数学实验.docx
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数学实验.docx
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数学实验
3.若多项式
求
的值,其中
4.求下列多项式的和、差、积:
(1)
(2)
5.求多项式
的商及余子式。
6.在钢线碳含量对于电阻的效应的研究中,得到以下数据。
分别用一次,三次,五次多项式拟合曲线来拟合这组数据并画出图形。
碳含量x
0.10
0.30
0.40
0.55
0.70
0.80
0.95
电阻y
15
18
19
21
22.6
23.8
26
7.用不同的方法对
在(-3.3)上的二维插值效果进行比较。
7/clc;clear;
[X,Y]=meshgrid(-3:
.5:
3);
Z=X.^2/16-Y.^2/9;
[X1,Y1]=meshgrid(-3:
.1:
3);
Z1=X1.^2/16-Y1.^2/9;
figure
(1)
subplot(1,2,1),mesh(X,Y,Z),title('数据点')
subplot(1,2,2),mesh(X1,Y1,Z1),title('函数图象')
[Xi,Yi]=meshgrid(-3:
0.125:
3);
Zi1=interp2(X,Y,Z,Xi,Yi,'*nearest');
Zi2=interp2(X,Y,Z,Xi,Yi,'*linear');
Zi3=interp2(X,Y,Z,Xi,Yi,'*spline');
Zi4=interp2(X,Y,Z,Xi,Yi,'*cubic');
figure
(2)
subplot(2,2,1),mesh(Xi,Yi,Zi1),title('最近点插值')
subplot(2,2,2),mesh(Xi,Yi,Zi2),title('线性插值')
subplot(2,2,3),mesh(Xi,Yi,Zi3),title('样条插值')
subplot(2,2,4),mesh(Xi,Yi,Zi4),title('立方插值')
1.
clc;clear;
p=[1800-10];
r=roots(p)
poly2sym([1800-10])
poly2sym(poly(r))
r=
-8.0194
1.0344
-0.5075+0.9736i
-0.5075-0.9736i
ans=
x^4+8*x^3-10
ans=
x^4+8*x^3-7/1125899906842624*x^2+15/1125899906842624*x-10
2.
clc;clear;
p=[10-102];
A=[123;456;235];
x=polyval(p,A)
Y=polyvalm(p,A)
x=
-7-10-1
2677158
-10-177
Y=
151205291
344495714
238335473
3.
clc;clear;
p=[4-31];
A=[12;-23];
x=polyval(p,-3)
y=polyval(p,7)
z=polyvalm(p,A)
x=
46
y=
176
z=
-1426
-2612
4.
(1)
clc;clear;
p1=[4-13];
p2=[5-2-1];
ph=p1+p2;
poly2str(ph,'s')
pc=p1-p2;
poly2str(ph,'t')
pj=conv(p1,p2);
poly2str(ph,'x')
ans=
9s^2-3s+2
ans=
9t^2-3t+2
ans=
9x^2-3x+2
5.
clc;clear;
p1=[860-14];
p2=[1-1-1];
[pq,pr]=deconv(p1,p2);
poly2str(pq,'s')
poly2str(pr,'t')
ans=
8s^2+14s+22
ans=
35t+26
6.
