数学实验课后习题解答

1、数学实验课后习题解答配套教材：王向东 戎海武 文翰 编著数学实验王汝军编写实验一曲线绘图【练习与思考】画出下列常见曲线的图形。以直角坐标方程表示的曲线：1. 立方曲线clear;x=-2:0.1:2;y=x.3;plot(x,y) 2. 立方抛物线clear;y=-2:0.1:2;x=y.3;plot(x,y)grid on 3. 高斯曲线clear;x=-3:0.1:3;y=exp(-x.2);plot(x,y);grid on%axis equal 以参数方程表示的曲线4. 奈尔抛物线clear;t=-3:0.05:3;x=t.3;y=t.2;plot(x,y)axis equalgrid

2、 on 5. 半立方抛物线clear;t=-3:0.05:3;x=t.2;y=t.3;plot(x,y)%axis equalgrid on 6. 迪卡尔曲线clear;a=3;t=-6:0.1:6;x=3*a*t./(1+t.2);y=3*a*t.2./(1+t.2);plot(x,y) 7. 蔓叶线clear;a=3;t=-6:0.1:6;x=3*a*t.2./(1+t.2);y=3*a*t.3./(1+t.2);plot(x,y) 8. 摆线clear;clc;a=1;b=1;t=0:pi/50:6*pi;x=a*(t-sin(t);y=b*(1-cos(t);plot(x,y);axi

3、s equalgrid on 9. 内摆线（星形线）clear;a=1;t=0:pi/50:2*pi;x=a*cos(t).3;y=a*sin(t).3;plot(x,y) 10. 圆的渐伸线（渐开线）clear;a=1;t=0:pi/50:6*pi;x=a*(cos(t)+t.*sin(t);y=a*(sin(t)+t.*cos(t);plot(x,y)grid on 11. 空间螺线cleara=3;b=2;c=1;t=0:pi/50:6*pi;x=a*cos(t);y=b*sin(t);z=c*t;plot3(x,y,z)grid on 以极坐标方程表示的曲线：12. 阿基米德线clea

4、r;a=1;phy=0:pi/50:6*pi;rho=a*phy;polar(phy,rho,r-*) 13. 对数螺线clear;a=0.1;phy=0:pi/50:6*pi;rho=exp(a*phy);polar(phy,rho) 14. 双纽线clear;a=1;phy=-pi/4:pi/50:pi/4;rho=a*sqrt(cos(2*phy);polar(phy,rho)hold onpolar(phy,-rho) 15. 双纽线clear;a=1;phy=0:pi/50:pi/2;rho=a*sqrt(sin(2*phy);polar(phy,rho)hold onpolar(p

5、hy,-rho) 16. 四叶玫瑰线clear;closea=1;phy=0:pi/50:2*pi;rho=a*sin(2*phy);polar(phy,rho) 17. 三叶玫瑰线clear;closea=1;phy=0:pi/50:2*pi;rho=a*sin(3*phy);polar(phy,rho) 18. 三叶玫瑰线clear;closea=1;phy=0:pi/50:2*pi;rho=a*cos(3*phy);polar(phy,rho) 实验二极限与导数【练习与思考】1 求下列各极限（1） （2） （3）clear;syms ny1=limit(1-1/n)n,n,inf)y2=

6、limit(n3+3n)(1/n),n,inf)y3=limit(sqrt(n+2)-2*sqrt(n+1)+sqrt(n),n,inf) y1 =1/exp(1)y2 =3y3 =0 （4） （5） （6）clear;syms x ;y4=limit(2/(x2-1)-1/(x-1),x,1)y5=limit(x*cot(2*x),x,0)y6=limit(sqrt(x2+3*x)-x,x,inf) y4 =-1/2y5 =1/2y6 =3/2 （7） （8） （9）clear;syms x my7=limit(cos(m/x),x,inf)y8=limit(1/x-1/(exp(x)-1)

7、,x,1)y9=limit(1+x)(1/3)-1)/x,x,0) y7 =1y8 =(exp(1) - 2)/(exp(1) - 1)y9 =1/3 2 考虑函数作出图形，并说出大致单调区间；使用diff求，并求确切的单调区间。clear;close;syms x;f=3*x2*sin(x3);ezplot(f,-2,2)grid on 大致的单调增区间：-2,-1.7,-1.3,1.2,1.7,2;大致的单点减区间：-1.7,-1.3,1.2,1.7; f1=diff(f,x,1)ezplot(f1,-2,2)line(-5,5,0,0)grid onaxis(-2.1,2.1,-60,1

