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13992 Reputation

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10 years, 38 days
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These are answers submitted by vv

Remove the singularity...

vv 13992
April 28 2018
1 4 
restart;
f:=exp(-sec(t))*cos(t)/(-1/4+sin(t)^2):
g:=convert(series(f,t=Pi/6,3), polynom):
int(f-g, t = 0 .. Pi/2, numeric) + evalf(int(g, t = 0 .. Pi/2, CPV));

                         -0.6159737938

math...

vv 13992
April 25 2018
0 1

In my opinion Maple cannot be strictly separated from maths, so math questions should be OK (provided that Maple can/will be used for a solution).

RootOf...

vv 13992
April 25 2018
0 2

First note that there is not such thing as "global inverse" of f, unless f (supposed to be C^1) is strictly monotonic.
If we know that f is strictly monotonic in [a,b], then its inverse will be
RootOf(f(x)-y, x, a..b),  where y is in the interval [min(f(a),f(b)), max(f(a),f(b))].

To find the intervals of monotonicity we need the real roots of f'(x). If solve is able to find all of them, we are done.
Of course the set of the roots could be infinite (even in a bounded interval). If a,b are two consecutive roots of f' then f is invertible (<==> strictly monotonic) in [a,b] as above.
 

 

 

f'...

vv 13992
April 24 2018
0 0

f is continuous in R (provided that f(0) = 0), differentiable in R \ {0}, so, f ' (0) does not exist [and you must use the definition for this].

Please note that the inexistence of the limit of f ' (x) at x=0 is not the same with the non-differentiability of f at 0.

Procedure for t...

vv 13992
April 22 2018
1 1

 

> 

restart;
> 

nor:=LinearAlgebra:-Norm:
> 

c := t  -> (1 - cos(Pi*t))/2:
> 

c1 := u -> solve(c(t_)-u,t_):
> 

g:= (d,m,t) -> <t, seq(c(m^k*t),k=1..d-1)>;

proc (d, m, t) options operator, arrow; `<,>`(t, seq(c(m^k*t), k = 1 .. d-1)) end proc

 (1)
> 

plot(c, 0..5);

 
> 

g(3,10,t);
plots:-spacecurve(g(3,10,t),t=0..1, numpoints=10000,color=red);

Vector(3, {(1) = t, (2) = 1/2-(1/2)*cos(10*Pi*t), (3) = 1/2-(1/2)*cos(100*Pi*t)})

  

 
> 

T := proc(x::{list,Vector},m::posint)
local d:=numelems(x), t:=Vector(d),td,N,i,j;
t[1]:=floor(m*x[1]);
for i from 2 to d do
  N:= add(t[j]*m^(i-1-j), j=1..i-1);
  td:=c1( (-1)^N*(x[i]-1/2)+1/2 );
  t[i]:=floor( td * m );
od;
t[d]:=td*m;
evalf(add(t[i]/m^i,i=1..d));
end:  
 
> 

Digits:=20:
d:=5: m:=100:   # check
to 50 do
  x:=LinearAlgebra:-RandomVector(d,generator=rand(0.0..1.0) ):
  tx:=T(x, m):
  gt:=g(d,m,tx):
  nor(x-gt,2)<sqrt(d-1)/m: evalf[10](%); print(Vector[row](x-gt),%,evalb(%));
od:

