数学实验高等数学分册习题(下)参案
高等数学II (下) 实验参考答案
唐诗权编写
第5章
1. (1)
>> syms x y;
>> f=(1-cos(x^2+y^2))/((x^2+y^2)*exp(x^2*y^2));
>> limit(limit(f,x,0),y,0)
ans =
(2)
>>syms x y ;
>>f=(log(x*exp(x)+exp(y)))/sqrt(x^2+y^2);
>>limit(limit(f,x,0),y,0)
ans =
NaN
(3)
>>syms x y ;
>>f=(2*x*sin(y))/(sqrt(x*y+1)-1);
>>limit(limit(f,x,0),y,0)
ans =
4
2. (1)
syms x y ;
z=((x^2+y^2)/(x^2-y^2))*exp(x*y);
zx=diff(z,'x' )
zx =
(2*x*exp(x*y))/(x^2 - y^2) - (2*x*exp(x*y)*(x^2 + y^2))/(x^2 - y^2)^2 + (y*exp(x*y)*(x^2 + y^2))/(x^2 - y^2)
zy=diff(z,'y' )
zy =
(2*y*exp(x*y))/(x^2 - y^2) + (x*exp(x*y)*(x^2 + y^2))/(x^2 - y^2) + (2*y*exp(x*y)*(x^2 + y^2))/(x^2 - y^2)^2
(2)syms x y z ;
u=log(3*x-2*y+z);
ux=diff(u,'x' )
ux =
3/(3*x - 2*y + z)
uy=diff(u,'y' )
uy =
-2/(3*x - 2*y + z)
uz=diff(u,'z' )
uz =
1/(3*x - 2*y + z)
(3)syms x y ;
z=sqrt(x)*sin(y/x);
zx=diff(z,'x' )
zx =
sin(y/x)/(2*x^(1/2)) - (y*cos(y/x))/x^(3/2)
zy=diff(z,'y' )
zy =
cos(y/x)/x^(1/2)
(4)syms x y ;
z=asin(y*sqrt(x));
zx=diff(z,'x' )
zx =
y/(2*x^(1/2)*(1 - x*y^2)^(1/2))
zy=diff(z,'y' )
zy =
x^(1/2)/(1 - x*y^2)^(1/2)
3. syms x y ;
f=x+y-sqrt(x^2+y^2);
fx=diff(f,'x' );
fy=diff(f,'y' );
x=2;
y=4;
fx0=subs(fx)
fx0 =
0.5528
fy0=subs(fy)
fy0 =
0.1056
4. syms x y z a ;
f=x^2+y^2+z^2-2*a*x*y*z;
fx=diff(f,'x' );
fy=diff(f,'y' );
fz=diff(f,'z' );
dzx=-fx/fz
dzx =
-(2*x - 2*a*y*z)/(2*z - 2*a*x*y)
dzy=-fy/fz
dzy =
-(2*y - 2*a*x*z)/(2*z - 2*a*x*y)
5. syms x y u v R dux dvx duy dvy ;
f=x^2+y^2+u^2+v^2-R^2;
g=x+y+u+v;
fx=diff(f,'x' );fy=diff(f,'y' );fu=diff(f,'u' );fv=diff(f,'v' ); gx=diff(g,'x' );gy=diff(g,'y' );gu=diff(g,'u' );gv=diff(g,'v' );
ffx=fu*dux+fv*dvx+fx;
ggx=gu*dux+gv*dvx+gx;
[dux,dvx]=solve(ffx,ggx,'dux' , 'dvx' )
dux =
(v - x)/(u - v)
dvx =
-(u - x)/(u - v)
ffy=fu*duy+fv*dvy+fy;
ggy=gu*duy+gv*dvy+gy;
[duy,dvy]=solve(ffy,ggy,'duy' , 'dvy' )
duy =
(v - y)/(u - v)
dvy =
-(u - y)/(u - v)
6. syms x y z ;
f=x+y+z-exp(-(x+y+z));
fx=diff(f,'x' );fz=diff(f,'z' );
dzx=-fx/fz;
g=dzx;
gy=diff(g,'y' );gz=diff(g,'z' );
dzxy=gy+gz*dzx
dzxy =
7. syms x y ;
z=x^2-x*y-2*y^2;
zx=diff(z,'x' );zy=diff(z,'y' );
