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1、实验三、符号运算的应用1、简化。（比较simple和simplify指令的区别）clear;syms x f = (1/x3+6/x2+8)(1/3) f =(1/x3+6/x2+8)(1/3) simple(f) %simple不返回指定的变量时，给出所有简化的结果，最后结果是最简的形式 simplify:(1+6*x+8*x3)/x3)(1/3)radsimp:1/x*(1+6*x+8*x3)(1/3)combine(trig):(1+6*x+8*x3)/x3)(1/3)factor:(1+6*x+8*x3)/x3)(1/3)expand:(1/x3+6/x2+8)(1/3)combine
2、:(1/x3+6/x2+8)(1/3)convert(exp):(1/x3+6/x2+8)(1/3)convert(sincos):(1/x3+6/x2+8)(1/3)convert(tan):(1/x3+6/x2+8)(1/3)collect(x):(1/x3+6/x2+8)(1/3)mwcos2sin:(1/x3+6/x2+8)(1/3)ans =(1/x3+6/x2+8)(1/3) B = simple(f) %把最简的形式赋给变量B B =(1/x3+6/x2+8)(1/3) C = simplify(f) %数学上的恒等式化简 C =(1+6*x+8*x3)/x3)(1/3) 2、c
3、lear;syms x a y = sin(a*x) y =sin(a*x) A = diff(y) A =cos(a*x)*a B = diff(y,a) B =cos(a*x)*x C = diff(y,2) C =-sin(a*x)*a2 3、计算不定积分、定积分、反常积分， ， clear;syms x f1 = (x2+1)/(x2-2*x+2)2 f1 =(x2+1)/(x2-2*x+2)2 I =int(f1) I =1/4*(2*x-6)/(x2-2*x+2)+3/2*atan(x-1) f2 = cos(x)/(sin(x)+cos(x) f2 =cos(x)/(sin(x)
4、+cos(x) J =int(f2,0,pi/2) J =1/4*pi f3 = exp(-x2) f3 =exp(-x2) K = int(f3,0,inf) K =1/2*pi(1/2) K = vpa(K,4) K =.8865 4、求级数的和S, 以及前十项的部分和S1clear;syms n S = symsum(1/n2,1,inf) S =1/6*pi2 S1 = symsum(1/n2,1,10) S1 =1968329/1270080 s1 = vpa(S1,4) s1 =1.550 f = sym(a*x2+b*x+c=0)5、求一元二次方程的根clear;syms a b
5、 c x f = a*x2+b*x+c f =a*x2+b*x+c X = solve(f) X = 1/2/a*(-b+(b2-4*a*c)(1/2) 1/2/a*(-b-(b2-4*a*c)(1/2) 6、求微分方程的通解dsolve(Dy = x,x) ans =1/2*x2+C1 7、 求微分方程的特解y=dsolve(D2y=Dy+x,y(0)=1,Dy(0)=0,x) y =-1/2*x2+exp(x)-x 8、求微分方程组的通解x,y=dsolve(Dx =y+x,Dy = 2*x) x =-1/2*C1*exp(-t)+C2*exp(2*t)y =C1*exp(-t)+C2*e
6、xp(2*t) 9、求下列联立方程的解 （用数值和符号运算两种方法）法一：数值方法A = 3 4 -7 -12;5 -7 4 2;1 0 8 -5;-6 5 -2 10; b = 4;-3;9;-8; x = Ab x = -1.4841 -0.6816 0.5337 -1.2429 法二：符号方法clear;syms x y z w f1 = 3*x+4*y-7*z-12*w-4 f2 = 5*x-7*y+4*z+2*w+3 f3 = x+8*z-5*w-9 f4 = -6*x+5*y-2*z+10*w+8 f1 =3*x+4*y-7*z-12*w-4f2 =5*x-7*y+4*z+2*w+
7、3f3 =x+8*z-5*w-9f4 =-6*x+5*y-2*z+10*w+8 w x y z=solve(f1,f2,f3,f4) w =-2932/2359x =-3501/2359y =-1608/2359z =1259/2359 X= vpa(x;y;z;w,4) X = -1.484 -.6816 .5337 -1.243 10、求极限 clear;syms x h n f1 = (log(x+h) - log(x) / h f1 =(log(x+h)-log(x)/h L = limit(f1,h,0) L =1/x f2 = (1 - x/n)n f2 =(1-x/n)n M =
8、 limit(f2,n,inf) M =exp(-x) 11、设a、b定义如下，试上机输出factor(a)和factor(b)的结果，并指出那个结果才是12345678901234567890的因式分解，为什么？ a=sym(12345678901234567890); %直接创建符号常量 b=sym(12345678901234567890); %将数值常量转换为符号常量，有误差 x1=factor(a) x1 =(2)*(3)2*(5)*(101)*(3803)*(3607)*(27961)*(3541) x2=factor(b) x2 =(2)11*(7)*(7324703)*(117
9、570121) isequal(x1,x2) ans = 0 12、求函数，其中x = 1:10（用数值和符号运算三种方法）法一：符号，置换f = sym(x4 -3*x + 8) f =x4 -3*x + 8 fx = subs(f,x,1:10) fx = Columns 1 through 5 6 18 80 252 618Columns 6 through 10 1286 2388 4080 6542 9978 法二：符号，转换f = sym(x4 -3*x + 8) f =x4 -3*x + 8 p = sym2poly(f) p = 1 0 0 -3 8 fx = polyval(
10、p,1:10) fx = 6 18 80 252 618 1286 2388 4080 6542 9978 法三：数值，多项式求值p =1 0 0 -3 8 p = 1 0 0 -3 8 fx = polyval(p,1:10) fx = 6 18 80 252 618 1286 2388 4080 6542 9978 13、把展成多项式的形式。（用数值和符号运算两种方法）数值:多项式相乘n = conv(conv(1 1,1 1),conv(1 1,3 4 0 7) n = 3 13 21 22 25 21 7 f = poly2sym(n,s) f =3*s6+13*s5+21*s4+22*s3+25*s2+21*s+7 符号:表达式展开syms s f = (s+1)3*(3*s3+4*s2+7) f =(s+1)3*(3*s3+4*s2+7) f =expand(f) f =3*s6+13*s5+21*s4+22*s3+25*s2+21*s+7 p = sym2poly(f) p = 3 13 21 22 25 21 7 isequal(n,p) ans = 1