Vector[row](5, {(1) = 0.2245175949597057137e-2, (2) = 0.178700890374408893e-2, (3) = 0.502818085808100077e-2, (4) = 0.316188321077255800e-2, (5) = -0.6998708000e-15}), 0.6596538793e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.614205816283349693e-2, (2) = 0.58272696109815946e-3, (3) = 0.495928784131864040e-2, (4) = 0.381567203511813577e-2, (5) = 0.9414680500e-12}), 0.8787396552e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.586928409452430485e-2, (2) = 0.185056101232407102e-2, (3) = -0.1082881038373104706e-1, (4) = -0.3942168588339884894e-2, (5) = -0.4083914300e-12}), 0.1306433693e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.319600571683232791e-2, (2) = 0.1940509658645814175e-2, (3) = -0.653193795808389261e-2, (4) = -0.371827178092912454e-2, (5) = 0.2294257300e-11}), 0.8394747691e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.403959205986585828e-2, (2) = -0.1079273520512657950e-1, (3) = 0.355004577119255623e-2, (4) = -0.552511049641019727e-2, (5) = -0.9044114000e-13}), 0.1326390245e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.1243254335959312306e-2, (2) = -0.135800350252244505e-2, (3) = 0.473531960625732755e-2, (4) = 0.190711504155047987e-2, (5) = -0.4099288000e-13}), 0.5426803333e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.261962070266442793e-2, (2) = 0.171242859980398808e-2, (3) = 0.472919563680099705e-3, (4) = 0.4077468707718304194e-2, (5) = 0.1172276170e-11}), 0.5161804753e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.70425142720743796e-3, (2) = 0.371442276276967716e-2, (3) = 0.3207496584664778946e-2, (4) = 0.327778569310831046e-2, (5) = -0.8894589100e-12}), 0.5943468678e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.619845830097949042e-2, (2) = -0.3816343176658510635e-2, (3) = 0.255797091847471485e-2, (4) = 0.754622514135589480e-2, (5) = 0.4083021500e-12}), 0.1079231623e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.275882001720814820e-2, (2) = -0.113703586090948656e-2, (3) = -0.314753579794591941e-2, (4) = -0.670461269520757691e-2, (5) = -0.1070629390e-11}), 0.7985158197e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.310748361632089640e-2, (2) = 0.957346041346005009e-2, (3) = 0.296568346652236863e-3, (4) = 0.13644158538005765e-3, (5) = 0.2925298400e-12}), 0.1007046016e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.511755200815169579e-2, (2) = -0.231197153602198405e-2, (3) = 0.122532239719994439e-2, (4) = 0.1042815589001761737e-1, (5) = 0.1504406490e-12}), 0.1190724154e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.9399143835186173e-4, (2) = -0.183745957326462722e-2, (3) = -0.427579604196086895e-2, (4) = 0.589628853634182031e-2, (5) = 0.1389082070e-11}), 0.7512239504e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.451747602775906765e-2, (2) = -0.18087631646913584e-3, (3) = -0.441753325942565300e-2, (4) = -0.483896941010406977e-2, (5) = 0.3939638200e-12}), 0.7960560970e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.296209474675120431e-2, (2) = 0.642782211537201029e-2, (3) = -0.499917399430419126e-2, (4) = 0.239882013789489116e-2, (5) = 0.1268748590e-11}), 0.8990938834e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.268336241605795273e-2, (2) = 0.247738254723468146e-2, (3) = -0.311760622046606805e-2, (4) = -0.1103602126608338809e-1, (5) = 0.1577367720e-11}), 0.1203540993e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.60591631531697656e-3, (2) = 0.973038432364687423e-3, (3) = 0.241205747973921853e-2, (4) = 0.891126284059033480e-2, (5) = -0.2088726900e-12}), 0.9302825650e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.320902998616073463e-2, (2) = -0.100163300251327827e-2, (3) = 0.83748952784675779e-3, (4) = -0.258877667797425055e-2, (5) = -0.1425632680e-11}), 0.4324846300e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.479767050784526983e-2, (2) = -0.641702791126071034e-2, (3) = -0.209589326829079752e-2, (4) = 0.516914285221134588e-2, (5) = -0.1013133900e-12}), 0.9762617269e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.470543115375124229e-2, (2) = -0.438760161362192393e-2, (3) = 0.251639916895022527e-2, (4) = -0.124394006337165176e-2, (5) = -0.5434022500e-12}), 0.7019386150e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.235804118185165489e-2, (2) = 0.607772219611439259e-2, (3) = -0.359359187776981942e-2, (4) = -0.362090024481772109e-2, (5) = -0.2549344700e-12}), 0.8277915588e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.222007002543814218e-2, (2) = -0.19266088126927887e-3, (3) = 0.1838329070756913547e-2, (4) = -0.16227993312995806e-3, (5) = -0.3340775030e-11}), 0.2893374791e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.422005134521964257e-2, (2) = -0.194584250992639647e-2, (3) = -0.92984029266901054e-3, (4) = -0.38046948905607169e-3, (5) = 0.1621784000e-12}), 0.4754418622e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.72923453878922316e-3, (2) = -0.139804634320469602e-2, (3) = -0.29718662794632406e-3, (4) = -0.560421045991363441e-2, (5) = -0.5399395000e-13}), 