Matlab符号微积分练习

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Matlab符号微积分练习

2024-02-24 17:49| 来源: 网络整理| 查看: 265

由于这篇博文公式实在太多，截图又不太好看，所以切换到Markdown编辑器下来写。还好CSDN的这个Markdown支持LaTeX公式，方便许多了。 1.计算下列各式： (1) limx→0tanx−sinx1−cos2x

%1(1) clear clc syms x; f=(tan(x)-sin(x))/(1-cos(2*x)); limit(f) %其它等价用法 limit(f,0) limit(f,x,0)

结果

ans = 0

(2) limx→+∞x2+xex

%1(2) clear clc syms x; f=(x^2+x)/exp(x); limit(f,inf) %其它等价用法 limit(f,x,inf)

结果

ans = 0

(3) y=x3−2x2+sinx ,求 y′

%1(3) clear clc syms x; y=x^3-2*x^2+sin(x); diff(y) %其它等价用法 diff(y,x) diff(y,1) diff(y,x,1)

结果

ans = cos(x) - 4*x + 3*x^2

(4) sin2xlnx ,求 y′

%1(4) clear clc syms x; y=sin(2*x)*log(x); diff(y) %其它等价用法 diff(y,x) diff(y,1) diff(y,x,1)

结果

ans = 2*cos(2*x)*log(x) + sin(2*x)/x

(5) f=ln(x+y2) ,求 ∂f∂x , ∂f∂y , ∂2f∂x∂y

clear clc syms x y; f=log(x+y^2); diff(f,x) %其它等价用法 diff(f) diff(f,1) diff(f,x,1) diff(f,y) %其它等价用法 diff(f,y,1) diff(diff(f,x),y)

结果

ans = 1/(y^2 + x) ans = (2*y)/(y^2 + x) ans = -(2*y)/(y^2 + x)^2

(6) y=xyln(x+y) ,求 ∂f∂x , ∂f∂y , ∂2f∂x∂y

%1(6)将表达式中等号左边的y改成f clear clc syms x y; f=x*y*log(x+y); df_dx=diff(f,x) df_dy=diff(f,y) dff_dxdy=diff(diff(f,x),y)

结果

df_dx = y*log(x + y) + (x*y)/(x + y) df_dy = x*log(x + y) + (x*y)/(x + y) dff_dxdy = log(x + y) + x/(x + y) + y/(x + y) - (x*y)/(x + y)^2 %1(6)把整个表达式视为f=0 clear clc syms x y; f=y-x*y*log(x+y); df_dx=diff(f,x) df_dy=diff(f,y) dff_dxdy=diff(diff(f,x),y) %此题还可以考隐函数求导,对于F(x,y)=0，dy/dx=-Fx/Fy(注意负号和顺序) dy_dx=-df_dx/df_dy dyy_dxx=diff(-df_dx/df_dy,x)

结果

df_dx = - y*log(x + y) - (x*y)/(x + y) df_dy = 1 - (x*y)/(x + y) - x*log(x + y) dff_dxdy = (x*y)/(x + y)^2 - x/(x + y) - y/(x + y) - log(x + y) dy_dx = -(y*log(x + y) + (x*y)/(x + y))/(x*log(x + y) + (x*y)/(x + y) - 1) dyy_dxx = ((y*log(x + y) + (x*y)/(x + y))*(log(x + y) + x/(x + y) + y/(x + y) - (x*y)/(x + y)^2))/(x*log(x + y) + (x*y)/(x + y) - 1)^2 - ((2*y)/(x + y) - (x*y)/(x + y)^2)/(x*log(x + y) + (x*y)/(x + y) - 1)

(7) ∫cos(4x+3)dx , ∫π60cos(4x+3)dx

%1(7) clear clc syms x; f=cos(4*x+3); int(f) int(f,0,pi/6) %其它等价用法int(f,x,0,pi/6)

