matlab第一次作业Word下载.docx
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matlab第一次作业Word下载.docx
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sin((pi*(n+1))/n)
i=n*3^n/2^n
1/2^n*3^n*n
f1=symsum(f,n,1,inf)
1
g1=symsum(g,n,1,inf)
g1=
sum((n/(n+1))^(1/2),n==1..Inf)
h1=symsum(h,n,1,inf)
h1=
(sum(exp(-pi*i)*exp(-(pi*i)/n),n==1..Inf)*i)/2-(sum(exp(pi*i)*exp((pi*i)/n),n==1..Inf)*i)/2>
i1=symsum(i,n,1,inf)
i1=
Inf
因而,第
(1)小题收敛,
(2)(3)(4)小题均发散.
三、
symsn
s=(-1)^n*(n+1)/3^n
s=
(-1)^n/3^n*(n+1)
f=(-1)^n/log(n)
(-1)^n/log(n)
h=n/2^n*cos(2*n*pi/3)
1/2^n*n*cos((2*pi*n)/3)
s1=symsum(s,n,1,inf)
s1=
-7/16
f1=symsum(f,n,2,inf)
sum((-1)^n/log(n),n==2..Inf)
((3^(1/2)*i)/2-1/2)/(4*(3^(1/2)*(i/4)-5/4)^2)-((3^(1/2)*i)/2+1/2)/(4*(3^(1/2)*(i/4)+5/4)^2)
因而,第
(1)、(3)小题收敛,第
(2)小题不收敛.
s2=symsum(abs(s),n,1,inf)
s2=
5/4
f2=symsum(abs(f),n,2,inf)
f2=
sum(abs((-1)^n)/log(n),n==2..Inf)
h2=symsum(abs(h),n,1,inf)
h2=
sum(1/2^n*n*abs(cos((2*pi*n)/3)),n==1..Inf)
因而,第
(1)小题绝对收敛,第
(2)小题不收敛,第(3)小题条件收敛.
四、
symsnx
s=((2*n-1)*(x^(2*n-2)))/(2^n)
s=
1/2^n*x^(2*n-2)*(2*n-1)
f=(x^n)/(n*2^n)
f=
(1/2^n*x^n)/n
g=n*(n+1)*x^(n-1)/2
g=
(n*x^(n-1)*(n+1))/2
h=x^n/(n*(n+1))
h=
x^n/(n*(n+1))
s1=
1/4*pi^(1/2)*2^(3/4)*(x^2)^(1/4)/(1-1/2*x^2)^(3/2)*LegendreP(1/2,1/2,(1/2*x^2+1)/(1-1/2*x^2))
f1=
-log(1-1/2*x)
g1=
-1/(x-1)^3
h1=
1-(x-1)/x*log(1-x)
五、
1、>
f=((3^n+5^n)/n)^(1/n)
((3^n+5^n)/n)^(1/n)
s=limit(f,n,inf)
5
R=1/5
R=
0.2000
2、>
f=(n/(2^n))^(1/(2*n))
(1/2^n*n)^(1/(2*n))
f1=limit(f,n,inf)
f1=
2^(1/2)/2
R=1/f1
2^(1/2)
六、
symsx
r=taylor(log(x+1),x,0,'
order'
9)
r=
-x^8/8+x^7/7-x^6/6+x^5/5-x^4/4+x^3/3-x^2/2+x
f=taylor(x^4-5*x^3+x^2-3*x+4,x,4,'
5)
21*x+37*(x-4)^2+11*(x-4)^3+(x-4)^4-140
x=-0.9:
1/50:
1.2
y1=log(x+1)
y2=-x.^8./8+x.^7./7-x.^6./6+x.^5./5-x.^4./4+x.^3./3-x.^2./2+x
plot(x,y1)
holdon
plot(x,y2,'
r'
)
x=0:
1/100:
8
s1=x.^4-5.*x.^3+x.^2-3.*x+4;
plot(x,s1)
s2=21.*x+37.*(x-4).^2+11.*(x-4).^3+(x-4).^4-140;
plot(x,s2,'
微分练习题
f=(cos(cos(2*x)))^2
cos(cos(2*x))^2
dfdx=diff(f,x,2)
dfdx=
8*sin(2*x)^2*sin(cos(2*x))^2-8*sin(2*x)^2*cos(cos(2*x))^2+8*cos(2*x)*cos(cos(2*x))*sin(cos(2*x))
g=sym('
atan(y/x)=log(sqrt(x^2+y^2))'
atan(y/x)==log((x^2+y^2)^(1/2))
atan(y(x)/x)=log(sqrt(x^2+y(x)^2))'
atan(y(x)/x)==log((x^2+y(x)^2)^(1/2))
dgdx=diff(g,x)