clf;
x=[0.100.300.400.550.700.800.95];
y=[1518192122.623.826];
p1=polyfit(x,y,1);
p3=polyfit(x,y,3);
p5=polyfit(x,y,5);
disp('一阶拟合和函数'),f1=poly2str(p1,'x')
disp('三阶拟合和函数'),f3=poly2str(p3,'x')
disp('五阶拟合和函数'),f5=poly2str(p5,'x')
y1=polyval(p1,x);
y3=polyval(p3,x);
y5=polyval(p5,x);
plot(x,y,'ro:
',x,y1,'b.--',x,y3,'k*-.',x,y5)
legend('拟合点','一次拟合点','三次拟合点','五次拟合点')
2.用MATLAB软件求下列函数极限:
(3)
3.求下列的导数
(3)
4.求高阶导数
(2)求
的40阶导数
7.求有方程
所确定的隐函数的导数
。
12.求函数的极值:
(5)
14.求下列不定积分:
(1)
(3)
15.求下列定积分:
(3)
(5)
19.分别求下列级数前15项,前40项的部分和:
(5)
(6)
21.求函数
在
处的前6项泰勒级数展开式。
2/(3)
clc;clear;
symsxn;
L=limit(sin(x)^tan(x),x,pi/2)
L=
1
3/(3)
clc;clear;
symsxy;
f=(1+sin(x))./(1+cos(x));
g=diff(f,x)
g=
cos(x)/(1+cos(x))+(1+sin(x))/(1+cos(x))^2*sin(x)
4/
(2)
clc;clear;
symsx;
y=x^4*cos(7*x)
diff(y,40)
y=
x^4*cos(7*x)
ans=
5816200368074729538861238724268369360*cos(7*x)-4401448927191687218597694169716603840*x*sin(7*x)-1216189835145071468296731283737482640*x^2*cos(7*x)+145526988820777782531232803182262880*x^3*sin(7*x)+6366805760909027985741435139224001*x^4*cos(7*x)
7/
clc;clear;
symsxy;
f=exp(y)+x*y-exp
(1);
dx=diff(f,x)
dy=diff(f,y)
pretty(dy/dx)
dx=
y
dy=
exp(y)+x
exp(y)+x
----------
y
12/
clc;clear;
x=-5:
0.1:
5;
y=(log(x)).^2./x;
plot(x,y)
xmin=fminbnd('(log(x)).^2./x',-5,5)
ymin=eval('(log(xmin)).^2./xmin')
xmax=fminbnd('(log(x)).^2./x',-5,5)
ymax=eval('(log(xmax)).^2./xmax')
xmin=
1.0030-0.0041i
ymin=
-7.2169e-006-2.4786e-005i
Exiting:
Maximumnumberoffunctionevaluationshasbeenexceeded
-increaseMaxFunEvalsoption.
Currentfunctionvalue:
-0.000007
xmax=
1.0030-0.0041i
ymax=
-7.2169e-006-2.4786e-005i
14/
(1)(3)
clc;clear;
symsx;
f1=(sin(x)*cos(x))./(1+sin(x)^4);
I=int(f1,x)
f2=asin(x)./(1-x)^1/2;
K=int(f2,x)
I=
1/2*atan(sin(x)^2)
K=
1/4*i*asin(x)^2-asin(x)*log(1+i*(i*x+(1-x^2)^(1/2)))+i*polylog(2,-i*(i*x+(1-x^2)^(1/2)))
19/
(6)clc;clear;
symsn;
f=cos(1/n);
S15=symsum(f,n,1,15)
S40=symsum(f,n,1,40)
S15=
cos
(1)+cos(1/2)+cos(1/3)+cos(1/4)+cos(1/5)+cos(1/6)+cos(1/7)+cos(1/8)+cos(1/9)+cos(1/10)+cos(1/11)+cos(1/12)+cos(1/13)+cos(1/14)+cos(1/15)
S40=
cos(1/35)+cos(1/16)+cos(1/5)+cos(1/23)+cos(1/24)+cos(1/30)+cos
