MATLAB作业一1、试求出如下极限。

（1）2325(2)(3)lim(5)x xxxx xx+++→∞+++，（2）23312lim()xyx y xyx y→-→++，（3）2222221cos()lim()x yxyx yx y e+→→-++解：（1）syms x；f=((x+2)^(x+2))*((x+3)^(x+3))/((x+5)^(2*x+5))limit（f，x，inf）=exp（-5）（2）syms x y；f=(x^2*y+x*y^3)/(x+y)^3；limit(limit(f,x,-1),y,2)=-6；（3）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)=02、试求出下面函数的导数。

（1）()y x=, (2)22atan ln()yx yx=+解; （1）syms x;f=sqrt(x*sin(x)*sqrt(1-exp(x)));g= diff(f,x);g== (sin(x)*(1 - exp(x))^(1/2) + x*cos(x)*(1 - exp(x))^(1/2) - (x*exp(x)*sin(x))/(2*(1 - exp(x))^(1/2)))/(2*(x*sin(x)*(1 - exp(x))^(1/2))^(1/2))pretty(g)=(2)syms x y;f=atan(y/x)-log(x^2+y^2)pretty(-simple(diff(f,x)/diff(f,y)))=2 x + y=-------x - 2 y(3)假设1cosu-=，试验证22u ux y y x∂∂=∂∂∂∂。

解：syms x y;u=1/cos(sqrt(x/y));diff(diff(u,x),y)-diff(diff(u,y),x)=0;所以：22u u x y y x ∂∂=∂∂∂∂3、 假设20(,)xyt f x y e dt -=⎰，试求222222x f f f y x x y y ∂∂∂-+∂∂∂∂。

syms s y t;f=exp(-t^2)；g=int(f,t,0,x*y)；k= (x/y)*diff(g,x,2)-2*diff(diff(g,x),y)+diff(g,y,2)；simple （k ）=-2*exp(-x^2*y^2)*(x^3*y - x^2*y^2 + 1)4、 假设已知函数矩阵323(,,)sin y x e z f x y z x y z ⎡⎤+=⎢⎥+⎣⎦，试求出其Jacobi 矩阵。

解：syms x y z;>> f=3*x+exp(y)*z;>> g=x^3+y^2*sin(z)；K=jacobian([f,g],[x,y,z])；K=[ 3, z*exp(y), exp(y) ][ 3*x^2, 2*y*sin(z), y^2*cos(z)]5、 试求解下面的不定积分问题。

（1）()I x =， （2）()cos ax I x xe bxdx =⎰解：（1）syms x;>> f=sqrt(x*(x+1))/(sqrt(x)+sqrt(1+x))f =(x*(x + 1))^(1/2)/((x + 1)^(1/2) + x^(1/2))（2）syms x a b;f=x*exp(a*x)*cos(b*x)；int(f,x)= (exp(a*x)*(b^2*cos(b*x) - a^2*cos(b*x) + a^3*x*cos(b*x) + b^3*x*sin(b*x) - 2*a*b*sin(b*x) + a*b^2*x*cos(b*x) + a^2*b*x*sin(b*x)))/(a^2 + b^2)^27、试求解下面的定积分或无穷积分。

（1）0I ∞=⎰， （2）214011x I dx x +=+⎰解：（1）syms x;f=cos(x)/sqrt(x)；g=int(f,x,0,inf)；g=(2*pi)^(1/2)/2；（2）syms x;>> f=(1+x^2)/(1+x^4);>> g=int(f,x,0,1)；g =(2^(1/2)*(atan(2^(1/2)*(1/2 - i/2)) + atan(2^(1/2)*(1/2 + i/2))))/2；simple （g ）=(pi*2^(1/2))/4；8、假设5()sin(3/3)x f x e x π-=+，试求出积分函数0()()()tR t f x f t x dx =+⎰。

解：syms x t ；f=exp(-5*x)*sin(3*x+pi/3);g=f*subs(f,(x+t))= sin(pi/3 + 3*t + 3*x)*sin(pi/3 + 3*x)*exp(- 5*t - 10*x)；9、试对下面函数进行Fourier 幂级数展开。