fzx=inline(zx);fzy=inline(zy);
a=pi/3;b=pi/6;
f1=fzx(1,2)*cos(a)+fzy(1,2)*cos(b)
f1 =
-7.7942
8. syms t ;
x=sin(t);
y=cos(t);
z=t;
dx=diff(x,'t' );dy=diff(y,'t' );dz=diff(z,'t' );
x1=inline(dx);x2=inline(dy);x3=inline(dz);
t=pi/4;a=sin(t);b=cos(t);c=t;
x10=x1(a)
x10 =
0.7602
x20=x2(b)
x20 =
-0.6496
x30=x3(c)
x30 =
1
9. syms x y ;
f=x^2+y^2;
fx=diff(f,'x' );fy=diff(f,'y' );
x=1;y=2;
fx0=subs(fx)
fx0 =
2
fy0=subs(fy)
fy0 =
4
10. syms x y ;
f=x^3-y^3+3*x^2+3*y^2-9*x;
fx=diff(f,'x' );fy=diff(f,'y' );
[x0,y0]=solve(fx,fy)
x0 =
1
-3
1
-3
y0 =
2
2
fxx=diff(diff(f,'x' ), 'x' )
fxx =
6*x + 6
fxy=diff(diff(f,'x' ), 'y' );
fyy=diff(diff(f,'y' ), 'y' );
delta=inline(fxy^2-fxx*fyy);
delta(x0,y0)
ans =
-72
72
72
-72
x=1;y=0;
fmin=subs(f)
fmin =
-5
x=-3;y=2;
fmax=subs(f)
fmax =
31
11. syms x y z lamda a ;
L=x*y*z+lamda*((x^2+y^2)/4+z^2-a^2);
Lx=diff(L,'x' );
Ly=diff(L,'y' );
Lz=diff(L,'z' );
Llamda=diff(L,'lamda' );
[lamda x y z]=solve(Lx,Ly,Lz,Llamda)
lamda =
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
x =
2*a
(-2)*a
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
y =
2*a
(-2)*a
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
-(2*3^(1/2)*a)/3
z =
a
-a
(3^(1/2)*a)/3
(3^(1/2)*a)/3
(3^(1/2)*a)/3
-(3^(1/2)*a)/3
-(3^(1/2)*a)/3
-(3^(1/2)*a)/3
(3^(1/2)*a)/3
-(3^(1/2)*a)/3
V=x.*y.*z
V =
(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
-(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
(4*3^(1/2)*a^3)/9
第6章
1. syms x y ;
f=x*y;
y1=2*x;
y2=x^2+1;
I=int(int(f,y,y1,y2),x,0,1)
I =
1/12
2. syms x y r t ;
x=r*cos(t);y=r*sin(t);
f=exp(-(x^2+y^2));
I=int(int(f*r,r,0,1),t,0,2*pi)
I =
-pi*(1/exp(1) - 1)
3. syms x y z a A t r s ;
x=r*sin(s)*cos(t);
y=r*sin(s)*sin(t);
z=r*cos(s);
f=x^2+y^2;
I1=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,a),t,0,2*pi); I2=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,A),t,0,2*pi); I=I2-I1
I =
(4*pi*A^5)/15 - (4*pi*a^5)/15
4. syms x y z t r s ;
x=r*sin(s)*cos(t);
y=r*sin(s)*sin(t);
z=r*cos(s);
f=(z*log(x^2+y^2+z^2+1))/(x^2+y^2+z^2+1);
M=int(int(int(f*r^2*sin(s),s,0,1/2*pi),r,0,1),t,0,2*pi) M =
