数学实验 第三章

1、第三章练习题 1.（1）syms t;f=exp(5*t);int(f,t)运行结果：ans =1/5*exp(5*t)（2）syms x;f=x/(x4+2*x2+5);int(f,x)运行结果：ans =1/4*atan(1/2*x2+1/2)（3）syms x;f=1/(x*log(x)*log(log(x);int(f,x)运行结果：ans =log(log(log(x)（4）syms x;f=sqrt(1+cos(x);int(f,x)运行结果：ans =-2*cos(1/2*x)*(-1+cos(1/2*x)2)/sin(1/2*x)*2(1/2)/(cos(1/2*x)2)(1/

2、2)（5）syms x a;f=x2/(sqrt(a2-x2);int(f,x)运行结果：ans =-1/2*x*(a2-x2)(1/2)+1/2*a2*atan(x/(a2-x2)(1/2)（6）syms x;f=1/(1+sqrt(2*x);int(f,x)运行结果：ans =-1/2*log(2*x-1)+2(1/2)*x(1/2)-atanh(2(1/2)*x(1/2)（7）syms x;f=cos(x)*sin(x)/(1+cos(x)2);int(f,x)运行结果：ans =-1/2*log(1+cos(x)2)（8）syms x;f=1/(x*sqrt(x2-1);int(f,x

3、)运行结果：ans =-atan(1/(x2-1)(1/2)（9）syms x;f=sqrt(x2-9)/x;int(f,x)运行结果：ans =(x2-9)(1/2)+3*atan(3/(x2-9)(1/2)2.（1）syms x ;f=x*exp(-x2/2);int(f,x,0,1)运行结果：ans =-exp(-1/2)+1（2）syms x ;f=(abs(x)+x)*exp(-x2);int(f,x,-2,2)运行结果：ans = -exp(-4)+1（3）syms x ;f=abs(x-1);int(f,x,0,2)运行结果：ans = 1（4）syms x ;f=1/(exp(

4、x)+1);int(f,x,0,1)运行结果：ans =1-log(1+exp(1)+log(2)（5）syms x ;f=sqrt(1+cos(2*x);int(f,x,0,pi)运行结果：ans = 2*2(1/2)（6）syms x ;f=1/(x2*sqrt(1+x2);int(f,x,1,sqrt(3)运行结果：ans =-2/3*3(1/2)+2(1/2)3.（1）syms x t;f=acos(t2);diff(int(f,t,1,x),x)运行结果为;Ans= = (acos(x2)*(1-x4)(1/2)-2*x4*acos(x2)/(1-x4)(1/2)+2/(1-x2)(

5、1/2)*(1+x2)(1/2)*EllipticF(x,i)*x-2*(1-x2)(1/2)/(1+x2)(1/2)*EllipticF(x,i)*x-2/(1-x2)(1/2)*(1+x2)(1/2)*EllipticE(x,i)*x+2*(1-x2)(1/2)/(1+x2)(1/2)*EllipticE(x,i)*x-4*EllipticK(i)/(1-x4)(1/2)*x3+4*EllipticE(i)/(1-x4)(1/2)*x3)/(1-x4)(1/2)+2*(x*acos(x2)*(1-x4)(1/2)-2*(1-x2)(1/2)*(1+x2)(1/2)*EllipticF(x,

6、i)+2*(1-x2)(1/2)*(1+x2)(1/2)*EllipticE(x,i)+2*EllipticK(i)*(1-x4)(1/2)-2*EllipticE(i)*(1-x4)(1/2)/(1-x4)(3/2)*x3（2）syms x t;f=sqrt(1+t4);diff(int(f,t,0,log(x),x)运行结果为;ans=(1/12-1/12*i)*(1/x*2(1/2)+5*log(x)4*2(1/2)/x+(2+3*i)/x*2(1/2)+5*i*log(x)4*2(1/2)/x-4*i/(1-i*log(x)2)(1/2)*(1+i*log(x)2)(1/2)*Elli

