MATLAB实验报告2.doc

年级、专业 09级数学与应用数学2班 姓名 卢海宽 学号 150142009040 名单序号 117 实验时间 2011年 10月27 日 使用设备、软件 PC, MATLAB 注： 实验报告的最后一部分是实验小结与收获 实验（2） Matlab基础操作实验：符号运算求解微积分与线性代数1. 求下列函数极限（Find the limits of the following functions） (1) syms x r;r=sin(x)/x;limit(r,x,0)ans = 1 (2) syms x r;r=(1+1/x)x;limit(r,x,inf)ans = exp(1)(3) syms x r a;r=(1+a/x)x;limit(r,x,inf)ans = exp(a) (4) syms x r;r=(1+1/x)x;limit(r,x,-inf)ans = exp(1)(5) syms x r;r=(1-cos(x)/(x2);limit(r,x,0)ans = 1/2(6) syms n r;r=n(1/n);limit(r,n,inf)ans = 1(7)syms x n r;r=(cos(sqrt(x)(n/x);limit(r,x,0,right)ans = 1/exp(n/2)2.求下列函数的导数或偏导数( Find the derivatives of the following functions) (1)syms x a b c f g;f=sqrt(a*x2+b*x+c);g=sqrt(exp(x2)+x*sin(x);a1=diff(f,x)a2=diff(g,x)a1 = (b + 2*a*x)/(2*(a*x2 + b*x + c)(1/2) a2 = (sin(x) + 2*x*exp(x2) + x*cos(x)/(2*(exp(x2) + x*sin(x)(1/2) (2)syms x f;f=log(x3);diff(f,x)ans = 3/x (3)Find the 3rd derivative of f(x).syms x f;f=x*exp(-x2);diff(f,x,3)ans = (24*x2)/exp(x2) - 6/exp(x2) - (8*x4)/exp(x2)(4) )Find syms x y f;f=x3-2*x2*y2+3*y-5;diff(f,x,2)ans = 6*x - 4*y2(5)Find .syms x y f;f=x3-2*x2*y2+3*y-5;fx=diff(f,x);fxy=diff(fx,y)fxy = -8*x*y3. 求下列函数的不定积分或定积分（Find indefinite integrals or definite integrals of the functions）(1)syms x s;s=sin(x)-2*cos(3*x)+1/x+exp(-x);int(s,x)ans = log(x) - (2*sin(3*x)/3 - cos(x) - 1/exp(x)(2)syms x s;s=exp(x)*sin(exp(x);int(s,x)ans = -cos(exp(x)(3) syms x s;s=x2/(sqrt(x6)+4);int(s,x)Warning: Explicit integral could not be found. ans = int(x2/(x6)(1/2) + 4), x) (4)syms x s;s=cos(3*x)*cos(5*x);int(s,x)ans = sin(2*x)/4 + sin(8*x)/16(5) syms x s a;s=(sqrt(x2-a2)/x;int(s,x)ans = (x2 - a2)(1/2) - log(-a2)(1/2) + (x2 - a2)(1/2)/x)*(-a2)(1/2)(6) syms x s a b;s=exp(a*x)*sin(b*x);int(s,x)ans = -(exp(a*x)*(b*cos(b*x) - a*sin(b*x)/(a2 + b2)(7) syms x s;s=sqrt(sin(x)-(sin(x)3);int(s,x,0,pi)Warning: Explicit integral could not be found. ans = int(sin(x) - sin(x)3)(1/2), x = 0.pi)(8) syms x s;s=1/(x2);int(s,x,1,inf)ans = 14. 解下列方程（Solve the equations.）(1)syms x;solve(sqrt(1-x2)-x,x)ans = 2(1/2)/2(2)syms x y z;s=solve(x+y+z-10,x-y+z,2*x-y-z+4,x,y,z)s.x,s.y,s.zs = x: 1x1 sym y: 1x1 sym z: 1x1 sym ans = 2 ans = 5 ans = 3(3)syms x y z;s=solve(x2+4*x*y+z,x+3*y*z-3,y+sin(z),x,y,z)s.x,s.y,s.zs = x: 1x1 sym y: 1x1 sym z: 1x1 sym ans = -6.4840528860501102693511549985753 ans = -0.071995758403084664887655346911159 ans = -43.9102390493223146672631539563435. 求解下列常微分方程 (Solve the following ordinary differential equations.)(1)syms x y;s=dsolve(x*Dy=y*log(x*y)-y,x)s = 1/x exp(exp(C8 + log(x)/x(2)syms v t;s=dsolve(Dv+2*t=0,v(1)=5,t)s = 6 - t2(3)syms a b x y;s=dsolve(D2y-(a+b)*Dy+a*b*y=0,x)s = C13*exp(a*x) + C14*exp(b*x) 6. 用MATLAB验证(Use MALTAB to prove the following identities)(1) syms x;s=solve(sin(x)2+cos(x)2-1,x)s = C_求解此方程所得的解是全体实数因此该等式在实数域上成立。 (2) syms x y;s=solve(sin(x+y)-sin(x)*cos(y)-cos(x)*sin(y),x,y)s = C_2求解此方程所得的解是全体二维空间点的全体因此该等式成立。7.（1）分别用数字和符号两种方法，编程计算100! 结果有何不同？那个计算得快？（2）用符号方法，编程计算200!，结果为多大数量级？能用数值方法计算吗？解：（1）tics=1;for n=1:100 s=s*n;endformat long;stocs = 9.332621544394410e+157Elapsed time is 0.000345 seconds.以上为用数字计算ticsyms n s;s=sym(1);for n=1: 100 s=s*n;endstocs = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000Elapsed time is 0.098745 seconds.以上为用符号运算，可见两者对同一运算的结果相同，但符号运算所消耗的时间要比数字运算的多，精度要大。（2） tics=1;for n=1:200 s=s*n;endstocs = InfElapsed time is 0.000512 seconds.数字运算，数据溢出。ticsyms n s;s=sym(1);for n=1:200 s=s*n;endstocs = 78865786736479050355236321393218506229513597768717326329474253324435944996340334292030428401198462390417721213891963883025764279024263710506192662495282993111346285727076331723739698894392244562145166424025403329186413122742829485327752424240757390324032125740557956866022603190417032406235170085879617892222278962370