-(pi*(log(2)^2 - log(16) + 2))/4
5.x=1/2:0.01:1;
y=1./x;
plot(x,y,'k' );
xlabel('x' );ylabel('y' )
syms x ;
f=1/x;
fx=diff(f,'x' );
I=int(x*sqrt(1+fx^2),x,1/2,1) I =
int(x*(1/x^4 + 1)^(1/2), x = 1/2..1)
6. syms x y r t ;
x=r*cos(t);y=r*sin(t);
z=sqrt(x^2+y^2);
zx=diff(z,'x' );zy=diff(z,'y' );
f=(sqrt(x^2+y^2))*sqrt(1+zx^2+zy^2); M=int(int(f*r,r,0,1),t,0,2*pi) M =
(2*pi)/3
10.syms x y z ;
P=4*x;Q=-2*x*y;R=z^2;
px=diff(P,'x' );
qy=diff(Q,'y' );
rz=diff(R,'z' );
x=1;y=1;z=3;
px0=subs(px)
px0 =
4
qy0=subs(qy)
qy0 =
-2
rz0=subs(rz)
rz0 =
6
第7章
1.(1)syms n ;
symsum((n+1)^(1/3)-n^(1/3),n,1,inf)
ans =
Inf
(2)syms n b c ;
b='factorial(n+1)';c='factorial(n)';
f='((b/((n+1)^(n+1)))/(c/(n^n)))';
p=limit(f,n,inf)
p =
(3)syms n ;
f='((n/(2*n+1))^n)^(1/n)';
p=limit(f,n,inf)
p =
1/2
(4)syms n ;
f='(sqrt(n)/sqrt(n^4+1))/(1/sqrt(n^3))';
p=limit(f,n,inf)
p =
1
(5)syms n;
f='(log(n)/(1+3^n))/(1/3^n)';
p=limit(f,n,inf)
p =
Inf
(6)syms n ;
symsum(sqrt(n+2)-2*sqrt(n+1)+sqrt(n),n,1,inf)
ans =
sum((n + 2)^(1/2) - 2*(n + 1)^(1/2) + n^(1/2), n = 1..Inf)
2.(1)syms n ;
p=limit('(1/((n+1)*2^(n+1)))/(1/(n*2^n))',n,inf);
r=1/p
r =
2
(2)syms n ;
p=limit('(1/factorial(n+1))/(1/factorial(n))',n,inf);
r=1/p
r =
Inf
(3)syms n ;
p=limit('(n+1)/n',n,inf);
r=1/p
r =
1
(4)syms n ;
p=limit('(2^(n+1)/((n+1)^2+1))/(2^n/(n^2+1))',n,inf);
r=1/p
r =
1/2
3.syms x ;
taylor('sin(x)',x,pi/4,6)
ans =
(2^(1/2)*(pi/4 - x)^3)/12 - (2^(1/2)*(pi/4 - x)^2)/4 + (2^(1/2)*(pi/4 - x)^4)/48 - (2^(1/2)*(pi/4 - x)^5)/240 + 2^(1/2)/2 - (2^(1/2)*(pi/4 - x))/2
4.syms x ;
taylor('1/(1+x^2)',x,13)
ans =
x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
5.syms x ;
taylor('1/(x^2+3*x+2)',x,-4,6)
ans =
(5*x)/36 + (19*(x + 4)^2)/216 + (65*(x + 4)^3)/1296 + (211*(x + 4)^4)/7776 + (665*(x +
4)^5)/46656 + 13/18
6.syms x ;
taylor('(1+x)*log(1+x)',x,7)
ans =
x^6/30 - x^5/20 + x^4/12 - x^3/6 + x^2/2 + x
7.syms x t ;
f=int(1/(sqrt(1-t^2)),t,0,x);
taylor(f,x,12)
ans =
(63*x^11)/2816 + (35*x^9)/1152 + (5*x^7)/112 + (3*x^5)/40 + x^3/6 + x
8.syms x y ;
y=taylor('log(1+x)',x,8);
x=2;
y0=eval(y)
y0 =
12.6857
9.