dgdx=
(diff(y(x),x)/x-y(x)/x^2)/(y(x)^2/x^2+1)==(2*x+2*y(x)*diff(y(x),x))/(2*(x^2+y(x)^2))
dgdx1=subs(dgdx,'
diff(y(x),x)'
'
dydx'
dgdx1=
-(y(x)/x^2-dydx/x)/(y(x)^2/x^2+1)==(2*x+2*dydx*y(x))/(2*(x^2+y(x)^2))
dydx=solve(dgdx1,'
dydx=
(x+y(x))/(x-y(x))
(exp(t)*cos(t)-exp(t)*sin(t))/(exp(t)*cos(t)+exp(t)*sin(t))
dydxdt=diff(dydx,t)
dydxdt=
-(2*exp(t)*sin(t))/(exp(t)*cos(t)+exp(t)*sin(t))-(2*exp(t)*cos(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))^2
dydx2=dydxdt/dxdt
dydx2=
-((2*exp(t)*sin(t))/(exp(t)*cos(t)+exp(t)*sin(t))+(2*exp(t)*cos(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))^2)/(exp(t)*cos(t)+exp(t)*sin(t))
L=(x+y)^2*dydx2
L=
-(exp(t)*cos(t)+exp(t)*sin(t))*((2*exp(t)*sin(t))/(exp(t)*cos(t)+exp(t)*sin(t))+(2*exp(t)*cos(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))^2)
R=2*(x*dydx-y)
(2*exp(t)*sin(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))-2*exp(t)*cos(t)
L-R
ans=
2*exp(t)*cos(t)-(exp(t)*cos(t)+exp(t)*sin(t))*((2*exp(t)*sin(t))/(exp(t)*cos(t)+exp(t)*sin(t))+(2*exp(t)*cos(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))^2)-(2*exp(t)*sin(t)*(exp(t)*cos(t)-exp(t)*sin(t)))/(exp(t)*cos(t)+exp(t)*sin(t))
H=simple(L-R)
H=
symsxy
z=cos(sqrt(x^2+y^2))
z=
cos((x^2+y^2)^(1/2))
dzdx=diff(z,x,1)
dzdx=
-(x*sin((x^2+y^2)^(1/2)))/(x^2+y^2)^(1/2)
dzdy=diff(z,y,1)
dzdy=
-(y*sin((x^2+y^2)^(1/2)))/(x^2+y^2)^(1/2)
dzdxdy=diff(diff(z,x),y)
dzdxdy=
(x*y*sin((x^2+y^2)^(1/2)))/(x^2+y^2)^(3/2)-(x*y*cos((x^2+y^2)^(1/2)))/(x^2+y^2)
x+2*y+z(x,y)-2*sqrt(x*y*z(x,y))=0'
x+2*y-2*(x*y*z(x,y))^(1/2)+z(x,y)==0
diff(z(x,y),x)-(x*y*diff(z(x,y),x)+y*z(x,y))/(x*y*z(x,y))^(1/2)+1==0
diff(z(x,y),x)'
dzdx'
dzdx-(y*z(x,y)+dzdx*x*y)/(x*y*z(x,y))^(1/2)+1==0
dzdx=solve(dgdx1,'
dzdx=
-((x*y*z(x,y))^(1/2)-y*z(x,y))/((x*y*z(x,y))^(1/2)-x*y)
dgdy=diff(g,y)
dgdy=
diff(z(x,y),y)-(x*y*diff(z(x,y),y)+x*z(x,y))/(x*y*z(x,y))^(1/2)+2==0
dgdy1=subs(dgdy,'
diff(z(x,y),y)'
dzdy'
dgdy1=
dzdy-(x*z(x,y)+dzdy*x*y)/(x*y*z(x,y))^(1/2)+2==0
dzdy=solve(dgdy1,'
-((x*z(x,y))/(x*y*z(x,y))^(1/2)-2)/((x*y)/(x*y*z(x,y))^(1/2)-1)
选址问题
y=20*sqrt(x^2+9)+15*(5-x)
y=
20*(x^2+9)^(1/2)-15*x+75
dydx=diff(y,x)
(20*x)/(x^2+9)^(1/2)-15
z=solve('
(20*x)/(x^2+9)^(1/2)-15'
x'
(9*7^(1/2))/7
因此,M应建在距离B点(9*7^(1/2))/7米处.
