{NotebookFileName[],DateString[]}

Falling object

style[
t_
]:=
Style[t,Bold,FontFamily->"Arial",FontSize->14]

g1=10;
t01=1;

y1[
t_,
g_:g1
]:=
-(1/2)g (t^2)

y1[1]

v1[
h_,
t0_:t01
]:=
((y1[t0+h]-y1[t0])/h)

D[y1[t],t]//Simplify

Block[{is=400,as=1/GoldenRatio,cr=0.015,rim=0.001,lim},
lim=Limit[v1[h],h->0];
Show[
Plot[v1[h],{h,((-1)/2),(1/2)}
,Epilog->{
Circle[{0,lim},ImageScaled[{as*cr,cr}]],
{White,Disk[{0,lim},ImageScaled[{as*(cr-rim),(cr-rim)}]]}
}
,ImageSize->is
,AspectRatio->as
]
]
]

y1[
t_
]:=
-(1/2)g (t^2)

vb1[
h_,
t0_
]:=
((y1[t0+h]-y1[t0])/h)

vb1[h,t0]//Simplify

Random plots

Show[
Plot[Exp[-(1/(x^2))],{x,-1,1}],
ListPlot[{{0,0},{0.8,Exp[-(1/(0.8^2))]}}
,PlotMarkers->Graphics[{{White,Disk[]},{Black,Circle[]}},ImageSize->10]
]
]

Show[
Plot[Exp[-(1/(x^2))],{x,-1,1}],
ListPlot[{{0,0},{0.8,Exp[-(1/(0.8^2))]}}
,PlotMarkers->"○"
]
]

Plot[Sign[1-ω]+Sign[1+ω],{ω,-2,2}
,Exclusions->None
]

Plot[Sin[(1/x)],{x,-0.1,0.1}
,PlotPoints->200
,PlotStyle->Directive[Thick,Red]
]

Plot[x Sin[(1/x)],{x,-0.1,0.1}
,PlotPoints->200
]

Plot[(Sqrt[x]) Sin[(1/x)],{x,-0.1,0.1}
,PlotPoints->200
]

GHK equation

Quantity[1,"Faradays"]

UnitConvert[Quantity[1,"Faradays"]]

UnitConvert[Quantity[1,"Faradays"],"Coulombs"]

Quantity[273+20,"Kelvins"]

Quantity[1,"MolarGasConstant"]

ϕ1[
p_,
z_,
vm_,
ci_,
co_
]:=Module[{e},
e=Exp[-((z vm F)/(R T))];
p (z^2)(((vm (F^2))/(R T)))(((ci-co e)/(1-e)))
]

ϕ1[p,z,vm,ci,co]

ϕtot2=ϕ1[(Subscript[P, 1]),(Subscript[z, 1]),(Subscript[V, m]),(Subscript[C, i,1]),(Subscript[C, o,1])]+ϕ1[(Subscript[P, 2]),(Subscript[z, 2]),(Subscript[V, m]),(Subscript[C, i,2]),(Subscript[C, o,2])]

ϕtot2//TraditionalForm

ϕtot3=ϕ1[p1,z1,Vm,ci1,co1]+ϕ1[p2,z2,Vm,ci2,co2]

Solve[0==ϕtot3,Vm]

ϕtot4=ϕtot3/.p2->r p1

Series[ϕtot4,{r,0,0}]

Normal[Series[ϕtot4,{r,0,0}]]

Solve[0==Normal[Series[ϕtot4,{r,0,0}]],Vm]

Infinite discontinuities

{
Plot[Cos[x],{x,-4π,4π}
,PlotLabel->style@TraditionalForm[Cos[x]]
],
Plot[Tan[x],{x,-4π,4π}
,ExclusionsStyle->{Dashed}
,PlotLabel->style@TraditionalForm[HoldForm[Tan[x]==((Sin[x])/(Cos[x]))]]
]
}//Column

{
Plot[((Sin[x])^2),{x,-4π,4π}
,PlotLabel->style@TraditionalForm[((Sin[x])^2)]
],
Plot[((Csc[x])^2),{x,-4π,4π}
,ExclusionsStyle->{Dashed}
,PlotLabel->style@TraditionalForm[HoldForm[((Csc[x])^2)==(1/((Sin[x])^2))]]
]
}//Column

Plot[Log[x],{x,-2,2}
,PlotLabel->style@TraditionalForm[ln[x]]
,AspectRatio->1
]

Plot[(x^(-1/2)),{x,-2,2}
,PlotLabel->style@TraditionalForm[(x^(-1/2))]
,PlotRange->{Full,{0,Automatic}}
,AspectRatio->1
]

The logistic equation

l1[
n_,
rv_:r,
kv_:k
]:=
r (n/k)(k-n)

lsol1=DSolve[{
n'[t]==l1[n[t]],
n[0]==n0
},n[t],t][[1]]

n[t]/.lsol1//Simplify

Swimming bugs

v1[
t_
]:=
(f/b)(1-(E^(-(b/m)t)))

Series[v1[t],{t,0,1}]

Series[v1[t],{t,∞,1}]