~v200 200 ~w189 28 442 736 160 460 0 0 ~f? 12 11 9 ? 3 1 1 0 ? ? ? "Georgia" ? ? ? 1 ? 0 1 "Times" 12 ? ? 8 0 c n 106 1 0 0 k 468 i"" -2 1 26177 26178 26115 26178 1 1 1 1 0 0 8405120 0 -1 0 0 -1 -1 -1 -1 -1 1 1 ? ? ~Q ]|Expr|[#b @`bb#_b#_b#_})b C# b'4" *|: ;bP8&c0!* `fb#@}&c0!) | |La funzione esponenziale naturale ,Hovvero,Z in base e ,] numero| | di Nepero,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 0 ~V?v0 (p)~p0 1 ~V?v0 (t)~p0 1 ~V?v0 ('a)~p0 1 ~V?f0 (C)~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?c1 ('p)~p0 2 ~V?f256 (sin)~p0 2 ~V?f257 (cos)~p0 2 ~V?f258 (tan)~p0 2 ~V?f261 (sec)~p0 2 ~V?f260 (csc)~p0 2 ~V?f262 (cot)~p0 2 ~V?f272 (arcsin)~p0 2 ~V?f273 (arccos)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f293 (sech)~p0 2 ~V?f288 (sinh)~p0 2 ~V?f289 (cosh)~p0 2 ~V?f290 (tanh)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f32 (log)~p0 2 ~V?f307 (ln)~p0 2 ~V?f291 (exp)~p0 2 ~V?c2 (e)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Standard Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Ht(?x^(-?y)):(1/?x^?y)~p0 3 ~Hs(exp(?z)):(e^?z)~p0 3 ~Hs(e^(ln(?x))):(?x)~p0 3 ~Hs(10^(log(?x))):(?x)~p0 3 ~Hs(?y^(log_?y(?x))):(?x)~p0 3 ~Hs(ln(e^?x)):(?x)~p0 3 ~Hs(log(10^?x)):(?x)~p0 3 ~Hs(log_?y(?y^?x)):(?x)~p0 3 ~He(ln(?u*?v)):(ln(?u)+ln(?v))~p0 3 ~He(log(?u*?v)):(log(?u)+log(?v))~p0 3 ~He(log_?y(?u*?v)):(log_?y(?u)+log_?y(?v))~p0 3 ~He(ln(?u/?v)):(ln(?u)-ln(?v))~p0 3 ~He(log(?u/?v)):(log(?u)-log(?v))~p0 3 ~He(log_?y(?u/?v)):(log_?y(?u)-log_?y(?v))~p0 3 ~He(ln(?u^?v)):(?v*ln(?u))~p0 3 ~He(log(?u^?v)):(?v*log(?u))~p0 3 ~He(log_?y(?u^?v)):(?v*log_?y(?u))~p0 3 ~He(ln(sqrt(?u))):(1/2*ln(?u))~p0 3 ~He(log(sqrt(?u))):(1/2*log(?u))~p0 3 ~He(log_?y(sqrt(?u))):(1/2*log_?y(?u))~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *|: ;bP8&c0!*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sin(-?x)):(-sin(?x))~p0 4 ~Hs(cos(-?x)):(cos(?x))~p0 4 ~Hs(tan(-?x)):(-tan(?x))~p0 4 ~Hs(sin('p)):(0)~p0 4 ~Hs(sin(?n*'p)):(0)~p0 4 ~Hs(cos(1/2*'p)):(0)~p0 4 ~Hs(cos(?n/2*'p)):(0)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *|: ;bP8&c0!*Transform to basic| | types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0 4 ~Ht(csc(?x)):(1/(sin(?x)))~p0 4 ~Ht(sin(?x)):(1/(csc(?x)))~p0 4 ~Ht(sec(?x)):(1/(cos(?x)))~p0 4 ~Ht(cos(?x)):(1/(sec(?x)))~p0 4 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trig Addition| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0 4 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0 4 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0 4 ~Ht(sin(2*?x)):(2*cos(?x)*sin(?x))~p0 4 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*sin((?n-1)*?x))~p0 4 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*sin((?n-1)*?