（1）()()sin ,;f x x x x πππ=--≤< （2）(),;x f x e x ππ=-≤<解：（1）syms x;>> f=(pi-abs(x))*sin(x);>> [A,B,F]=fseries(f,x,8,-pi,pi)[A,B,F]=fseries(f,x,8,-pi,pi)A =[ 0, 0, 0, 0, 0, 0, 0, 0, 0]B =[ pi/2, 16/(9*pi), 0, 32/(225*pi), 0, 48/(1225*pi), 0, 64/(3969*pi)]F =(16*sin(2*x))/(9*pi) + (32*sin(4*x))/(225*pi) + (48*sin(6*x))/(1225*pi) + (64*sin(8*x))/(3969*pi) + (pi*sin(x))/2（2）syms x;[A,B,F]=fseries(f,x,8,-pi,pi)A =[ (2*exp(pi) - 2)/pi, -(exp(pi) + 1)/pi, ((2*exp(pi))/5 - 2/5)/pi, -(exp(pi)/5 + 1/5)/pi, ((2*exp(pi))/17 - 2/17)/pi, -(exp(pi)/13 + 1/13)/pi, ((2*exp(pi))/37 - 2/37)/pi, -(exp(pi)/25 + 1/25)/pi, ((2*exp(pi))/65 - 2/65)/pi]B =[ 0, 0, 0, 0, 0, 0, 0, 0]F =(2*exp(pi) - 2)/(2*pi) - (cos(3*x)*(exp(pi)/5 + 1/5))/pi + (cos(2*x)*((2*exp(pi))/5 - 2/5))/pi - (cos(5*x)*(exp(pi)/13 + 1/13))/pi + (cos(4*x)*((2*exp(pi))/17 - 2/17))/pi - (cos(7*x)*(exp(pi)/25 + 1/25))/pi + (cos(6*x)*((2*exp(pi))/37 - 2/37))/pi + (cos(8*x)*((2*exp(pi))/65 - 2/65))/pi - (cos(x)*(exp(pi) + 1))/pi>> syms x;10、试求出下面函数的Taylor 幂级数展开。