0.5829392023e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.368685977669783485e-2, (2) = -0.373956751827731446e-2, (3) = -0.639498151574874456e-2, (4) = 0.1340857540050587714e-1, (5) = -0.4525906100e-12}), 0.1575636326e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.3078985582025345382e-2, (2) = 0.71049171296069372e-3, (3) = 0.542269145667356283e-2, (4) = -0.31978153021381634e-3, (5) = -0.1669304400e-12}), 0.6284329204e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.125321801628151357e-2, (2) = 0.688496424534676266e-3, (3) = -0.1590834201e-5, (4) = -0.28561582084002077e-3, (5) = 0.7879995500e-12}), 0.1458136362e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.187235009111001751e-2, (2) = 0.1393850477571501164e-2, (3) = 0.366115015658015357e-2, (4) = 0.361009340740225135e-2, (5) = -0.6332305600e-12}), 0.5646707793e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.156828627027680846e-2, (2) = -0.130132327224703670e-2, (3) = -0.104340521870953124e-2, (4) = 0.67928102725709501e-3, (5) = -0.3627550000e-14}), 0.2388112487e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.74353786025047025e-3, (2) = 0.317101011389826762e-2, (3) = 0.180693705013200166e-2, (4) = -0.2894566065450748756e-2, (5) = -0.4520790200e-12}), 0.4717169480e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.224321675021188086e-2, (2) = -0.166030766538155720e-2, (3) = 0.530173489916569672e-2, (4) = 0.261555980367242049e-2, (5) = 0.6999531000e-13}), 0.6537445140e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.177489617880375083e-2, (2) = 0.112559822704104403e-2, (3) = 0.155856332083461695e-2, (4) = 0.3523428502787770390e-2, (5) = 0.1577433130e-11}), 0.4388723716e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.142248928200009240e-2, (2) = -0.366317043660385106e-2, (3) = -0.275863834767106840e-2, (4) = 0.266928572960701061e-2, (5) = -0.7193998600e-12}), 0.5493401974e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.903253218527791813e-3, (2) = 0.760425612883232736e-2, (3) = -0.809816385216884054e-2, (4) = 0.200233217527449387e-2, (5) = 0.1313319900e-12}), 0.1132387608e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.269912109790725060e-2, (2) = 0.683589706601734925e-2, (3) = 0.750749259310287591e-2, (4) = -0.776618315016855834e-2, (5) = 0.9435919300e-12}), 0.1306486851e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.418256226678040169e-2, (2) = -0.550978964770481340e-2, (3) = -0.47706898189747461e-3, (4) = 0.1048590566434820137e-1, (5) = 0.1794301100e-12}), 0.1257113446e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.367774623133392882e-2, (2) = 0.338290688229710182e-2, (3) = -0.14497184375394752e-3, (4) = -0.231018536402677748e-2, (5) = -0.5805422100e-12}), 0.5507072686e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.308025432076038944e-2, (2) = 0.453505174176875314e-2, (3) = 0.333904919004368431e-2, (4) = 0.7255898944310806e-4, (5) = -0.7694659500e-12}), 0.6419437303e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.485697503452520250e-2, (2) = 0.686391456316770274e-2, (3) = 0.6784068578533270851e-3, (4) = 0.408053528967211488e-2, (5) = -0.8724029300e-12}), 0.9370940920e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.473633837121329318e-2, (2) = -0.1045583377707223185e-1, (3) = -0.3228661437095516013e-2, (4) = -0.2685323647052749672e-2, (5) = 0.1515649540e-12}), 0.1222262570e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.377933027792209362e-2, (2) = -0.551129857571372329e-2, (3) = -0.26092730504601421e-3, (4) = 0.95950207365360741e-3, (5) = 0.1770566570e-11}), 0.6756217627e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.76145764706622248e-3, (2) = -0.2390831790194682196e-2, (3) = -0.497098638379546841e-2, (4) = 0.58261853861024700e-3, (5) = -0.2536705870e-11}), 0.5598753824e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.30312958994790132e-3, (2) = 0.30543629043494512e-3, (3) = 0.577377433296877592e-2, (4) = 0.33606623697169290e-3, (5) = 0.2461563000e-12}), 0.5799533554e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.154052538043282555e-2, (2) = 0.337000037279360873e-2, (3) = 0.32661409377263359e-3, (4) = -0.560879683824746114e-2, (5) = 0.3804276500e-12}), 0.6730185711e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.392761758034173186e-2, (2) = -0.217276325877687460e-2, (3) = -0.935392453205133960e-2, (4) = -0.460279671189654340e-2, (5) = -0.3172334940e-11}), 0.1135027408e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.144432976285397077e-2, (2) = 0.108651745255581567e-2, (3) = 0.86678974009380951e-3, (4) = -0.569753959951737504e-2, (5) = -0.2314541300e-11}), 0.6039858490e-2 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.128131422607133849e-2, (2) = 0.680749855560426033e-2, (3) = 0.416695032117804010e-2, (4) = 0.6912351285026329273e-2, (5) = 0.2371383200e-12}), 0.1063615899e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = 0.193745529333707771e-3, (2) = 0.283962100082164068e-2, (3) = 0.201340260013797986e-2, (4) = 0.991047039966587666e-2, (5) = 0.2539255400e-13}), 0.1050581736e-1 < 0.2000000000e-1, true