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trig Subtraction| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?x-?y)):(cos(?x)*cos(?y)+sin(?x)*sin(?y))~p0 4 ~Ht(sin(?x-?y)):(sin(?x)*cos(?y)-cos(?x)*sin(?y))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})/# b$L" *|: ;bP8&c0!*Transform ,M into| | another flavor of trig function}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0 4 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0 4 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0 4 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0 4 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0 4 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0 4 ~Hs((sin(?x))^2+(cos(?x))^2):(1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})b @# b%4" *|: ;bP8&c0!*substituting | |z,]tan,Hx,O2,I into a rational function in sin,Hx,I and cos,H| |x,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(cos(2*sin(?a+?b)*?z)):((1-?z^2)/(1+?z^2))~p0 4 ~Hs(sin(2*sin(?a+?b)*?z)):(2*?z/(1+?z^2))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Other rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0 4 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0 4 ~Ht(cos(?x)*sin(?x)):(1/2*sin(2*?x))~p0 4 ~Hs(sin(arccos(?x))):(sqrt(-?x^2+1))~p0 4 ~Hs(cos(arcsin(?x))):(sqrt(-?x^2+1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b!T" *|: ;bP8&c0!*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *|: ;bP8&c0!*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sinh(-?x)):(-sinh(?x))~p0 4 ~Hs(cosh(-?x)):(cosh(?x))~p0 4 ~Hs(tanh(-?x)):(-tanh(?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *|: ;bP8&c0!*Transform into | |other types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sinh(?x))^2):((cosh(?x))^2-1)~p0 4 ~Ht((cosh(?x))^2):(1+(sinh(?x))^2)~p0 4 ~Ht((tanh(?x))^2):(1-(sech(?x))^2)~p0 4 ~Ht(sinh(?x)):((e^?x-e^(-?x))/2)~p0 4 ~Ht(cosh(?x)):((e^?x+e^(-?x))/2)~p0 4 ~Ht(tanh(?x)):((e^?x-e^(-?x))/(e^?x+e^(-?x)))~p0 4 ~Ht(tanh(?x)):((sinh(?x))/(cosh(?x)))~p0 4 ~Hs((cosh(?x))^2-(sinh(?x))^2):(1)~p0 4 ~Hs(-(cosh(?x))^2+(sinh(?x))^2):(-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Other hyperbolic| | rules}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cosh(?x))^2):(1/2*(cosh(2*?x)+1))~p0 4 ~Ht((sinh(?x))^2):(1/2*(cosh(2*?x)-1))~p0 4 ~Ht(sinh(2*?x)):(2*cosh(?x)*sinh(?x))~p0 4 ~Ht(cosh(2*?x)):(2*(cosh(?x))^2-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Integration Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_}))# b$8" *|: ;bP8&c0!*after Partial Fraction| | Decomposition integration}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(ln(?x+?b*i)):((-i*sin(?a+?b)*?x/?b+1/2*ln(?x^2+?b^2))+1/2*~ 'p*i)~p0 4 ~Hs(ln(?x+i)):((-i*sin(?a+?b)*?x+1/2*ln(?x^2+1))+1/2*'p*i)~p0 4 ~Hs(ln(?x-?b*i)):((i*sin(?a+?b)*?x/?b+1/2*ln(?x^2+?b^2))+1/2*~ 'p*i)~p0 4 ~Hs(ln(?x-i)):((i*sin(?a+?b)*?x+1/2*ln(?x^2+1))+1/2*'p*i)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$4" *|: ;bP8&c0!