实验2符号运算求解微积分与线性代数.doc

年级、专业 09信息与计算科学 姓名 学号 名单序号 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 f=sin(x)/x; limit(f,x,0)ans = 1(2) syms x f=(1+1/x)x; limit(f,x,inf)ans = exp(1)(3) syms a x f=(1+a/x)x; limit(f,x,inf) ans = exp(a) (4) syms x f=(1+1/x)x; limit(f,x,-inf) ans = exp(1) (5) syms x f=(1-cos(x)/(x2); limit(f,x,0) ans =1/2 (6) syms n f=n(1/n); limit(f,n,inf)ans = 1 (7) syms x f=(cos(x0.5)(pi/x); limit(f,x,0,right) ans = 1/exp(pi/2)2.求下列函数的导数或偏导数( Find the derivatives of the following functions) (1) syms a b c x f=(a*x*x+b*x+c)0.5; g=(exp(x2)+x*sin(x)0.5; f1=diff(f,x) f1 = (b + 2*a*x)/(2*(a*x2 + b*x + c)(1/2) f2=diff(g,x) f2 = (sin(x) + 2*x*exp(x2) + x*cos(x)/(2*(exp(x2) + x*sin(x)(1/2)(2) syms x f=log(x3); diff(f,x)ans = 3/x(3)Find the 3rd derivative of f(x). syms x 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=x3-2*x*x*y*y+3*y-5; diff(f,x,2) ans =6*x - 4*y2(5)Find . syms x y f=x3-2*x*x*y*y+3*y-5; diff(diff(f,y),x)ans = (-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 f=sin(x)-2*cos(3*x)+1/x+exp(-x); int(f,x)ans = log(x) - (2*sin(3*x)/3 - cos(x) - 1/exp(x) (2) syms x f=exp(x)*sin(exp(x); int(f,x) ans = -cos(exp(x)(3) syms x f=(x2)/(x6+4)0.5); int(f,x)Warning: Explicit integral could not be found. ans = int(x2/(x6 + 4)(1/2), x) (4) syms x f=cos(3*x)*cos(5*x); int(f,x) ans = sin(2*x)/4 + sin(8*x)/16(5) syms x f=(x2-a2)0.5)/x; int(f,x)ans =(x2 - a2)(1/2) - log(-a2)(1/2) + (x2 - a2)(1/2)/x)*(-a2)(1/2) (6) syms a b x f=exp(a*x)*sin(b*x); int(f,x)ans =-(exp(a*x)*(b*cos(b*x) - a*sin(b*x)/(a2 + b2)(7) syms x f=(sin(x)-(sin(x)3)0.5; int(f,x,0,pi) ans =4/3 (8) syms x f=1/(x2); int(f,x,1,inf)ans = 14. 解下列方程（Solve the equations.）(1) syms x f=(1-x2)0.5-x; solve(f)ans = 0.70710678118654752440084436210485(2) f1=x+y+z=10; f2=x-y+z=0; f3=2*x-y-z=-4; x,y,z=solve(f1,f2,f3) x =2 y =5z = 3(3) f1=x*x+4*x*y+z=0; f2=x+3*y*z-3=0; f3=y+sin(z)=0; x,y,z=solve(f1,f2,f3) x =-6.4840528860501102693511549985753y = -0.071995758403084664887655346911159z =-43.9102390493223146672631539563435. 求解下列常微分方程 (Solve the following ordinary differential equations.)(1) dsolve(x*Dy=y*log(x*y)-y)ans = exp(1)/x exp(exp(C6 + t/x) + 1)/x(2) dsolve(Dv+2*t=0,v(1)=5) ans = 6 - t2(3) dsolve(D2y-(a+b)*Dy+a*b*y=0) ans =C11*exp(a*t) + C12*exp(b*t) 6. 用MATLAB验证(Use MALTAB to prove the following identities)(1) function y=eg6_1(x)y=(sin(x)2+(cos(x)2; eg6_1(6)ans = 1(2) function f=eg6_2(x,y)f=sin(x+y)-sin(x)*cos(y)-cos(x)*sin(y); eg6_2(2,3)ans = -4.8572e-0177.（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？ 数字方法： tick=1;for i=1:100; k=k*i; i=i+1;endfprintf(s=%dn,s);toc 输出结果： s=9.426900e+157Elapsed time is 0.000169 seconds.符号方法： tic syms x f=x100; diff(f,100) ans =93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000 tocElapsed time is 46.860357 seconds.数值运算和符号运算都能计算100！但是计算结果的表示方法不同且大小有些差异，从计算时间上看，数值计算明显快于符号运算。（2）用符号方法，编程计算100!，结果为多大数量级？能用数值方法计算吗？ syms x f=x100; diff(f,100) ans =93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000 结果为e+157数量级，可以用数值方法运算。小结：通过这次作