1 求函数在指定点的导数值
>> syms a x y;
a=[x x^2 x^3;1 2*x 3*x^2;0 2 6*x];
y=det(a);
dy=diff(y);
x=[1 2 3];
eval(dy)
ans =
6 24 54
2 符号法求下列函数的导数或积分
1）y=x10+10x+logx10,求y’
>> y=sym('x^10+10^x+log(10)/log(x)')
Dy=diff(y)
y =
log(10)/log(x) + 10^x + x^10
Dy =
10^x*log(10) + 10*x^9 - log(10)/(x*log(x)^2)
2）y=ln(1+x), 求y’’∣x=1
>> syms x y;
y=log(1+x);
D2y=diff(y,2);
x=1;
eval(D2y)
ans =
-0.2500
3) y=ex/cosx,求y’
>> syms x y;
y=exp(x)/cos(x);
Dy=diff(y)
Dy =
exp(x)/cos(x) + (exp(x)*sin(x))/cos(x)^2
4)
>> f=sym('t*sin(t)');
int(f,0,pi)
ans =
pi
5) 已知函数z=sin(xy), 计算
>> syms x y z;
z=sin(x*y);
D3z=diff(diff(z,'x'),'y',2)
D3z =
- 2*x*sin(x*y) - x^2*y*cos(x*y)
3 用数值方法求定积分
1）
>> fun=inline('x.^2.*sqrt(2.*x.^2+3)');
[f,n]=quad(fun,1,5)
f =
232.8057
n =
21
2）
fun=inline('x./sin(x.^2)');
[f,n]=quadl(fun,pi/4,pi/3)
f =
0.3254
n =
18
4 已知数据[x,y]如下表，试求2次拟合多项式f(x)，然后求x=0.05,0.25,0.45,0.65,0.85,1.05各点的函数近似值，并绘出拟合曲线及求得的函数点。
| x | 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
| y | -0.447 | 1.978 | 3.28 | 6.16 | 7.08 | 7.34 | 7.66 | 9.56 | 9.48 | 9.3 | 11.2 |
y=[-0.447 1.978 3.28 6.16 7.08 7.34 7.66 9.56 9.48 9.3 11.2];
a=polyfit(x,y,2)
x1=0.05:0.2:1.05;
y1=a(1)*x1.^2+a(2)*x1+a(3);
yfit=polyval(a,x);
plot(x1,y1,'or',x,yfit,'-g')
a =
-9.8108 20.1293 -0.0317
5 求方程exsinx=0在区间[-4,-3]上的一个根。
>> fzero('exp(x)*sin(x)',[-4,-3])
ans =
-3.1416
6 已知某函数的离散点列如下
| X | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 0.97279 | 3.4234 | 3.8186 | 1.8415 | 0 | -0.15853 | -0.18141 | -2.5766 | -7.0272 |
>> x=[-4 -3 -2 -1 0 1 2 3 4];
y=[ 0.97279 3.4234 3.8186 1.8415 0 -0.15853 -0.18141 -2.5766 -7.0272];
x1=-2.5:1:2.5;
c1=interp1(x,y,x1,'pehip');
c2=interp1(x,y,x1,'splin');
c3=interp1(x,y,x1,'cubic');
plot(x1,c1,'o-r',x1,c2,'*--g',x1,c3,':bH')
legend('hermite插值','三次样条插值','三次插值')
7 用经典的RK方法计算常微分方程：y’=e-2t-2y 初值条件y(0)=0.1
>> fun=inline('exp(-2*t)-2*y','t','y');
[t,y]=ode45(fun,[0,1],0.1)
t =
0
0.0063
0.0126
0.0188
0.0251
0.0501
0.0751
0.1001
0.1251
0.1501
0.1751
0.2001
0.2251
0.2501
0.2751
0.3001
0.3251
0.3501
0.3751
0.4001
0.4251
0.4501
0.4751
0.5001
0.5251
0.5501
0.5751
0.6001
0.6251
0.6501
0.6751
0.7001
0.7251
0.7501
0.7751
0.8001
0.8251
0.8501
0.8751
0.9001
0.9251
0.9438
0.9626
0.9813
1.0000
y =
0.1000
0.1050
0.1098
0.1144
0.1190
0.1358
0.1507
0.1638
0.1753
0.1852
0.1938
0.2011
0.2073
0.2123
0.21
0.2195
0.2219
0.2235
0.2244
0.2247
0.2244
0.2236
0.2224
0.2207
0.2187
0.21
0.2137
0.2108
0.2077
0.2044
0.2009
0.1973
0.1935
0.16
0.1857
0.1817
0.1776
0.1735
0.1694
0.1653
0.1611
0.1581
0.1550
0.1519
0.14
8 求二阶微分方程
初值条件为x(0)=1, x’(0)=0
function dy = fun(x,y)
dy = zeros(2,1);
dy(1) = y(2);
dy(2) = (1-y(1)^2)*y(2)-y(1);
>> [x,y]=ode45('fun',[0,1],[1 0])
x =
0
0.0001
0.0001
0.0002
0.0002
0.0005
0.0007
0.0010
0.0012
0.0025
0.0037
0.0050
0.0062
0.0125
0.0188
0.0251
0.0313
0.0563
0.0813
0.1063
0.1313
0.1563
0.1813