*Derivatives of | |Integrals}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(Diff(?x)*(Integral(?y*d*?x))):(?y)~p0 4 ~V?c4 (i)~p0 1 ~V?d16 (d)~p0 1 ~V?v0 (k)~p0 1 ~V?c0 (n)~p0 1 ~V?c0 (c)~p0 1 ~V?c0 (b)~p0 1 ~V?c0 (a)~p0 1 ~V?v0 (z)~p0 1 ~V?v0 (y)~p0 1 ~V?v0 (x)~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_}`fb!D})b ]# b'4" *|: ;bP8&c0!*Dato un| | capitale iniziale c ,L un tasso di interesse "!Symbol^:!&c0 a| |: &c0!* ,L la funzione che esprime il capitale valutato al tempo| | &c0!%t&c0!* con ricapitalizzazione di intervallo &c0!%p&c0!* | | /H ,Z}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~A(C(?p,?t)=c*(1+'a*?p)^(?t/?p))~p1 1 ~d~He(C(?p,?t)):(c*(1+'a*?p)^(?t/?p))~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_}) # b'4}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~Q ]|Expr|[#b @`bb#_b#_b#_}`f#})## b!#" *|: ;bP8&c0!*Ponendo ,Z| |}% b!( b#8 b$@ b%H b&P!WW}]|[~p0 1 ~A(c=1)~p0 255 ~d~A('a=1)~p0 255 ~d~Q ]|Expr|[#b @`bb#_b#_b#_}`f#})%# b!#" *|: ;bP8&c0!*si ha ,Z| |}% b!( b#8 b$@ b%H b&P!WW}]|[~p0 255 ~A(C(p,t)=(p+1)^(t/p))~p1 2 ~sb/_!! } b00*! c#T"!c"H"_c/__c/__! ""} ^ _~A(C(p,t))~p0 3 ~A(C(p,t)=c*('a*p+1)^(t/p))~p0 4 ~sb/^!! } "&! c#T"!c#L"_c/__c/__} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_}`fb#@})!# b'C`f }["! `fb#@})b D# b'C" *|: ;bP8&c0!*la| | funzione esponenziale naturale /H la funzione che associa a | |t il valore $(!lim}(' p ,M,^ 0}_ C,Hp,Lt,I,L| |}& b!( b"0 b#8 b$@ b%H b&P!WW})4# b'Cossia quella in cui l,Gintervallo| | di ricapitalizzazione diventa istantaneo,N| |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_}`f#})## b!#" *|: ;bP8&c0!*Ponendo ,Z| |}% b!( b#8 b$@ b%H b&P!WW}]|[~p0 1 ~A(n=100000000)~p0 255 ~d~Q ]|Expr|[#b @`bb#_b#_b#_}`f#})4# b#9`fb"#}" *|: ;bP8&c0!*,Hpuoi| | cambiare il valore di n,I`f#} si ha ,Z| |}% b!( b#8 b$@ b%H b&P!WW}]|[~p0 255 ~A(y=C(1/n,t))~p0 1 ~d~A(y=c*('a/n+1)^(n*t))~p0 255 ~sb/_!! } "&! c#T"!c#L"_c/__c/__} ^ _~A(y=(1/n+1)^(n*t))~p0 255 ~sb/_!! } b00*! c#T"!c"H"_c/__c/__! ""} ^ _~Q ]|Expr|[#b @`bb#_b#_b#_}`fb!D})b E# b'4" *|: ;bP8&c0!*Calcoliamo| | la y per t ,] 1 ,Hossia dopo l,Gunit/@ di tempo,L ad esempio| | l,Ganno,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 2 ~A(EvaluateAt(y):(t=1)=(1/n+1)^n)~p0 2 ~sb/^!! } b00*! c#T"!c"D" c'8 _c/__c/__!"} ^ _~A(EvaluateAt(~ y):(t=1)=2.7182817863957975)~p1 3 ~sb/_!! } $&! c#T"!c%X"_c/__c/__} ^ _~A(EvaluateAt(y):(t=1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_}`fb"#})=# b'4" *|: ;bP8&c0!)il numero| | e /H il valore cui tende $^