8、20)f1 =6*x*sin(x3) + 9*x4*cos(x3) 用fzero函数找的零点，即原函数的驻点x1=fzero(6*x*sin(x3) + 9*x4*cos(x3),-2,-1.7)x2=fzero(6*x*sin(x3) + 9*x4*cos(x3),-1.7,-1.5)x3=fzero(6*x*sin(x3) + 9*x4*cos(x3),-1.5,-1.1)x4=fzero(6*x*sin(x3) + 9*x4*cos(x3),0)x5=fzero(6*x*sin(x3) + 9*x4*cos(x3),1,1.5)x6=fzero(6*x*sin(x3) + 9*x4*co

9、s(x3),1.5,1.7)x7=fzero(6*x*sin(x3) + 9*x4*cos(x3),1.7,2) x1 = -1.9948x2 = -1.6926x3 = -1.2401x4 = 0x5 = 1.2401x6 = 1.6926x7 = 1.9948 确切的单调增区间：-1.9948,-1.6926,-1.2401,1.2401,1.6926,1.9948确切的单调减区间：-2,-1.9948,-1.6926,-1.2401,1.2401,1.6926,1.9948,23 对于下列函数完成下列工作，并写出总结报告，评论极值与导数的关系，(i) 作出图形，观测所有的局部极大、局部极

10、小和全局最大、全局最小值点的粗略位置；(ii) 求所有零点（即的驻点）；(iii) 求出驻点处的二阶导数值;(iv) 用fmin求各极值点的确切位置;(v) 局部极值点与有何关系?(1) (2) (3) clear;close;syms x;f=x2*sin(x2-x-2)ezplot(f,-2,2)grid on f =x2*sin(x2 - x - 2) 局部极大值点为：-1.6,局部极小值点为为：-0.75，-1.6全局最大值点为为：-1.6，全局最小值点为：-3f1=diff(f,x,1)ezplot(f1,-2,2)line(-5,5,0,0)grid onaxis(-2.1,2.1

11、,-6,20) f1 =2*x*sin(x2 - x - 2) + x2*cos(x2 - x - 2)*(2*x - 1) 用fzero函数找的零点，即原函数的驻点x1=fzero(2*x*sin(x2-x-2)+x2*cos(x2-x-2)*(2*x-1),-2,-1.2)x2=fzero(2*x*sin(x2-x-2)+x2*cos(x2-x-2)*(2*x-1),-1.2,-0.5)x3=fzero(2*x*sin(x2-x-2)+x2*cos(x2-x-2)*(2*x-1),-0.5,1.2)x4=fzero(2*x*sin(x2-x-2)+x2*cos(x2-x-2)*(2*x-1

12、),1.2,2)x1 = -1.5326x2 = -0.7315x3 = -3.2754e-027x4 = 1.5951 ff=(x) x.2.*sin(x.2-x-2)ff(-2),ff(x1),ff(x2),ff(x3),ff(x4),ff(2) ff = (x)x.2.*sin(x.2-x-2)ans = -3.0272ans = 2.2364ans = -0.3582ans = -9.7549e-054ans = -2.2080ans = 0 实验三级数【练习与思考】1. 用taylor命令观测函数的maclaurin展开式的前几项, 然后在同一坐标系里作出函数和它的taylor展开式

13、的前几项构成的多项式函数的图形，观测这些多项式函数的图形向的图形的逼近的情况(1) clear;syms xy=asin(x);y1=taylor(y,0,1)y2=taylor(y,0,5)y3=taylor(y,0,10)y4=taylor(y,0,15)x=-1:0.1:1;y=subs(y,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y4=subs(y4,x);plot(x,y,x,y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =0y2 =x3/6 + xy3 =(35*x9)/1152 + (5*x

14、7)/112 + (3*x5)/40 + x3/6 + xy4 =(231*x13)/13312 + (63*x11)/2816 + (35*x9)/1152 + (5*x7)/112 + (3*x5)/40 + x3/6 + x (2) clear;syms xy=atan(x);y1=taylor(y,0,3)y2=taylor(y,0,5),y3=taylor(y,0,10),y4=taylor(y,0,15)x=-1:0.1:1;y=subs(y,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y4=subs(y4,x);plot(x,y,x,

