Question

微积分应用问题

Answer 1

如图

Answer 2

• a(n)=n/2^n    a(n+1)/a(n)=(n+1)2^n/[n2^(n+1)]--->1/2  ∴R=2

• z=x^3y+xy^3   偏z/偏x=3x^2y+y^3   偏^2z/偏x^2=6xy∫

• y'+p(x)y=0

• dy/dx=x/y  ydy=xdx   y^2/2=x^2/2+C

• ∫[0,1]x(1-x)dx=∫[0,1](x-x^2)dx=[x^2/2-x^3/3]|[0,1]=1/6

• ∫[0,π/4]cos^2(x)dx=1/2∫[0,π/4][1-cos(2x)]dx=x/2|[0,π/4]-1/4sin(2x)|[0,π/4]=π/8-1/4

• ∫[0,π/4]xcosxdx=xsinx|[0,π/4]-∫[0,π/4]sinxdx=√2π/8+cosx|[0,π/4]=√2π/8+√2/2-1

• ∫[0,1]xe^xdx=xe^x|[0,1]-∫[0,1]e^xdx=e-e+1=1

• ∫[0,+∞]2/(1+x^2)dx=2arctanx|[0,+∞]=π

• ∫[0,1]dx/(1+e^x)=∫[0,1]e^(-x)dx/[1+e^(-x)]=-ln[1+e^(-x)]|[0,1]=ln2-ln(1+e)=ln[2/(1+e)]

• z=x^y+y^2  dz=y*x^(y-1)dx+(x^ylnx+2y)dy

• 递减，一般项趋近于0，交错级数收敛

• 绝对收敛

• sinx=x-x^3/3!+x^5/5!-x^7/7!+……   逐项微分： cosx=1-x^2/2!+x^4/4!-x^6/6!+……

• z=uv+50t    u=e^t  v=t^2   z'(t)=t^2*e^t+2t*e^t+50

• yy''+2y'^2=0  设dy/dx=p(x)  则 y''=dp/dx=dp/dy*dy/dx=pdp/dy

  ypdp/dy+2p^2=0     ypdp/dy=-2p^2    ydp/dy=-2p    dp/p=-2dy/y

  lnp=-2lny+lnC1    p=C1y^(-2)

  dy/dx=C1y^(-2)

  y^2dy=c1dx    y^3/3=C1x+C2   通解：y^3=3C1x+3C2