Mtalab 练习题（数学实验作业）

第四周数学实验作业

1、已知 z = log(x^2 +1) * y^2 +sin(x^3 y)+xy
（1）求Zx,Zxy,Zyy

syms x y
z = log(x^2+1)*y^2+sin(x^3*y)+x*y;
zx = diff(z,x)
zxy = diff(zx,y)
zyy = diff(z,y,2)

(2)（2）当x = 1,y = 2,dx = 0.02,dy = 0.03时，求dz

syms x y
z = log(x^2+1)*y^2+sin(x^3*y)+x*y;
zx = diff(z,x);
zy = diff(z,y);
x = 1;y = 2;
zx = eval(zx);
zy = eval(zy);
dx = 0.02;dy = 0.03;
dz = zx*dx+zy*dy

2、求曲面x^2 + 2y^2 + 3z^2 = 6在点(1,-1,-1)处的切平面和法线方程，并将曲面和切平面及其法线画在同一坐标窗口中。

syms x y z t
F = x^2+2*y^2+3*z^2-6;
%F = z-4/(x^2+y^2+1);
Fx=diff(F,x);Fy=diff(F,y);Fz=diff(F,z);
M = [x y z];M0=[1 -1 -1];
%M = [x y z];M0=[1/4 1/2 64/21]
Fx0=subs(Fx,M,M0);Fy0=subs(Fy,M,M0);Fz0=subs(Fz,M,M0);
N0 = [Fx0,Fy0,Fz0]; %计算法向量
eqTplane = dot(N0,M-M0) %计算切平面
eqNcurvex = M0(1)+Fx0*t %法线方程
eqNcurvey = M0(2)+Fy0*t
eqNcurvez = M0(3)+Fz0*t
[x, y, z] = ellipsoid(0,0,0,sqrt(6),sqrt(3),sqrt(2),50);
surfl(x, y, z);
colormap copper;
axis equal;
hold on 
syms z 
z = solve(eqTplane,z); %求出切平面方程
ezsurf(z) %绘制切平面
axis([-4 4 -2 2 -2 2]);
hold on
t = 0:0.2:10;
x = eval(eqNcurvex);y = eval(eqNcurvey);z = eval(eqNcurvez);
plot3(x,y,z) %绘制法线
view(30,5)

我也不知道怎么做到这么丑的

3、对于二元函数z = 2xy - 3x^2 - 2y^2 + 10
（1）利用导数求其极值

syms x y
z = 2*x*y - 3*x^2 - 2*y^2 + 10;
zx = diff(z,x);zy = diff(z,y);
[x0,y0]=solve(zx,zy,x,y);
%驻点[x0 y0]=[0,0]
zxx=diff(zx,x);zxy=diff(zx,y);zyy=diff(zy,y);
xmax=[];xmin=[];
for k=1:length(x0)
    A=subs(zxx,[x,y],[x0(k),y0(k)]);
    B=subs(zxy,[x,y],[x0(k),y0(k)]);
    C=subs(zyy,[x,y],[x0(k),y0(k)]);
    delta=double(A*C-B^2);
    if delta>0
        if double(A)<0
            xmax=[xmax;[x0(k),y0(k)]];
        else
            xmin=[xmin;[x0(k),y0(k)]];
        end
    end
    if ~isempty(xmax)
        zmax=subs(z,[x y],[xmax(:,1),xmax(:,2)]);
    else
        zmax=[];
    end
    if ~isempty(xmin)
        zmin=subs(z,[x y],[xmin(:,1),xmax(:,2)])
    else
        zmin=[];
    end
end
disp('极大值点和极大值是：')
    [xmax zmax]
disp('极小值点和极小值是：')
    [xmin zmin]

（2）画出等高线图和梯度向量场观察其极大值

x = -4:0.2:3;y=-3:0.2:3;
[x,y]=meshgrid(x,y);
z = 2.*x.*y - 3.*x.^2 - 2.*y.^2 + 10;
figure
mesh(x,y,z)
figure
contour(x,y,z,50)
hold on
grid on
[px,py]=gradient(z,0.2,0.2);
quiver(x,y,px,py)

高线图和梯度向量场可以看出，函数的极大值点为（0,0）

（3）利用 fminsearch 求其极大值。

z = @(x)2*x(1).*x(2) - 3*x(1).^2 - 2*x(2).^2 + 10;
[xmin,zmin]=fminsearch(z,rand(1,2));
z = @(x)-(2.*x(1).*x(2) - 3.*x(1).^2 - 2.*x(2).^2 + 10);
[xmax,z1]=fminsearch(z,rand(1,2));
zmax=-z1