第三章 微积分练习

第三章微积分练习

Limit[Sin[x]/x,x→0]

1

Limit[(x^2-1)/(4x^2-7x+1),x→Infinity]

1/4

Limit[Sin[1/x],x→0]

Interval[{-1,1}]

Limit[1/x,x→0,Direction→1]

-∞

Limit[1/x,x→0,Direction→-1]

∞

D[x^x,x]

x x (1+Log[x])

D[z*Sin[x^2*y^2],x,y]

4 x y z Cos[x2 y2]-4 x3 y3 z Sin[x2 y2]

D[c*x^n,{x,2}]

c (-1+n) n x-2+n

D[z[1]^2+Sin[z[1]+z[2]],z[1]]

Cos[z[1]+z[2]]+2 z[1]

D[x^2+y^2,x,NonConstants→{y}]

2 x+2 y D[y,x,NonConstants→{y}]

D[x^2+y^2,x]

2 x

Dt[x^2+y^2]

2 x Dt[x]+2 y Dt[y]

Dt[x^2+y^2,x]

2 x+2 y Dt[y,x]

Dt[x^2+y^2+z^2,x]

2 x+2 y Dt[y,x]+2 z Dt[z,x]

Dt[x^2+y^2+z^2,x,Constants→{z}]

2 x+2 y Dt[y,x,Constants→{z}]

u=Exp[a*θ]Cos[a*Log[r]];D[u,{r,2}]+1/r^2D[u,{θ,2}]+1/r*D[u,r]

(a2©a θCos[a Log[r]])/r2-(a ©a θSin[a Log[r]])/r2+©a θ(-((a2Cos[a Log[r]])/r2)+(a Sin[a Log[r]])/r2)

Simplify[%]

f[x_]:=Sin[x^2];f[x]+f'[x]+f''[x]

2 Cos[x2]+2 x Cos[x2]+Sin[x2]-4 x2 Sin[x2]

D[x*g[2x],x]

g[2 x]+2 x g'[2 x]

D[h[x,y],x,x,y]

h(2,1)[x,y]

f'[x_]:=hf[x]+gf[x];D[f[x^2],x]

2 x (gf[x2]+hf[x2])

Derivative[0,2][g][x_,y_]:=gy2[x,y];D[g[x,y],{y,3}] gy2(0,1)[x,y]

Integrate[Cos[x]^2+Sin[x]^3,{x,0,1}]

1/12 (14-9 Cos[1]+Cos[3]+3 Sin[2])

Expand[%]

7/6-(3 Cos[1])/4+Cos[3]/12+Sin[2]/4

N[%]

0.906265

NIntegrate[Cos[x]^2+Sin[x]^3,{x,0,1}]

0.906265

Integrate[2x+2y,{x,0,a},{y,0,b}]

a b (a+b)

Expand[%]

a2 b+a b2

Integrate[Sqrt[x],{x,3,9}]

-2 (-9+3)

NIntegrate[Sqrt[x],{x,3,9}]

14.5359

Integrate[Sin[Sin[x]],x]

\[Integral]Sin[Sin[x]]®x

Unprotect[Integrate]

{Integrate}

Integrate[Sin[Sin[a_.+b_.]],x]:=F[a,b]

Protect[Integrate]

{Integrate}

Integrate[Sin[Sin[7x]],x]

F[7 x,0]

Series[Sin[2x],{x,0,6}]

2 x-(4 x3)/3+(4 x5)/15+O[x]7

Series[f[x],{x,0,3}]

f[0]+f'[0] x+1/2 f''[0] x2+1/6 f(3)[0] x3+O[x]4

Series[Exp[1/x],{x,Infinity,3}]

1+1/x+1/(2 x2)+1/(6 x3)+O[1/x]4

Series[Exp[Sqrt[x]],{x,0,3}]

1+x+x/2+x3/2/6+x2/24+x5/2/120+x3/720+O[x]7/2 Series[Cos[x]Cos[y],{x,0,3},{y,0,3}]

(1-y2/2+O[y]4)+(-(1/2)+y2/4+O[y]4) x2+O[x]4

Expand[%]

(1-y2/2+O[y]4)+(-(1/2)+y2/4+O[y]4) x2+O[x]4

Sum[x^n/n!,{n,0,Infinity}]

©x

Sum[x^n/(n!^2),{n,0,Infinity}]

BesselI[0,2 x]

Sum[x^k/k!,{k,0,n}]

(©x Gamma[1+n,x])/n!

t=Series[Log[x+1],{x,0,4}]

x-x2/2+x3/3-x4/4+O[x]5

%^2

x2-x3+(11 x4)/12-(5 x5)/6+O[x]6

D[%,x]

2 x-

3 x2+(11 x3)/3-(25 x4)/6+O[x]5

Normal[%]

2 x-

3 x2+(11 x3)/3-(25 x4)/6

SeriesCoefficient[%,2]

SeriesCoefficient[2 x-3 x2+(11 x3)/3-(25 x4)/6,2]

t=Series[Sin[x],{x,0,5}]

x-x3/6+x5/120+O[x]6

%/.x→Series[Sin[x],{x,0,5}]

x-x3/3+x5/10+O[x]6

Series[Sin[Sin[x]],{x,0,5}]

x-x3/3+x5/10+O[x]6

Series[x^5,{x+1,0,6}]

General::ivar: 1+x is not a valid variable. General::ivar: 1+x is not a valid variable. Series[x5,{1+x,0,6}]

InverseSeries[t]

x+x3/6+(3 x5)/40+O[x]6

%/.x→t

x+O[x]6

InverseSeries[t,y]

y+y3/6+(3 y5)/40+O[y]6

{Residue[Cos[x]/x^3,{x,0}],Limit[Cos[x]/x^3,x→0]} {-(1/2),∞}

Series[Cos[x]/x^3,{x,0,3}]

1/x3-1/(2 x)+x/24-x3/720+O[x]4

{Residue[Sin[x]/x,{x,0}],Limit[Sin[x]/x,x→0]} {0,1}

DSolve[y'[x] a*y[x],y[x],x]

{{y[x]→©a x C[1]}}

DSolve[{y'[x] a*y[x],y[0] 1},y[x],x]

{{y[x]→©a x}}

DSolve[{x[t] -y'[t],y[t] -x'[t]},{x[t],y[t]},t]