（1）0sin ,xt dt t⎰ （2）ln(x （3）5sin(3/3)x e x π-+分别关于0x =、x a =的幂级数展开。

（4）对2222221cos()(,)()x y x y f x y x y e +-+=+关于1x =、0y =进行二维Taylor 幂级数展开。

解：（1）syms x t;>> g=sin(t)/tg =sin(t)/tf=int(g,t,0,x)f =sinint(x)taylor(f,x,0,'Order',9)ans =- x^7/35280 + x^5/600 - x^3/18 + x（2）syms x；f=log(x+sqrt(1+x^2));taylor(f,x,0,'Order',9)ans =- (5*x^7)/112 + (3*x^5)/40 - x^3/6 + x>> taylor(f,x,0,'Order',15)ans =(231*x^13)/13312 - (63*x^11)/2816 + (35*x^9)/1152 - (5*x^7)/112 + (3*x^5)/40 - x^3/6 + x(3)>> syms x a;>> f=exp(-5*x)*sin(3*x+pi/3);>> taylor(f,x,0,'Order',9)ans =(46/3 - (31679*3^(1/2))/5040)*x^8 + ((4591*3^(1/2))/252 - 5713/420)*x^7 + (11/4 - (1222*3^(1/2))/45)*x^6 + ((305*3^(1/2))/12 + 239/20)*x^5 + (- (161*3^(1/2))/12 - 20)*x^4 + ((5*3^(1/2))/6 + 33/2)*x^3 + (4*3^(1/2) - 15/2)*x^2 + (3/2 - (5*3^(1/2))/2)*x + 3^(1/2)/2>> taylor(f,x,a,'Order',9)ans =(a - x)^6*((11*cos(pi/3 + 3*a)*exp(-5*a))/2 - (2444*sin(pi/3 + 3*a)*exp(-5*a))/45) - (a - x)^3*(33*cos(pi/3 + 3*a)*exp(-5*a) + (5*sin(pi/3 + 3*a)*exp(-5*a))/3) - (a - x)^4*(40*cos(pi/3 + 3*a)*exp(-5*a) + (161*sin(pi/3 + 3*a)*exp(-5*a))/6) - (a - x)^5*((239*cos(pi/3 + 3*a)*exp(-5*a))/10 + (305*sin(pi/3 + 3*a)*exp(-5*a))/6) - (a - x)^2*(15*cos(pi/3 + 3*a)*exp(-5*a) - 8*sin(pi/3 + 3*a)*exp(-5*a)) + (a - x)^7*((5713*cos(pi/3 + 3*a)*exp(-5*a))/210 - (4591*sin(pi/3 + 3*a)*exp(-5*a))/126) + (a - x)^8*((92*cos(pi/3 + 3*a)*exp(-5*a))/3 - (31679*sin(pi/3 + 3*a)*exp(-5*a))/2520) + sin(pi/3 + 3*a)*exp(-5*a) - (a - x)*(3*cos(pi/3 + 3*a)*exp(-5*a) - 5*sin(pi/3 + 3*a)*exp(-5*a))(4); syms x y;>> f=(1-cos(x^2+y^2))/(x^2+y^2)*exp(x^2+y^2);taylor(f,[x y],[1 0],'Order',9)ans =y^4*((cos(1)*exp(1))/2 - (exp(1)*(cos(1) - 1))/2) - (x - 1)^6*((52*exp(1)*sin(1))/15 - (5*exp(1)*(2*cos(1) + sin(1)))/2 + (26*exp(1)*(cos(1) - 1))/9 + (2*exp(1)*(2*cos(1) - (4*sin(1))/3))/3 + 2*exp(1)*(cos(1)/6 + 2*sin(1)) + exp(1)*((41*cos(1))/45 - sin(1)/2)) + y^6*((exp(1)*sin(1))/3 + (exp(1)*(cos(1) - 1))/3) + (x - 1)^3*(4*exp(1)*sin(1) + (2*exp(1)*(cos(1) - 1))/3 + exp(1)*(2*cos(1) - (4*sin(1))/3)) - y^8*((exp(1)*sin(1))/3 - (5*cos(1)*exp(1))/24 + (3*exp(1)*(cos(1) - 1))/8) + (x - 1)^5*(5*exp(1)*sin(1) - (2*exp(1)*(2*cos(1) + sin(1)))/3 + (26*exp(1)*(cos(1) - 1))/15 + 2*exp(1)*(2*cos(1) - (4*sin(1))/3) - exp(1)*((4*cos(1))/3 + (11*sin(1))/15)) - exp(1)*(cos(1) - 1) - (x - 1)^8*((578*exp(1)*sin(1))/105 - (26*exp(1)*(2*cos(1) + sin(1)))/9 + (1231*exp(1)*(cos(1) - 1))/360 + (26*exp(1)*(2*cos(1) - (4*sin(1))/3))/15 + (5*exp(1)*(cos(1)/6 + 2*sin(1)))/2 - (2*exp(1)*((4*cos(1))/3 + (11*sin(1))/15))/3 + 2*exp(1)*((41*cos(1))/45 - sin(1)/2) - exp(1)*((719*cos(1))/2520 + (11*sin(1))/45)) - (x - 1)^4*((4*exp(1)*sin(1))/3 - 2*exp(1)*(2*cos(1) + sin(1)) + (5*exp(1)*(cos(1) - 1))/2 + exp(1)*(cos(1)/6 + 