2010数学软件实践作业题Maple部分2010.9
数学软件实践 课程 考核形式： 大型作业
学号 姓名 专业
一、 基础题：
1、 x=5,y=11,计算x+y;x*y;x/y; > x:=5:y:=11:
> x+y;
16
> x*y;
55> x/y;
511>
2、 计算25！； > 25!;
15511210043330985984000000
3、 求圆周率 50位有效数字的近似值；
> evalf(Pi,50);
3.1415926535897932384626433832795028841971693993751
4、 求角度为Pi/4各种三角函数的值；
> sin(Pi/4);cos(Pi/4);tan(Pi/4);cot(Pi/4); sec(Pi/4);csc(Pi/4);
2222
1 1 2 2
1
5、 f(x)=x^2+x-1,g(x)=x^3-2, 求f(x)+g(x)； > f:=x->x^2+x-1;g:=x->x^3-2;
f := x???x???x???1 g := x???x???2
32> f(x)+g(x);
x???x???3???x
23>
二、 直接函数题（可根据课件把Maple函数或程序直接拷备各题后面）
1、求F31?2231?1除以3的余数, 并判别它是否为素数
> 2&^(2^31)+1 mod 3;
2
>
2、 将10进制数1705124778833转换为2进制数
> convert(1705124778833, binary);
11000110100000001010110110101001101010001
>
3、求和13?33?53??313
> sum((2*k+1)^3,k=0..15);
13081632
4、设f(x)?x?15x?23x?7,求 f(1)，f(3)，f(1687)，f(5432109876)
> f:=x->x^3+15*x^2-23*x+7;
f := x???x???15x???23x???7
32> f(1),f(3),f(1687),f(5432109876);
0,100,4843800444,160289708416811759640280676875
5、求极限 limx?0sin2x3xcosx 2>
Limit(sin(2*x)/(3*x)*cos(2*x),x=0)=limit(sin(2*x)/(3*x)*cos(2*x),x=0);
limx???01sin(2x)cos(2x)2???3x3
>
2
6、求
sin2xcosx2 的导数 3x>
Diff(sin(2*x)/(3*x)*cos(2*x),x)=diff(sin(2*x)/(3*x)*cos(2*x),x);
ddx?1sin(2x)cos(2x)????2cos(2x)???1sin(2x)cos(2x)???2sin(2x)??2?3?3xx33xx??22
>
7、求由 x2?y?sinx?exy?0确定的隐函数y对x得导数
> f:=x^2+y+sinx+e^(x*y)=0;
f := x???y???sinx???e2(xy)???0
> implicitdiff(f,y,x);
?2x???e1???e(xy)ln(e)y(xy)
ln(e)x>
8、求函数 f(x)?x?5x?5x?242 的极值，并画图
x???5x???5x???242> f:=(x+5)/(x^4+5*x^2+2);
f :=
> plot((x+5)/(x^4+5*x^2+2),x = -10 .. 10);
> plot(f,x=-10..10,y=-0.005..0.005);
3
> d:=diff(f,x);
d := 1x???5x???242???(x???5)(4x???10x)(x???5x???2)4223
> simplify(d);
?3x???5x???2???20x???50x(x???5x???2)422423
> xmin:=fsolve(d=0,{x=-5});
xmin := {x???-6.785181839}
}> xmax:=fsolve(d=0,{x=0});
xmax := {x???0.03981606880
> Digits:=4:
> Xmin:=eval(f,xmin);
Xmin := -0.0007593
> Xmax:=eval(f,xmax);
Xmax := 2.510
>
9、求函数f(x,y)?x?2y在条件q(x,y)?x?y?2x?2y?1?0下的条件极值
2222> f:=x^2+2*y^2:
> q:=x^2+y^2+2*x-2*y+1: > g:=f+mu*q;
g := x???2y????(x???y???2x???2y???1)
2222> exp1:=diff(g,x); exp2:=diff(g,y);
exp1 := 2x????(2x???2)exp2 := 4y????(2y???2)
4
> exp3:=solve({q=0,exp1,exp2},{x,y,mu});
exp3 := {y???RootOf(?2_Z???_Z???4_Z???1),x???24124RootOf(?2_Z???_Z???4_Z???1)2321112424???RootOf(?2_Z???_Z???4_Z???1)???RootOf(?2_Z???_Z???4_Z???1)???,?22231112424???RootOf(?2_Z???_Z???4_Z???1)???RootOf(?2_Z???_Z???4_Z???1)???2222124???RootOf(?2_Z???_Z???4_Z???1)}2
> allvalues(exp3);
10、求解线性规划问题 目标函数f?x?3y?2z在约束条件
3x+2y-4z<=22, 5x-4y-3z<=11,7x+4y+9z<=27下的的最大值
> with(simplex):
cnsts:={3*x+2*y-4*z<=22,
5*x-4*y-3*z<=11,7*x+4*y+9*z<=27};
cnsts := {3x???2y???4z???22,5x???4y???3z???11,7x???4y???9z???27}
> obj:=x+3*y-2*z;
obj := x???3y???2z
> maximize(obj,cnsts union {x>=0,y>=0,z>=0});
{x???0,z???0,y???274}
> f:=(x,y,z)->x+3*y-2*z;
f := (x,y,z)???x???3y???2z
> f(0,27/4,0);
814
>
11、求不定积分
?exsinxdx
> Int(e^x*sinx,x)=int(e^x*sinx,x);
?exsinxdx???sinxe??ln(e)??/2x
12、求定积分
?0xsin(sinx)dt，并求近似值
2>
Int(x^2*sin(sinx),t=0..Pi/2)=int(x^2*sin(sinx),t=0..Pi/2);
5