《微分积分》word版.docx

I. 一重积分A. 用int积分int(sin(x),x,0,pi) ans =2 B. trapz 计算X = 0:pi/100:pi;Y = sin(X); Z = trapz(X,Y)Z = 1.999835503887444C. 符号分析syms x;IS=int(sym(exp(-x*x),x,0,1)vpa(IS) IS =(pi(1/2)*erf(1)/2ans =0.74682413281242702539946743613185 D. 用quad或quadlfun=inline(exp(-x.*x),x)a=quad(fun,0,1)b=quadl(fun,0,1) fun = Inline function: fun(x) = exp(-x.*x)a = 0.7468b = 0.7468 F = (x) exp(-x.*x);Q = quad(F,0,1)Ql = quadl(F,0,1) Q = 0.7468Ql = 0.7468 II. 多重积分A. 用MUPAD工具箱计算F2=int(int(int(x2+y2+z2,z=sqrt(x*y).x2*y),y=sqrt(x).x2),x=1.2)B. 符号法syms x y;s=vpa(int(int(xy,x,0,1),y,1,2) Warning: Explicit integral could not be found.s =0.69314718055994530941723212145818 C. dblquad 数值法F = (x,y) y*sin(x)+x*cos(y);Q = dblquad(F,pi,2*pi,0,pi) Q = -9.8696 D. 其他不常用III. 极限与导数A. 极限syms x a b;f=x*(1+a/x)x*sin(b/x);L=limit(f,x,inf) L =b*exp(a) inf:为无穷B. 单边极限sym x;limit(exp(x3)-1)/(1-cos(sqrt(x-sin(x),x,0,right) ans =12 C. 多变量函数极限syms x y a;f=exp(-1/(y2+x2)*sin(x)2./x2*(1+1/y2)(x+a2*y2);L=limit(limit(f,x,1/sqrt(y),y,inf) L =exp(a2) 三个或三个以上的黑点表示“续行”IV. 微分A. 函数的导数解syms x;f=sin(x)/(x2+4*x+3);df1=diff(f)df4=diff (f,4) text(0.1,0.5, $ latex(df4) $,interpreter,latex,FontSize,10)df1 =cos(x)/(x2 + 4*x + 3) - (sin(x)*(2*x + 4)/(x2 + 4*x + 3)2df4 =sin(x)/(x2 + 4*x + 3) + (12*sin(x)/(x2 + 4*x + 3)2 + (24*sin(x)/(x2 + 4*x + 3)3 - (24*cos(x)*(2*x + 4)3)/(x2 + 4*x + 3)4 - (12*sin(x)*(2*x + 4)2)/(x2 + 4*x + 3)3 - (48*sin(x)*(2*x + 4)2)/(x2 + 4*x + 3)4 + (24*sin(x)*(2*x + 4)4)/(x2 + 4*x + 3)5 + (4*cos(x)*(2*x + 4)/(x2 + 4*x + 3)2 + (16*cos(x)*(2*x + 4)/(x2 + 4*x + 3)3 + (8*cos(x)*(8*x + 16)/(x2 + 4*x + 3)3 - (6*sin(x)*(2*x + 4)*(8*x + 16)/(x2 + 4*x + 3)4 syms x y;z=(x2-2*x)*exp(-x2-y2-x*y);zx=simple(diff(z,x)zy=diff(z,y) zx =(2*x + 2*x*y - x2*y + 4*x2 - 2*x3 - 2)/exp(x2 + x*y + y2)zy =(2*x - x2)*(x + 2*y)/exp(x2 + x*y + y2) x,y=meshgrid(-3:.2:3,-2:.2:2);z=(x.2-2*x).*exp(-x.2-y.2-x.*y);surf(x,y,z), axis(-3 3 -2 2 -0.7 1.5) contour(x,y,z,30), hold onzx=-exp(-x.2-y.2-x.*y).*(-2*x+2+.2*x.3+x.2.*y-4*x.2-2*x.*y);zy=-x.*(x-2).*(2*y+x).*exp(-x.2-y.2-x.*y);quiver(x,y,zx,zy) B. 隐函数的偏导数syms x y;f=(x2-2*x)*exp(-x2-y2-x*y);pretty(-simple(diff(f, x)/d