2*sin(1))) + (exp(1)*(2*cos(1) + sin(1)) - 2*exp(1)*(cos(1) - 1))*(x - 1)^2 + (x - 1)^7*((52*exp(1)*sin(1))/9 - (26*exp(1)*(2*cos(1) + sin(1)))/15 + (289*exp(1)*(cos(1) - 1))/105 + (5*exp(1)*(2*cos(1) - (4*sin(1))/3))/2 + (2*exp(1)*(cos(1)/6 + 2*sin(1)))/3 - 2*exp(1)*((4*cos(1))/3 + (11*sin(1))/15) - exp(1)*(cos(1)/15 - (202*sin(1))/315)) +2*exp(1)*sin(1)*(x - 1) + y^2*exp(1)*sin(1) + y^2*(2*cos(1)*exp(1) - 2*exp(1)*(cos(1) - 1))*(x - 1) - y^6*(x - 1)^2*((11*cos(1)*exp(1))/2 - (35*exp(1)*sin(1))/3 - (exp(1)*(cos(1) - 2*sin(1)))/2 + (exp(1)*(2*cos(1) + sin(1)))/3 - (79*exp(1)*(cos(1) - 1))/6 + exp(1)*(cos(1)/6 - sin(1)/3)) + y^4*(2*exp(1)*sin(1) + 2*exp(1)*(cos(1) - 1))*(x - 1) - y^4*(x - 1)^4*((157*exp(1)*sin(1))/3 - (69*cos(1)*exp(1))/4 + 3*exp(1)*(cos(1) - 2*sin(1)) - 8*exp(1)*(2*cos(1) + sin(1)) + 2*exp(1)*(cos(1) + sin(1)/2) - exp(1)*(cos(1)/12 + sin(1)) + (559*exp(1)*(cos(1) - 1))/12 + 2*exp(1)*(2*cos(1) - (4*sin(1))/3) + (exp(1)*(cos(1)/6 + 2*sin(1)))/2 + 2*exp(1)*((4*cos(1))/3 + 2*sin(1))) - y^2*(x - 1)^5*((132*exp(1)*sin(1))/5 - 5*cos(1)*exp(1) + (2*exp(1)*(cos(1) - 2*sin(1)))/3 - 8*exp(1)*(2*cos(1) + sin(1)) + 21*exp(1)*(cos(1) - 1) + 3*exp(1)*(2*cos(1) - (4*sin(1))/3) + 2*exp(1)*(cos(1)/6 + 2*sin(1)) + 2*exp(1)*((4*cos(1))/3 + 2*sin(1)) + exp(1)*((11*cos(1))/15 - (4*sin(1))/3)) - y^6*(x - 1)*((8*exp(1)*sin(1))/3 - (5*cos(1)*exp(1))/3 + 3*exp(1)*(cos(1) - 1)) - y^4*(x - 1)^2*(7*exp(1)*sin(1) - 5*cos(1)*exp(1) - (exp(1)*(2*cos(1) + sin(1)))/2 + exp(1)*(cos(1) + sin(1)/2) + 8*exp(1)*(cos(1) - 1)) + y^4*(x - 1)^3*(22*exp(1)*sin(1) - (19*cos(1)*exp(1))/3 + 2*exp(1)*(cos(1) - 2*sin(1)) - 2*exp(1)*(2*cos(1) + sin(1)) - exp(1)*(cos(1) - (2*sin(1))/3) + (61*exp(1)*(cos(1) - 1))/3 + (exp(1)*(2*cos(1) - (4*sin(1))/3))/2) + y^2*(x - 1)^6*((404*exp(1)*sin(1))/9 - (52*cos(1)*exp(1))/15 + (5*exp(1)*(cos(1) - 2*sin(1)))/2 - (37*exp(1)*(2*cos(1) + sin(1)))/3 + (919*exp(1)*(cos(1) - 1))/30 + 8*exp(1)*(2*cos(1) - (4*sin(1))/3) - 2*exp(1)*(2*cos(1) - sin(1)/6) + 3*exp(1)*(cos(1)/6 + 2*sin(1)) + (2*exp(1)*((4*cos(1))/3 + 2*sin(1)))/3 - 2*exp(1)*((4*cos(1))/3 + (11*sin(1))/15) + exp(1)*(cos(1)/2 + (41*sin(1))/45)) - y^2*(x - 1)^3*((20*exp(1)*sin(1))/3 - 4*cos(1)*exp(1) - 2*exp(1)*(2*cos(1) + sin(1)) + 8*exp(1)*(cos(1) - 1) + exp(1)*((4*cos(1))/3 + 2*sin(1))) + y^2*(x - 1)^4*((37*exp(1)*sin(1))/2 - (4*cos(1)*exp(1))/3 + 2*exp(1)*(cos(1) - 2*sin(1)) - 3*exp(1)*(2*cos(1) + sin(1)) + (37*exp(1)*(cos(1) - 1))/3 + 2*exp(1)*(2*cos(1) - (4*sin(1))/3) - exp(1)*(2*cos(1) - sin(1)/6)) + y^2*(x - 1)^2*(6*exp(1)*sin(1) + exp(1)*(cos(1) - 2*sin(1)) + 3*exp(1)*(cos(1) - 1))解：x=[0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2];>> y=[0 2.208 3.206 3.444 3.241 2.816 2.311 1.81 1.36 0.982 0.679 0.447 0.227];>> S1=trapz(x,y)S1 =2.2618；。