3=--'

Zadanie 1

D[x Log[x],{x,2}]

Plot[p^p(1-p)^(1-p),{p,0,1}]

Maximize[{p^p(1-p)^(1-p),0≤p≤1},p]

Minimize[{p^p(1-p)^(1-p),0≤p≤1},p]

D[p^p(1-p)^(1-p),p]//FullSimplify

D[p Log[p]+(1-p) Log[1-p],p]

Log[(p/(1-p))]

2.

f[x_]:=(((Pi^2-x^2)Sin[x])/(1+Cos[x]))

Plot[f[x],{x,-Pi,Pi}]

f'[x]//FullSimplify

Plot[(π^2)-(x^2)-2 x Sin[x],{x,-Pi,Pi}]

D[(π^2)-(x^2)-2 x Sin[x],x]//Simplify

Plot[-2 x-2 x Cos[x]-2 Sin[x],{x,-Pi,Pi}]

(π^2)-(x^2)-2 x Sin[x]== (π^2)+Sin[x]^2 -(x+Sin[x])^2≥0

π≥x+Sin[x]

Limit[f[x],x->Pi]

Limit[f[x],x->-Pi]

(((Pi^2-x^2)Sin[x])/(1+Cos[x]))/.x->y+Pi//Expand//Together

((-2 π y Sin[y]-(y^2) Sin[y])/(-1+Cos[y]))+O[y]

-2 π y Sin[y]-(y^2) Sin[y]+O[y]^3

-1+Cos[y]+O[y]^3

((-2 π (y^2)+((O[y])^3))/(-((y^2)/2)+((O[y])^3)))

3.

Reduce[Sqrt[x^3/(x+3)]≥0,x]

f[x_]:=1-x+Sqrt[x^3/(x+3)]

Plot[f[x],{x,-5,5}]

Reduce[D[f[x],x]≥0,x]

Reduce[D[f[x],x]<=0,x]

4.

f[x_]:=(1/(x-a))+(1/(x-b))+(1/(x-c))

Simplify[f'[x]]

f[x]==0

ff[x_]:=(1/(x-2))+(1/(x-3))+(1/(x-4))

Plot[ff[x],{x,0,5}]

n - 2

FullSimplify[D[ArcSin[x]-ArcTan[(x/(Sqrt[1-x^2]))],x]]

5.

f[x_]:=((2 ArcSin[1-Cos[x]]-Tan[x]^2)/(Log[1+x^2]-Sin[x]^2))

((2 ArcSin[1-Cos[x]]-Tan[x]^2)/(Log[1+x^2]-Sin[x]^2))+O[x]^2

Log[1+x^2]-Sin[x]^2 +O[x]^6

2 ArcSin[1-Cos[x]]+O[x]^6

2 ArcSin[1-Cos[x]]-Tan[x]^2+O[x]^6

1-Cos[x]+O[x]^6

2 ArcSin[1-Cos[x]]-Tan[x]^2

2ArcSin[((x^2)/2)+((O[x])^4)]

2ArcSin[x]+O[x]^4

6.

f[x_]:=Piecewise[{{(((ArcTan[x])^2)/(Log[1+x^2])),x≠0},{a,x==0}}]

Limit[(((ArcTan[x])^2)/(Log[1+x^2])),x->0]

SeriesCoefficient[ArcTan[x],{x,0,n}]

Table[((I^(n-1))/n),{n,1,20,2}]

ArcTan[x]+O[x]^30

SeriesCoefficient[ArcSin[x],{x,0,n}]

f[x_]:=Piecewise[{{(((ArcTan[x])^2)/(Log[1+x^2])),x≠0},{1,x==0}}]

Limit[(((((ArcTan[x])^2)/(Log[1+x^2]))-1)/x),x->0]

Limit[D[(((ArcTan[x])^2)/(Log[1+x^2])),x],x->0]

7.

Tan[Sin[(1/(Sqrt[n]))]]-(1/(Sqrt[n]))

Tan[y]+O[y]^5

Sin[x]+O[x]^5

y+((y^3)/3)+((O[y])^5)/.y->x-((x^3)/6)+((O[x])^5)

((x^3)/6)/.x->(1/(Sqrt[n]))

8.

9.

f[x_]:=Piecewise[{{((1-Cos[x])/(x^2)),x≠0}},1/2]

Plot[f[x],{x,-2,2}]

((1-Cos[x])/(x^2))+O[x]^10

FullSimplify[n!SeriesCoefficient[((1-Cos[x])/(x^2)),{x,0,n}]==((Cos[((n π)/2)] )/((n+1)(n+2))),Assumptions->Element[n,Integers]&&n>1]

i

FullSimplify[Gamma[3+n],Assumptions->Element[n,Integers]&&n>1]

?Gamma

((Cos[((n π)/2)] )/((n+1)(n+2)))

Table[Cos[Pi n/2],{n,1,20}]

Table[ \,{n,1,20}]

10.

(∑_(n=1)^∞)(3^((((-1))^n)-n)) (x^n)

Table[(3^((((-1))^n)-n)),{n,1,10}]

{(1/(3^(2k-1))),(1/(3^(2k+2)))

Table[(1/(3^(2k)))