结果

ans = sin(4*x + 3)/4 ans = (3^(1/2)*cos(3))/8 - (3*sin(3))/8

(8) y=∫ln(1+t)dx , ∫270ln(1+t)dx

%1(8)把题中所有dx改为dt clear clc syms t; y=log(1+t); int(y) int(y,0,27) %其它等价用法int(y,t,0,27)

结果

ans = (log(t + 1) - 1)*(t + 1) ans = 28*log(28) - 27

2.计算下列定积分 (1)计算积分 ∫1−1x+x3+x5dx 的值 (2)计算积分 ∫101sinx+cosxdx 的值 (3)计算积分 ∫62ex2dx 的值 (4)计算积分 ∫101xx4+4dx 的值 (5)计算积分 ∫101∫101sinyx+y4+x2dxdy 的值 (6)计算积分 ∫101∫y1yx+y4dxdy 的值 (7)计算积分 ∫30z∫101∫y1yx+y4dxdydz 的值

%2 clear clc syms x y z; int(x+x^3+x^5,-1,1) int(sin(x)+cos(x),1,10) int(exp(x/2),2,6) int(x/(x^4+4),1,10) int(int(sin(y)*(x+y)/(4+x^2),x,1,10),y,1,10) int(int(y*(x+y)/4,x,1,y),y,1,10) int(z*int(int(y*(x+y)/4,x,1,y),y,1,10),0,3)

结果

ans = 0 ans = cos(1) - cos(10) - sin(1) + sin(10) ans = 2*exp(1)*(exp(2) - 1) ans = atan(50)/4 - atan(1/2)/4 ans = log(520^(1/2)/5)*(cos(1) - cos(10)) - (atan(1/2)*(cos(1) - 10*cos(10) - sin(1) + sin(10)))/2 + (atan(5)*(cos(1) - 10*cos(10) - sin(1) + sin(10)))/2 ans = 27135/32 ans = 244215/64

3.求解下列非线性方程（组） (1)

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪0.4x+0.3124y+2.6598z+6.9785w=0.243.142x+8.22y+6.16z+0.254w=3.2510.1785x+5.358y+9.7932z+3.846w=0.212.643x+8.321y+0.283z+6.735w=−2.354

clear clc syms x y z w; f1=0.4*x+0.3124*y+2.6598*z+6.9785*w-0.24; f2=3.142*x+8.22*y+6.16*z+0.254*w-3.251; f3=0.1785*x+5.358*y+9.7932*z+3.846*w-0.21; f4=2.643*x+8.321*y+0.283*z+6.735*w+2.354; [x,y,z,w]=solve(f1,f2,f3,f4)

结果

x = -4629578672873047/19238652687953436 y = 3345137846701581/1603221057329453 z = -29542916663552315/38477305375906872 w = 19161390580068175/38477305375906872

(2)

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪0.9501x+0.8913y+0.8214z+0.9218w=0.240.2311x+0.7621y+0.4447z+0.7382w=0.54280.6068x+0.4565y+0.6154z+0.1763w=0.53760.4860x+0.0185y+0.7919z+0.4057w=−0.5714

%3(2) clear clc syms x y z w; f1=0.9501*x+0.8913*y+0.8214*z+0.9218*w-0.24; f2=0.2311*x+0.7621*y+0.4447*z+0.7382*w-0.5428; f3=0.6068*x+0.4565*y+0.6154*z+0.1763*w-0.5376; f4=0.4860*x+0.0185*y+0.7919*z+0.4057*w+0.5714; [x,y,z,w]=solve(f1,f2,f3,f4)

结果

x = -825043647512601/577559329613032 y = -349805538096493/577559329613032 z = 1204288618243359/577559329613032 w = 96242555204173/288779664806516

4.求解下列非线性方程（组） (1) x2−x−1=0

clear clc syms x; solve('x^2-x-1=0')

结果

ans = 1/2 - 5^(1/2)/2 5^(1/2)/2 + 1/2

(2) 2x3+x2+3x−1=0

%4(2) clear clc syms x; solve('2*x^3+x^2+3*x-1=0')