(1)+cos(1/7)+cos(1/39)+cos(1/40)+cos(1/14)+cos(1/25)+cos(1/26)+cos(1/9)+cos(1/10)+cos(1/19)+cos(1/31)+cos(1/17)+cos(1/27)+cos(1/28)+cos(1/33)+cos(1/34)+cos(1/11)+cos(1/3)+cos(1/20)+cos(1/4)+cos(1/21)+cos(1/8)+cos(1/18)+cos(1/2)+cos(1/29)+cos(1/22)+cos(1/6)+cos(1/12)+cos(1/13)+cos(1/36)+cos(1/15)+cos(1/32)+cos(1/37)+cos(1/38)
(5)clc;clear;
symsn;
f=log(n)/(n^3);
S15=symsum(f,n,1,15)
S40=symsum(f,n,1,40)
S15=
83/512*log
(2)+29/729*log(3)+1/125*log(5)+1/216*log(6)+1/343*log(7)+1/1000*log(10)+1/1331*log(11)+1/1728*log(12)+1/2197*log(13)+1/2744*
log(14)+1/3375*log(15)
S40=
1/12167*log(23)+5349/32768*log
(2)+1/1000*log(10)+1/8000*log(20)+1/1728*log(12)+1/27000*log(30)+1/17576*log(26)+1/21952*log(28)+109/23328*log(6)+1/5832*log(18)+1/29791*log(31)+1/59319*log(39)+1/9261*log(21)+1/2197*log(13)+1/50653*log(37)+1/6859*log(19)+1/4913*log(17)+1/1331*log(11)+1/13824*log(24)+1/2744*log(14)+1/10648*log(22)+1/3375*log(15)+1/35937*log(33)+1/39304*log(34)+127/15625*log(5)+262/6561*log(3)+1/42875*log(35)+1/343*log(7)+1/54872*log(38)+1/64000*log(40)+1/24389*log(29)
21/
clc;clear;
symsx;
f=x.^2*exp(-x)
y=taylor(f,x,6,0)
f=
x^2*exp(-x)
y=
x^2-x^3+1/2*x^4-1/6*x^5
24.求解微分方程
25.求解微分方程
27.用数值方法求解下列微分方程,用不同的颜色和线型将y和
画在同一个图形窗口里:
初始时间:
;终止时间:
;初始条件:
28.用数值方法求解下列微分方程,用不同的颜色和线型将y和
画在同一个图形窗口里:
初始时间:
;终止时间:
;初始条件:
24/
y=dsolve('Dy=x*sin(x)/cos(y)','x')
结果
y=
-asin(-sin(x)+x*cos(x)-C1)
25/
y=dsolve('Dy=y/x^2','x')
结果
y=
C1*exp(-1/x)
27/
将导数表达式的右端写成exf.m函数文件:
functionxdot=exf(t,x)
u=1-2*t;
xdot=[01;10]*x+[01]'*u;
主程序如下:
clf,
t0=0;tf=pi;x0t=[00];
[t,x]=ode23('exf',[t0,tf],x0t)
y=x(:
1)
Dy=x(:
2)
plot(t,y,'-',t,Dy,'o')
legend('数值积分解','解析解')
28/
将导数表达式的右端写成exf.m函数文件:
functionxdot=exf(t,x)
u=sin(2*t);
xdot=[01;t0]*x+[01]'*u;
主程序如下:
clf,
t0=0;tf=3;x0t=[00];
[t,x]=ode23('exf',[t0,tf],x0t)
y=x(:
1)
Dy=x(:
2)
plot(t,y,'-',t,Dy,'o')
legend('数值积分解','解析解')
2/
r=dsolve('D2y+4*Dy+29*y=0','y(0)=0,Dy(0)=15','x')
结果
r =
3*exp(-2*x)*sin(5*x)
3/
[x,y,z]=dsolve('Dx=2*x-3*y+3*z','Dy=4*x-5*y+3*z','Dz=4*x-4*y+2*z',
't');
x=simple(x)
y=simple(y)
z=simple(z)
结果
x=
C3*exp(2*t)+exp(-t)*C1
y=
C2*exp(-2*t)+C3*exp(2*t)+exp(-t)*C1
z=
C2/exp(t)^2+C3*exp(t)^2
4/
r=dsolve('D2y-5*(1-2*y^4)*Dy+7*y=0','y(0)=0,Dy(0)=1','x')
结果
r=
[emptysym]
1,分别用quad,trapz,quadl,int积分法计算下面的定积分,并比较结果精度。
2、计算下列积分
3、计算重积分
其中D是由所围成的区域
4、计算重积分
5、计算重积分:
1/
(1)clc
x=0:
1/100:
pi;