10.(1)syms x n ;
f=3*x^2+1
a0=int(f,x,-pi,pi)/pi
an=int(f*cos(n*x),x,-pi,pi)/pi
bn=int(f*sin(n*x),x,-pi,pi)/pi
f =
3*x^2 + 1
a0 =
2*pi^2 + 2
an =
((6*(pi^2*n^2*sin(pi*n) - 2*sin(pi*n) + 2*pi*n*cos(pi*n)))/n^3 + (2*sin(pi*n))/n)/pi
bn =
(2)syms x n ;
f=exp(2*x)
a0=int(f,x,-pi,pi)/pi
an=int(f*cos(n*x),x,-pi,pi)/pi
bn=int(f*sin(n*x),x,-pi,pi)/pi
f =
exp(2*x)
sinh(2*pi)/pi
an =
((exp(2*pi)*(2*cos(pi*n) + n*sin(pi*n)))/(n^2 + 4) - (2*cos(pi*n) - n*sin(pi*n))/(exp(2*pi)*(n^2 +
4)))/pi
bn =
((2*sin(pi*n) + n*cos(pi*n))/(exp(2*pi)*(n^2 + 4)) + (exp(2*pi)*(2*sin(pi*n) - n*cos(pi*n)))/(n^2 +
4))/pi
第8章
1.(1)dsolve('x*Dy-y*log(y)=0','x')
ans =
1
exp(exp(C3 + log(x)))
(2)dsolve('x^2+y^2=x*y*Dy','x')
ans =
2^(1/2)*x*(C8 + log(x))^(1/2)
-2^(1/2)*x*(C8 + log(x))^(1/2)
(3)dsolve('Dy+y*cos(x)=exp(-sin(x))','x')
ans =
C14/exp(sin(x)) + x/exp(sin(x))
(4)dsolve('(x-2)*Dy=y+2*(x-2)^3','x')
ans =
(x - 2)^3 + C16*(x - 2)
(5) dsolve('D2y+6*Dy+13*y=0','x')
ans =
(C18*cos(2*x))/exp(3*x) + (C19*sin(2*x))/exp(3*x)
(6) dsolve('D4y-2*D3y+D2y=0','x')
ans =
C22 - 2*C21 - C21*x + C23*exp(x) + C24*x*exp(x)
(7)dsolve('2*D2y+5*Dy=5*x^2-2*x-1','x'
ans =
C26 + (7*x)/25 - (3*x^2)/5 + x^3/3 + C27/exp((5*x)/2) - 14/125
(8)dsolve('D2y-6*Dy+9*y=exp(2*x)*sin(2*x)','x')
ans =
(exp(2*x)*(4*cos(2*x) - 3*sin(2*x) + 10*x*cos(2*x) + 5*x*sin(2*x)))/25 + C29*exp(3*x) + C30*x*exp(3*x) - (x*exp(2*x)*(2*cos(2*x) + sin(2*x)))/5
2.(1)f=dsolve('cos(y)*sin(y)*Dy=cos(y)*sin(x)','y(0)=pi/4')
f =
pi - acos(sin(x)*(t - 1/2*2^(1/2)/sin(x)))
(2) f=dsolve('Dy=exp(2*x-y)','y(pi/2)=exp(1)')
f =
2*x + log(t - pi/2 + exp(exp(1) - 2*x))
(3)f=dsolve('Dy-y*tan(x)=sec(x)','y(0)=0')
f =
(exp(t*tan(x)) - 1)/(cos(x)*tan(x))
(4) f=dsolve('Dy+(2-3*x^2)/x^3*y=1','y(1)=0')
f =
-(x^3 - (x^3*exp((t*(3*x^2 - 2))/x^3))/exp((3*x^2 - 2)/x^3))/(3*x^2 - 2)
(5)f=dsolve('D2y=exp(2*y)','y(0)=0+eps,Dy(0)=0+eps')
f =
[ empty sym ]
(6)f=dsolve('D2y-3*Dy-4*y=0','y(0)=2,Dy(0)=0')
f =
8/(5*exp(t)) + (2*exp(4*t))/5
(7)f=dsolve('D2y-10*Dy+9*y=exp(2*x)','y(0)=6/7,Dy(0)=33/7')
f =
exp(2*x)/9 - exp(t)*(exp(2*x)/8 - 3/8) + exp(9*t)*(exp(2*x)/72 + 27/56)