积分练习题
s=1/((asin(x))^2*sqrt(1-x^2))
1/(asin(x)^2*(1-x^2)^(1/2))
f=cos(x)*sin(x)/(1+cos(x))^2
(cos(x)*sin(x))/(cos(x)+1)^2
g=log(x+1)/sqrt(x+1)
log(x+1)/(x+1)^(1/2)
h=cos(log(x))
cos(log(x))
ints=int(s,x)
ints=
-1/asin(x)
intf=int(f,x)
intf=
-log(cos(x)+1)-1/(cos(x)+1)
intg=int(g,x)
intg=
2*(log(x+1)-2)*(x+1)^(1/2)
inth=int(h,x)
inth=
(x*(cos(log(x))+sin(log(x))))/2
s=(1+log(x))/x
(log(x)+1)/x
f=x*atan(x)
x*atan(x)
g=abs(x-1)
abs(x-1)
h=(x*sin(x))^2
x^2*sin(x)^2
intf=int(f,x,0,1)
pi/4-1/2
ints=int(s,x,1,exp
(1))
(log(3060513257434037/1125899906842624)*(log(3060513257434037/1125899906842624)+2))/2
intg=int(g,x,0,2)
1
inth=int(h,x,0,x)
sin(2*x)/8-(x*cos(2*x))/4-(x^2*sin(2*x))/4+x^3/6
symsxt
f=cos(t^2)
cos(t^2)
g=1/(1+t^2)
1/(t^2+1)
f1=int(f,t,sin(x),0)
-(2^(1/2)*pi^(1/2)*fresnelC((2^(1/2)*sin(x))/pi^(1/2)))/2
g1=int(g,t,x^2,x^3)
atan(x^3)-atan(x^2)
df1dx=diff(f1,x)
df1dx=
-cos(sin(x)^2)*cos(x)
dg1dx=diff(g1,x)
dg1dx=
(3*x^2)/(x^6+1)-(2*x)/(x^4+1)
symstx
f=sin(t^2)
sin(t^2)
g=x^3+x^4
x^4+x^3
symsa
h=t/sqrt(a+t)
t/(a+t)^(1/2)
s=x^4
x^4
f1=int(f,t,0,sin(x))
(2^(1/2)*pi^(1/2)*fresnelS((2^(1/2)*sin(x))/pi^(1/2)))/2
h1=int(h,t,0,x^2)
Warning:
Explicitintegralcouldnotbefound.
piecewise([ainDom:
:
ImageSet(y*i,y,R_)orainR_,(4*a^(3/2))/3-(2*(-x^2+2*a)*(x^2+a)^(1/2))/3],[notainDom:
ImageSet(y*i,y,R_)andnotainR_,int(t/(a+t)^(1/2),t==0..x^2)])
f2=f1/g
f2=
(2^(1/2)*pi^(1/2)*fresnelS((2^(1/2)*sin(x))/pi^(1/2)))/(2*(x^4+x^3))
h2=h1/s
ImageSet(y*i,y,R_)orainR_,-((2*(-x^2+2*a)*(x^2+a)^(1/2))/3-(4*a^(3/2))/3)/x^4],[notainDom:
ImageSet(y*i,y,R_)andnotainR_,int(t/(a+t)^(1/2),t==0..x^2)/x^4])
f3=limit(f2,x,0)
f3=
1/3
h3=limit(h2,x,0)
h3=
1/(2*a^(1/2))
symsxyt
s1=int(exp(t),t,0,y)
s1=
exp(y)-1
s2=int(cos(t),t,0,x)
s2=
sin(x)
exp(y(x))-1+sin(x)=0'
exp(y(x))+sin(x)-1==0
exp(y(x))*diff(y(x),x)+cos(x)==0
cos(x)+dydx*exp(y(x))==0
-exp(-y(x))*cos(x)
七、
f=sin(sin(x))/x
sin(sin(x))/x
g=exp(-x)*x^5.1
x^(51/10)*exp(-x)
f1=int(f,x,0,x/2)
int(sin(sin(x))/x,x==0..x/2)
g1=int(g,x,0,inf))
g1=int(g,x,0,inf)
(14973651*gamma(1/10))/1000000
八、
symsxp
f=1/x^p
1/x^p
s=1/x^2/(1+x)
1/(x^2*(x+1))
g=1/(x*sqrt(1-(log(x))^2))
1/(x*(1-log(x)^2)^(1/2))
h=exp(-2^x)/(2-x)^2
exp(-2^x)/(x-2)^2
f1=int(f,x,1,inf)
piecewise([p==1,Inf],[p<
1,Inf+1/(p-1)],[1<
real(p),1/(p-1)],[real(p)==1andnot1<
=p,int(1/x^p,x==1..Inf)],[real(p)<
1andnotp<
1,limit(-x^(1-p)/(p-1),x==Inf)+1/(p-1)])
g1=int(g,x,1,2)
asin(log
(2))
s1=int(s,x,1,inf)
1-log
(2)
h1=int(h,x,0,2)
int(exp(-2^x)/(x-2)^2,x==0..2)
九、
symsxy
F=int(int(x*y,x,y^2,y+2),y,-1,2)
F=
45/8
十、
F=int(int(exp(-x^2-y^2),y,-sqrt(1-x^2),sqrt(1-x^2)),x,-1,1)
int(pi^(1/2)*erf((1-x^2)^(1/2))*exp(-x^2),x==-1..1)
十一、
symsxyz
int(int(int(1,z,x^2+2*y^2,6-2*x^2-y^2),y,-sqrt(2-x^2),sqrt(2-x^2)),x,-2^(1/2),2^(1/2))
6*pi
十二、
symsRAB
x=R*cos(A)*cos(B)
x=
R*cos(A)*co
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