f=1./sqrt(2*pi).*exp(-x.^2/2).*sin(x.^3).*2.*x.^2;
trapz(x,f)
ans=
0.1940
clc
x=0:
1/100:
pi;
f=1./sqrt(2*pi).*exp(-x.^2/2).*sin(x.^3).*2.*x.^2;
quad('1./sqrt(2*pi).*exp(-x.^2/2).*sin(x.^3).*2.*x.^2',-0.5,1.5,10e-10)
clc
x=0:
1/100:
pi;
f=1./sqrt(2*pi).*exp(-x.^2/2).*sin(x.^3).*2.*x.^2;
quadl('1./sqrt(2*pi).*exp(-x.^2/2).*sin(x.^3).*2.*x.^2',-0.5,1.5,10e-16)
ans=
0.2769
symsxt
f=exp(t)*cos(2.*t);
int(f,t,x,x^2./2)
ans=
-1/5*exp(x)*cos(2*x)-2/5*exp(x)*sin(2*x)+1/5*exp(1/2*x^2)*cos(x^2)+2/5*exp(1/2*x^2)*sin(x^2)
(2)clc;clear;
x=0:
0.001:
1.5;
y=1./(1+x).*x.^5.*sin(x.^3+cos(x));
It=trapz(x,y)
Iq=quad('myfun',0,pi)
IQ=quadl('myfun',0,pi)
symst
yy=1/(1+t)*t^5*sin(t^3+cos(t));
Ii=int(yy,t,0,pi);
Ii=double(Ii);
errt=abs(It-Ii)
errq=abs(Iq-Ii)
errQ=abs(IQ-Ii)
myfun
functiony=myfun(x)
y=1./(1+x).*x.^5.*sin(x.^3+cos(x));
结果
It=0.4084
Iq=-0.5291
IQ=-0.5291
errt=0.9375
errq=2.1644e-007
errQ=9.7233e-013
2/
(1)
symsxt
f=1./exp(x)*(1+x.^4);
int(f,x)
ans=
-(25+24*x+12*x^2+4*x^3+x^4)/exp(x)
(2)
symsxt
f=1/exp(x)*(1+x^4);
m=int(f,x,0,inf)
m=
25
3/
symsxy
y1=('2*x*y=1');y2=('y-sqrt(2*x)=0');
[x,y]=solve(y1,y2,x,y)
结果:
x=1/2
y=1
symsxy
f=cos(x+y);y1=1/(2*x);y2=sqrt(2*x);
jfy=int(f,y,y1,y2);
jfx=int(jfy,x,0.5,2.5);
jf2=double(jfx)
结果:
jf2=-1.8321
4/
functionz=integrnd(x,y)
z=exp(-(x.^2+y.^2))./(x+y);
I=dblquad(@integrnd,0,4,1,2)
symsxy
bjh=exp(-(x^2+y^2))/(x+y);
jfx=int(bjh,y,0,4);jfy=int(jfx,x,1,2);
结果:
I=0.0689
5/
functionw=integrnd(x,y,z)
w=x+exp(y)+sin(z);
I=triplequad(@integrnd,-2,2,-2,2,0,4)
symsxyz
f=x+exp(y)+sin(z);
jfz=int(f,z,0,4);
jfy=int(jfz,y,-2,2);
jfx=int(jfy,x,-2,2);
结果:
I=142.5178
1.
clc;formatlong
n=10000;
m=0;
fori=1:
n
x=randperm(6);
y=x
(1);
ify==1
m=m+1;
end
end
fprintf('各点数出现的概率%5.32f\n',m/n);
各点数出现的概率为:
0.166********000000000000000000000
2.
n=100000;a=2;m=0;
fori=1:
n
x=rand
(1)*a/2;y=rand
(1)*a/2;z=rand
(1)*a/2;
if(x^2+y^2+z^2<=(a/2)^2)
m=m+1;
end
end
fprintf('计算出来的pi为%5.16f\n',6*m/n);
计算出来的pi为:
3.151********00001
3.
clc;clear
n=100000;a=2;m=0;
fori=1:
n
x=rand
(1)*sqrt
(2)/2;
y=rand
(1);
if(y>=x^2&y<=1-x^2)
m=m+1;
end
end
fprintf('计算出s的面积为:
%f\n',m/n);
x=0:
0.01:
sqrt
(2)/2;
y1=x.^2;
y2=1-x.^2;
plot(x,y1,x,y2)
计算出来的面积s为:
0.666470
0.471265
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