15、y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =xy2 =x - x3/3y3 =x9/9 - x7/7 + x5/5 - x3/3 + xy4 =x13/13 - x11/11 + x9/9 - x7/7 + x5/5 - x3/3 + x (3) clear;syms xy=exp(x2);y1=taylor(y,0,3)y2=taylor(y,0,5)y3=taylor(y,0,10)y4=taylor(y,0,15)x=-1:0.1:1;y=subs(y,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y

16、4=subs(y4,x);plot(x,y,x,y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =x2 + 1y2 =x4/2 + x2 + 1y3 =x8/24 + x6/6 + x4/2 + x2 + 1y4 =x14/5040 + x12/720 + x10/120 + x8/24 + x6/6 + x4/2 + x2 + 1 (4) clear;syms xy=sin(x)2;y1=taylor(y,0,1)y2=taylor(y,0,5)y3=taylor(y,0,10)y4=taylor(y,0,15)x=-pi:0.1:pi;y=subs(y

17、,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y4=subs(y4,x);plot(x,y,x,y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =0y2 =x2 - x4/3y3 =- x8/315 + (2*x6)/45 - x4/3 + x2y4 =(4*x14)/42567525 - (2*x12)/467775 + (2*x10)/14175 - x8/315 + (2*x6)/45 - x4/3 + x2 (5) clear;syms xy=exp(x)/(1-x);y1=taylor(y,0,3)y

18、2=taylor(y,0,5)y3=taylor(y,0,10)y4=taylor(y,0,15)x=-1:0.1:0;y=subs(y,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y4=subs(y4,x);plot(x,y,x,y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =(5*x2)/2 + 2*x + 1y2 =(65*x4)/24 + (8*x3)/3 + (5*x2)/2 + 2*x + 1y3 =(98641*x9)/36288 + (109601*x8)/40320 + (685*x7)/2

19、52 + (1957*x6)/720 + (163*x5)/60 + (65*x4)/24 + (8*x3)/3 + (5*x2)/2 + 2*x + 1y4 =(47395032961*x14)+ (8463398743*x13)/3113510400 + (260412269*x12)/95800320 + (13563139*x11)/4989600 + (9864101*x10)/3628800 + (98641*x9)/36288 + (109601*x8)/40320 + (685*x7)/252 + (1957*x6)/720 + (163*x5)/60

20、 + (65*x4)/24 + (8*x3)/3 + (5*x2)/2 + 2*x + 1 (6) clear;syms xy=log(x+sqrt(1+x2);y1=taylor(y,0,3)y2=taylor(y,0,5)y3=taylor(y,0,10)y4=taylor(y,0,15)x=-1:0.1:1;y=subs(y,x);y1=subs(y1,x);y2=subs(y2,x);y3=subs(y3,x);y4=subs(y4,x);plot(x,y,x,y1,:,x,y2,-.,x,y3,-,x,y4,:,linewidth,3) y1 =xy2 =x - x3/6y3 =(3

21、5*x9)/1152 - (5*x7)/112 + (3*x5)/40 - x3/6 + xy4 =(231*x13)/13312 - (63*x11)/2816 + (35*x9)/1152 - (5*x7)/112 + (3*x5)/40 - x3/6 + x 2. 求公式中的数的值.k=4 5 6 7 8;syms nsymsum(1./n.(2*k),1,inf) ans = pi8/9450, pi10/93555, (691*pi12)/638512875, (2*pi14)/18243225, (3617*pi16)/325641566250 3. 利用公式来计算的近似值。精确到

22、小数点后100位,这时应计算到这个无穷级数的前多少项?请说明你的理由.解：matlab代码为clear;clc;closeepsl=1.0e-100;ep=1;fn=1;a=1;n=1;while epepsla=a+fn;n=n+1;fn=fn/n;ep=fn;endfnvpa(a,100)n fn = 8.3482e-101ans =2.71828182845904553488480814849026501178741455078125n = 70 精确到小数点后100位,这时应计算到这个无穷级数的前71项,理由是误差小于10的负100次方，需要最后一项小于10的负100次方，由上述循环知

23、n=70时最后一项小于10的负100次方，故应计算到这个无穷级数的前71项.4. 用练习3中所用观测法判断下列级数的敛散性(1) clear;clc;epsl=0.000001;n=50000;p=1000;syms nun=1/(n2+n3);s1=symsum(un,1,n);s2=symsum(un,1,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if saepsl disp(收敛)else disp(发散)end 级数1/(n3 + n2)收敛 clear;closesyms ns=;for k=

