos(x2*y)-10*cos(x2*y)*y*x2+.,4*sin(x2*y)*x4*y2+4*cos(x2*y)*x4*y2-sin(x2*y);,f1=int(f0,z);f1=int(f1,y);f1=int(f1,x);,f1=simple(int(f1,x),f1=,exp(-x2*y-z2)*sin(x2*y),f2=int(f0,z);f2=int(f2,x);f2=int(f2,x);,f2=simple(int(f2,y),f2=,2*exp(-x2*y-z2)*tan(1/2*x2*y)/(1+tan(1/2*x2*y)2),simple(f1-f2),ans=,0,顺序的改变使化简结果不同于原函数，但其误差为0，表明二者实际完全一致。这是由于积分顺序不同，得不出实际的最简形式。,例：,syms x y z,int(int(int(4*x*z*exp(-x2*y-z2),x,0,1),y,0,pi),z,0,pi),ans=,(Ei(1,4*pi)+log(pi)+eulergamma+2*log(2)*pi2*hypergeom(1,2,-pi2),Ei(n,z)为指数积分，无解析解，但可求其数值解：,vpa(ans,60),ans=,3.2 函数的级数展开与 级数求和问题求解,3.2.1 Taylor 幂级数展开,3.2.2 Fourier 级数展开,3.2.3 级数求和的计算,3.2.1 Taylor 幂级数展开,3.2.1.1 单变量函数的 Taylor 幂级数展开,例:,syms x;f=sin(x)/(x2+4*x+3);,y1=taylor(f,x,9);pretty(y1),2 23 3 34 4 4087 5 3067 6 515273 7 386459 8,1/3 x-4/9 x +-x -x +-x -x +-x -x,54 81 9720 7290 1224720 918540,taylor(f,x,9,2),ans=,syms a;taylor(f,x,5,a)%结果较冗长，显示从略,ans=,sin(a)/(a2+3+4*a)+(cos(a)-sin(a)/(a2+3+4*a)*(4+2*a)/(a2+3+4*a)*(x-a)+(-sin(a)/(a2+3+4*a)-1/2*sin(a)-(cos(a)*a2+3*cos(a)+4*cos(a)*a-4*sin(a)-2*sin(a)*a)/(a2+3+4*a)2*(4+2*a)/(a2+3+4*a)*(x-a)2+,例:对y=sinx进行Taylor幂级数展开,并观察不同阶次的近似效果。,x0=-2*pi:0.01:2*pi;y0=sin(x0);syms x;y=sin(x);,plot(x0,y0,r-.),axis(-2*pi,2*pi,-1.5,1.5);hold on,for n=8:2:16,p=taylor(y,x,n),y1=subs(p,x,x0);line(x0,y1)end,p=,x-1/6*x3+1/120*x5-1/5040*x7,p=,x-1/6*x3+1/120*x5-1/5040*x7+1/362880*x9,p=,x-1/6*x3+1/120*x5-1/5040*x7+1/362880*x9-1/39916800*x11,p=,x-1/6*x3+1/120*x5-1/5040*x7+1/362880*x9-1/39916800*x11+1/6227020800*x13,p=,3.2.1.2 多变量函数的Taylor 幂级数展开,多变量函数 在,的Taylor幂级数的展开,例：,？,syms x y;f=(x2-2*x)*exp(-x2-y2-x*y);,F=maple(mta