结果

ans = ((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3) - 17/(36*((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)) - 1/6 17/(72*((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)) - ((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)/2 - 1/6 - (3^(1/2)*(17/(36*((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)) + ((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3))*i)/2 17/(72*((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)) - ((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)/2 - 1/6 + (3^(1/2)*(17/(36*((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3)) + ((419^(1/2)*1728^(1/2))/1728 + 10/27)^(1/3))*i)/2

(3)

{x−0.7sinx−0.2cosy=0y−0.7cosx+0.2siny=0

%4(3) clear clc syms x y; f1=x-0.7*sin(x)-0.2*cos(y); f2=y-0.7*cos(x)+0.2*sin(y); [x,y]=solve(f1,f2)%4(3) clear clc syms x y; f1=x-0.7*sin(x)-0.2*cos(y); f2=y-0.7*cos(x)+0.2*sin(y); [x,y]=solve(f1,f2)

结果

x = 0.52652262191818418730769280519209 y = 0.50791971903684924497183722688768

(4)

⎧⎩⎨x2y2=0x−y2=α

%4(3) clear clc syms x y alpha; [x,y]=solve('x^2*y^2=0','x-y/2=alpha',x,y)

结果

x = alpha 0 y = 0 -2*alpha

5.求解微分方程，初始值都设为0。 (1) y′=−2y+3x2+1 (2) y′=−y+4x

clear clc dsolve('Dy = -2*y+3*x^2+1','y(0)=0') %等价dsolve('Dy = -2*y+3*x^2+1','y(0)=0','t') dsolve('Dy = -y+4*x','y(0)=0') %等价dsolve('Dy = -y+4*x','y(0)=0','t')

结果

ans = (3*x^2)/2 - (3*x^2 + 1)/(2*exp(2*t)) + 1/2 ans = 4*x - (4*x)/exp(t)

6.已知函数 f=sin2xlnx ，绘制一个子图，上面两个子图分别为f导函数及其积分函数在区间[-5，5]，步长为0.2，颜色分别为红色，黑色虚线的图像，下面的子图为f的图形，区间步长同上，颜色为蓝色点划线，加图例分别为导函数，积分函数，原函数。

clear clc syms x; f=sin(2*x)*log(x); df=diff(f); intf=int(f); x1=-5:0.2:5; y1=real(subs(df,x,x1)); %因为结果中含有虚部，画图时会出现警告Warning: Imaginary parts of complex X and/or Y arguments ignored %加real()函数忽略警告 x2=-5:0.2:5; y2=real(subs(intf,x,x2)); subplot(221);plot(x1,y1,'r--');legend('导函数'); subplot(222);plot(x2,y2,'k--');legend('积分函数'); x3=-5:0.2:5; y3=real(subs(f,x,x3)); subplot(2,2,[3 4]);plot(x3,y3,'b-.');legend('原函数');

结果 6题图片

7.绘制方程 4−x29−y24−−−−−−−−−−√ 在区间 x∈[−2,2] , y∈[−1,1] 的网格图形，步长取0.2。

%7 clear clc x=-2:0.2:2; y=-1:0.2:1; [X,Y]=meshgrid(x,y); Z=sqrt(4-X.^2/9-Y.^2/4); plot3(X,Y,Z);

7题结果图 8.绘制方程 5−x23−y27−−−−−−−−−−√ 在区间 x∈[−2,2] , y∈[−1,1] 的三维曲面图形，步长取0.1。

%8 clear clc x=-2:0.1:2; y=-1:0.1:1; [X,Y]=meshgrid(x,y); Z=sqrt(5-X.^2/3-Y.^2/7); mesh(X,Y,Z);

8题结果图 9.绘制方程 f=y1+x2+y2 在区间 x∈[−4,4] , y∈[−2,2] 的三维网格加网格线图形，步长取0.25。

%9 clear clc x=-4:0.25:4; y=-2:0.25:2; [X,Y]=meshgrid(x,y); Z=Y./(1+X.^2+Y.^2); mesh(X,Y,Z); grid on

9题结果图注：Matlab2010b中surf*函数，mesh*函数都自动带网格补充：shading interp对图像进行插值处理使其连续光滑

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