24、1:100s(k)=symsum(1/(n3 + n2),1,k);endplot(s,.) (2) clear;clc;epsl=0.000001;n=50000;p=1000;syms nun=1/(n*2n);s1=symsum(un,1,n);s2=symsum(un,1,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if saepsl disp(收敛)else disp(发散)end 级数1/(2n*n)收敛 clear;closesyms ns=;for k=1:100s(k)=symsum(1

25、/(2n*n),1,k);endplot(s,.) (3) clear;clc;epsl=0.00000000000001;n=50000;p=100;syms nun=1/sin(n);s1=symsum(un,1,n);s2=symsum(un,1,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if abs(sa)epsl disp(收敛)else disp(发散)end 级数1/sin(n)发散 clear;closesyms ns=;for k=1:100s(k)=symsum(1/sin(n),1

26、,k);endplot(s,.) 发散 (4) clear;clc;epsl=0.0000001;n=50000;p=1000;syms nun=log(n)/(n3);s1=symsum(un,1,n);s2=symsum(un,1,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if saepsl disp(收敛)else disp(发散)end 级数log(n)/n3收敛 clear;closesyms ns=;for k=1:100s(k)=symsum(log(n)/n3,1,k);endplot(

27、s,.) (5) clear;closesyms ns=;he=0;for k=1:100he=he+factorial(k)/kk;s(k)=he;endplot(s,.) (6) clear;clc;epsl=0.0000001;n=50000;p=1000;syms nun=1/log(n)n;s1=symsum(un,3,n);s2=symsum(un,3,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if saepsl disp(收敛)else disp(发散)end 级数1/log(n)n收敛

28、clear;closesyms ns=;for k=3:100s(k)=symsum(1/log(n)n,3,k);endplot(s,.) (7) clear;clc;epsl=0.0000001;n=50000;p=100;syms nun=1/(log(n)*n);s1=symsum(un,3,n);s2=symsum(un,3,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if (sa)epsl disp(收敛)else disp(发散)end 级数1/(n*log(n)发散 clear;close

29、syms ns=;for k=3:300s(k)=symsum(1/(n*log(n),2,k);endplot(s,.) (8) clear;clc;epsl=0.0000001;n=50000;p=100;syms nun=(-1)n*n/(n2+1);s1=symsum(un,3,n);s2=symsum(un,3,n+p);sa=vpa(s2-s1);sa=setstr(sa);sa=str2num(sa);fprintf(级数)disp(un)if (sa)epsl disp(收敛)else disp(发散)end 级数(-1)n*n)/(n2 + 1)收敛 clear;closes

30、yms ns=;for k=3:300s(k)=symsum(-1)n*n/(n2+1),2,k);endplot(s,.) 实验四积分【练习与思考】1(不定积分)用int计算下列不定积分，并用diff验证，解：matlab代码为：syms xy1=x*sin(x2);y2=1/(1+cos(x);y3=1/(exp(x)+1);y4=asin(x);y5=sec(x)3;f1=int(y1)f2=int(y2)f3=int(y3)f4=int(y4)f5=int(y5) dy=simplify(diff(f1;f2;f3;f4;f5) dy = x*sin(x2) tan(x/2)2/2 +

31、 1/2 1/(exp(x) + 1) asin(x) (cot(pi/4 + x/2)*(tan(pi/4 + x/2)2/2 + 1/2)/2 + 1/(2*cos(x) + tan(x)2/cos(x) f1 =-cos(x2)/2f2 =tan(x/2)f3 =x - log(exp(x) + 1)f4 =x*asin(x) + (1 - x2)(1/2)f5 =log(tan(pi/4 + x/2)/2 + tan(x)/(2*cos(x) 2(定积分)用trapz,quad,int计算下列定积分,解：matlab代码为clear;x=(0+eps):0.05:1;y1=sin(x)

32、./x;f1=trapz(x,y1) f1 =0.9460 fun1=(x)sin(x)./x;f12=quad(fun1,0+eps,1) f12 = 0.9461 f13=vpa(int(sin(x)/x,0,1),5) f13 =0.94608 3(椭圆的周长) 用定积分的方法计算椭圆的周长解：椭圆的参数方程为由参数曲线的弧长公式得matlab代码为s=vpa(int(sqrt(5*sin(t)2+4),t,0,2*pi),5) s =15.865 4（二重积分）计算数值积分解：fxy=(x,y)1+x+y;ylow=(x)1-sqrt(1-x.2);yup=(x)1+sqrt(1-x.