7、pticF(1/2+1/2*i)*log(x)*2(1/2),i)*log(x)/x+4*i*(1-i*log(x)2)(1/2)/(1+i*log(x)2)(1/2)*EllipticF(1/2+1/2*i)*log(x)*2(1/2),i)*log(x)/x)*2(1/2)/(1+log(x)4)(1/2)+(-1/6+1/6*i)*(log(x)*2(1/2)+log(x)5*2(1/2)+i*log(x)*2(1/2)+i*log(x)5*2(1/2)+4*(1-i*log(x)2)(1/2)*(1+i*log(x)2)(1/2)*EllipticF(1/2+1/2*i)*log(x)

8、*2(1/2),i)*2(1/2)/(1+log(x)4)(3/2)*log(x)3/x（3）syms x t;f=sin(t2+1);diff(int(f,t,-x2,1),x)运行结果为;2*cos(1)*sin(x4)*x+2*sin(1)*cos(x4)*x（4）syms x t;f=sqrt(1+log(t);diff(int(f,t,exp(x),x),x)运行结果为;(1+log(x)(1/2)-exp(x)*(1+log(exp(x)(1/2)4. syms x t;f=t*exp(-t2);f1=diff(int(f,t,0,x),x);f2=diff(x2);ff=f1/f

9、2;limit(ff,x,0)ans = 1/25. syms x t y;f=sin(t);g=exp(t2);y1=diff(int(f,t,0,y),y);y2=diff(int(g,t,1,x),x);ff=-y2/y1运行结果：ff = -exp(x2)/sin(y6. x=linspace(-1,2.5,60);y1=x.2;y2=x;y3=2*x;plot(x,y1,x,y2,x,y3);legend(y=x2,y=x,y=2x);syms x;f=x-(x.2);fprintf(y=x2与y=x所围面积为： );v=int(f,x,0,1)*pif=2*x-(x.2);fpri

10、ntf(y=x2与y=2x所围面积为： );v=int(f,x,0,2)*pi运行结果为：y=x2与y=x所围面积为： v = 1/6*piy=x2与y=2x所围面积为： v =4/3*pi7x=0:0.01:2*pi;y=0:0.01:2;y1=1+sin(x);y2=0;x1=0;x2=pi;plot(x,y1,b,x,y2,y,x1,y,r,x2,y,g);axis(-0.1 2*pi+0.1 -1 3);syms x;f=1+sin(x);fprintf(D绕x轴的体积为：);V=int(f,x,0,pi)*pisyms y;f=pi;fprintf(D绕y轴的体积为：);V=int(

11、f,x,0,asin(y-1)*pi运行结果为：D绕x轴的体积为： V = (pi+2)*piD绕y轴的体积为： V =pi2*asin(y-18. syms t;x=t;y=log(t);f=sqrt(diff(x,t)2+diff(y,t)2);int(f,t,sqrt(3),8)运行结果：ans = 65(1/2)-atanh(1/65*65(1/2)-2+atanh(1/2) = 6.48699.syms t R;x=R*cos(t);y=R*sin(t);f1=sqrt(diff(x,t)2+diff(y,t)2);f=y*f1;s=2*pi*int(f,t,0,pi)运行结果：s

12、= 4*pi*R3/(R2)(1/2)思考与提高1. 鱼塘面积为：2S;鱼塘体积为：V=6S; 钓鱼数：;钓鱼证:代码为：x=0:5:45;y=0,260,400,500,570,580,550,430,270,0;p=polyfit(x,y,2);xx=0:0.01:45;yy=p(1)*xx.2+p(2)*xx+p(3);plot(x,y,rp,xx,yy);syms x;f=p(1)*x.2+p(2)*x+p(3);S=int(f,x,0,45)cout=S*2*6*0.75/3/20运行结果：S = 1.7943e+004Cout= 2.6914e+003结论：故最多可卖出钓鱼证269