0.2063
0.2313
0.2563
0.2813
0.3063
0.3313
0.3563
0.3813
0.4063
0.4313
0.4563
0.4813
0.5063
0.5313
0.5563
0.5813
0.6063
0.6313
0.6563
0.6813
0.7063
0.7313
0.7563
0.7813
0.8063
0.8313
0.8563
0.8813
0.9063
0.9313
0.9485
0.9657
0.9828
1.0000
y =
1.0000 0
1.0000 -0.0001
1.0000 -0.0001
1.0000 -0.0002
1.0000 -0.0002
1.0000 -0.0005
1.0000 -0.0007
1.0000 -0.0010
1.0000 -0.0012
1.0000 -0.0025
1.0000 -0.0037
1.0000 -0.0050
1.0000 -0.0062
0.9999 -0.0125
0.9998 -0.0188
0.9997 -0.0251
0.9995 -0.0313
0.9984 -0.0563
0.9967 -0.0813
0.9943 -0.1062
0.9914 -0.1310
0.9878 -0.1559
0.9836 -0.1806
0.9788 -0.2053
0.9733 -0.2300
0.9673 -0.2546
0.9606 -0.2792
0.9533 -0.3037
0.9454 -0.3282
0.9369 -0.3527
0.9278 -0.3772
0.9180 -0.4018
0.9077 -0.4263
0.67 -0.4509
0.8851 -0.4756
0.8729 -0.5003
0.8601 -0.5252
0.8467 -0.5502
0.8326 -0.5753
0.8179 -0.6006
0.8026 -0.6261
0.7866 -0.6519
0.7700 -0.6779
0.7527 -0.7042
0.7348 -0.7308
0.7162 -0.7578
0.6969 -0.7851
0.6769 -0.8128
0.6562 -0.8410
0.6349 -0.8696
0.6128 -0.87
0.59 -0.9283
0.5663 -0.9584
0.5497 -0.9795
0.5327 -1.0008
0.5153 -1.0224
0.4976 -1.0442
9 用三点公式和五点公式求F(x)=1/(1+x)2在x=1.1处的导数值。
| x | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 |
| F(x) | 0.2500 | 0.2268 | 0.2066 | 0.10 | 0.1736 |
if nargin == 3
h = 0.1;
else if (nargin == 4 && h == 0.0)
disp('h不能为0！');
return;
end
end
y0 = subs(sym(func), findsym(sym(func)),x0);
y1 = subs(sym(func), findsym(sym(func)),x0+h);
y2 = subs(sym(func), findsym(sym(func)),x0+2*h);
y_1 = subs(sym(func), findsym(sym(func)),x0-h);
y_2 = subs(sym(func), findsym(sym(func)),x0-2*h);
switch type
case 1,df = (-3*y0+4*y1-y2)/(2*h); %用第一个公式求导数
case 2,df = (3*y0-4*y_1+y_2)/(2*h); %用第二个公式求导数
case 3,df = (y1-y_1)/(2*h); %用第三个公式求导数
end
function df=FivePoint(func,x0,type,h)
if nargin == 3
h = 0.1;
else if (nargin == 4 && h == 0.0)
disp('h不能为0！');
return;
end
end
y0 = subs(sym(func), findsym(sym(func)),x0);
y1 = subs(sym(func), findsym(sym(func)),x0+h);
y2 = subs(sym(func), findsym(sym(func)),x0+2*h);
y3 = subs(sym(func), findsym(sym(func)),x0+3*h);
y4 = subs(sym(func), findsym(sym(func)),x0+4*h);
y_1 = subs(sym(func), findsym(sym(func)),x0-h);
y_2 = subs(sym(func), findsym(sym(func)),x0-2*h);
y_3 = subs(sym(func), findsym(sym(func)),x0-3*h);
y_4 = subs(sym(func), findsym(sym(func)),x0-4*h);
switch type
case 1,
df = (-25*y0+48*y1-36*y2+16*y3-3*y4)/(12*h); %用第一个公式求导数
case 2,
df = (-3*y_1-10*y0+18*y1-6*y2+y3)/(12*h); %用第二个公式求导数
case 3,
df = (y_2-8*y_1+8*y1-y2)/(12*h); %用第三个公式求导数
case 4,
df = (3*y1+10*y0-18*y_1+6*y_2-y_3)/(12*h); %用第四个公式求导数
case 5,
df = (25*y0-48*y_1+36*y_2-16*y_3+3*y_4)/(12*h); %用第五个公式求导数
end
x =1.1;
dy3=TherePoint('1/(1+x)^2',x,1)
dy5=FivePoint('1/(1+x)^2',x,1)
10
求
>> y=sym('(x^3+x^2+x+1)^(1/3)-sqrt(x^2+x+1)*log(exp(x)+x)/x');
limit(y,inf)
ans =
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