33、2);s=quad2d(fxy,-1,1,ylow,yup) s =6.2832 或符号积分法：syms x yxi=int(1+x+y,y,1-sqrt(1-x2),1+sqrt(1-x2);s=int(xi,x,-1,1) s =2*pi 5（假奇异积分）用trapz,quad8计算积分，会出现什么问题？分析原因，并求出正确的解。解：matlab代码为clearx=-1:0.05:1;y=x.(1/3).*cos(x);s1=trapz(x,y)fun5=(x)x.(1/3).*cos(x);s2=quad(fun5,-1,1)int(x(1/3)*cos(x),x,-1,1) s1 =

34、0.9036 + 0.5217is2 = 0.9114 + 0.5262iwarning: explicit integral could not be found. ans =int(x(1/3)*cos(x), x = -1.1) ，原函数不存在，不能用int函数运算。用梯形法和辛普森法计算数值积分时，由于对负数的开三次方运算结果为复数，所以导致结果错误且为复数；显然被积函数为奇函数，在对称区间上的积分等于0，此时可以这样处理：（1）重新定义被积函数%fun5.mfunction y=fun5(x)m,n=size(x);for k=1:mfor l=1:ny(k,l)=nthroot(x

35、(k,l),3)*cos(x(k,l);endendend用辛普森法：s=quad(fun5,-1,1) s = 0 用梯形法clear;x=-1:0.01:1;y=fun5(x);s=trapz(x,y) s = -1.3878e-017 6（假收敛现象）考虑积分，（1）用解析法求;clear;syms x k;ik=int(abs(sin(x),0,k*pi) warning: explicit integral could not be found. ik =int(abs(sin(x), x = 0.pi*k) （2）分别用trapz,quad和quad8求和,发现什么问题？clear

36、;for k=4:2:8;x=0:pi/1000:k*pi;y=abs(sin(x);trapz(x,y)end ans = 8.0000ans = 12.0000ans = 16.0000 for k=4:2:8fun6=(x)abs(sin(x);quad(fun6,0,k*pi)end ans = 8.0000ans = 12.0000ans = 16.0000 7(simpson积分法)编制一个定步长simpson法数值积分程序.计算公式为其中为偶数,解：matlab代码为%fun7.mfunction y=fun7(f_name,a,b,n)%f_name为被积函数%a,b为积分区间

37、%n为偶数，用来确定步长h=(b-a)/nif mod(n,2)=0 disp(n必须为偶数) return;endif nargin4n=100;endif nargin in fmincon at 445local minimum possible. constraints satisfied.fmincon stopped because the predicted change in the objective functionis less than the default value of the function tolerance and constraints were sa

38、tisfied to within the default value of the constraint tolerance.no active inequalities.x = 161.9676 182.0320fval = -715.4403 heigh和height两个函数分别定义如下：(应写在m文件中)%heigh.mfunction f=heigh(beta,xdata)xx1=xdata(:,1);xx2=xdata(:,2);f=beta(1)+beta(2)*xx1+beta(3)*xx2+beta(4)*xx1.2+beta(5)*xx2.2;end%height.mfun

39、ction y=height(x)y=-(538.4375+1.4901*x(1)+0.6189*x(2)-0.0046*x(1).2-0.0017*x(2).2);end实验六多元函数的极值【练习与思考】1.求的极值，并对图形进行观测。解：maltab代码为syms x y;z=x4+y4-4*x*y+1;dzx=diff(z,x);dzy=diff(z,y);s=solve(dzx,dzy,x,y);x=s.x.y=s.y. x = 0, 1, -1, (-1)(3/4), -(-1)(3/4), -i, i, -(-1)(3/4)*i, (-1)(3/4)*iy = 0, 1, -1, (-1)(1/4), -(-1)(1/4), i, -i, (-1)(1/4)*i, -(-1)(1/4)*i 经计算可知，函数的驻点为(0,0)、(1,1)、(-1,-1)ezmeshc(z,-2,2,-2,2) 从图形上观测可知，(1,1)、(-1,-1)为极值点，(0,0)不是极值点。clearsyms x y;z=x4+y4-4*x*y+1;dzx=diff(z