13、1.4即2691个。2根据悬链式方程，以及已知条件，大概估计其方程为：代码为：x=-50:0.01:50;y=10*cosh(x/38);plot(x,y);syms t;x=t;y=10*cosh(t/38);l=int(sqrt(diff(x,t)2+diff(y,t)2),t,-50,50);double(l)*2运行结果为:ans = 205.4801结论:故此缆车索道的长度约为205.4801m3. 代码为:x=0.1 0.3 0.4 0.55 0.7 0.8 0.95;y=15 18 19 21 22.6 23.8 26;p=polyfit(x,y,1);xx=0:0.01:0.9

14、5;yy=p(1)*xx+p(2);plot(x,y,rp,xx,yy);syms x;f=p(1)*x+p(2)运行结果：f = 1870/149*x+10399/745 结论：即此拟合方程为：y= 12.5503*x +13.9584验证性实验实验一 不定积分（1）syms x;f=1/x4;int(f,x)运行结果：ans = -1/3/x3（2）syms x;f=exp(x)/(1+exp(x);int(f,x)运行结果：ans = log(1+exp(x)（3）syms x;f=x2*exp(3*x);int(f,x)运行结果：ans =1/3*x2*exp(3*x)-2/9*x*e

15、xp(3*x)+2/27*exp(3*x)（4）syms x;f=1/sin(x)*cos(x);int(f,x)运行结果：ans =log(sin(x)（5）syms x;f=(1+x)2/(x*(1+x2);int(f,x)运行结果：ans =log(x)+2*atan(x)（6）syms x;f=exp(x+exp(x);int(f,x)运行结果：ans =exp(exp(x)实验二 定积分1.（1）syms x ;f=sqrt(1-x2);int(f,x,0,1)运行结果：ans = 1/4*pi（2）syms x ;f=sqrt(x)*(1+sqrt(x); int(f,x,0,1)

16、运行结果：ans =40/3（3）syms x ;f=(1+log(x)/x;int(f,x,1,2)运行结果：ans =1/2*log(2)2+log(2)（4）syms x ;f=cos(x)*cos(2*x);int(f,x,-pi/2,pi/2)运行结果：ans = 2/32.（1）syms x t;f=sin(t)/t;diff(int(f,t,0,x),x)运行结果为;ans = sin(x)/x（2）syms x t;f=exp(-t2);diff(int(f,t,1,sqrt(log(x),x)运行结果为;ans =1/2/x2/log(x)(1/2)（3）syms x t;f

17、=atan(t)/(1+t2);diff(int(f,t,0,x),x)运行结果为;ans =atan(x)/(1+x2)（4）syms x t;f=exp(-t2);diff(int(f,t,x,x2),x)运行结果为;ans =2*exp(-x4)*x-exp(-x2)（5）syms x t y;f=exp(-t2);g=cos(t2);y1=diff(int(f,t,0,y),y);y2=diff(int(g,t,0,x),x);ff=-y2/y1运行结果为;ff = -cos(x2)/exp(-y2)3.(1) syms x t;f=cos(t2);f1=diff(int(f,t,si

18、n(x),0),x);f2=f1/1;limit(f2,x,0)ans = -1(2) f=atan(t)2;f1=diff(int(f,t,0,2*x),x);f2=diff(sqrt(x2+1);ff=f1/f2;limit(ff,x,inf)ans = 1/2*pi2实验三 定积分的应用1. y=linspace(-1,1,60);x1=5*y.2;x2=1+y.2;plot(x1,y,x2,y);legend(x1=5y2,x2=1+y2);syms y;f=(5*y.2)-(1+y.2);A=int(f,y,-1,1)运行结果：A = 2/32.x=linspace(-0.5,1.5,60);y1=x.2;y2=x.3;plot(x,y1,x,y2);legend(y1=x2,y2=x3);syms x;f=(x.2)-(x.3);v=int(f,x,0,1)*pi运行结果：v = 1/12*pi3. t=linspace(0,2*pi,60);x=cos(t).3;y=sin(t).3;plot(x,y);syms t;f1=diff(cos(t)3);f2=diff(sin(t)3);f3=sqrt(f12+f22);s=4*int(f3,t,0,2*pi)运行结果：s = 244. syms t a;x=a*(t-sin