  

Vector[row](5, {(1) = -0.105796763334716159e-2, (2) = -0.300733439200737383e-2, (3) = 0.561062685289715871e-2, (4) = 0.911453541107139415e-2, (5) = -0.1589016430e-11}), 0.1116768754e-1 < 0.2000000000e-1, true

  

Vector[row](%id = 18446744074331819414), 0.7833455316e-2 < 0.2000000000e-1, true

 (2)
> 

 


Download space-fill-quasi-sent.mw

F...

vv 13992
April 21 2018
0 1

F := k -> sum(i^k, i=1..n);

seems to be faster.

Check...

vv 13992
April 20 2018
0 0

Check this FAQ:

https://desktopfaq.maplesoft.com/customer/en/portal/articles/2840902-maple-does-not-start-or-takes-a-long-time-to-load-

Exact...

vv 13992
April 19 2018
2 1
 

The exact solution (just for fun).

> 

restart;
> 

h:=1: g:=1:  r:=1/2:
> 

v0:=sqrt(2*g*h):
> 

T:= n -> v0/g*(1+r-2*r^n)/(1-r):
> 

Too := v0/g*(1+r)/(1-r): # = T(infinity)
> 

Y:=proc(t)
local n := floor( ln(-t*sqrt(2)*g*(1-r)/(4*sqrt(g*h))+1/2+(1/2)*r)/ln(r) ),
      tau:= t - T(n);
piecewise(t<Too, v0*r^n*tau - g/2*tau^2, 0)
end:
> 

plot(Y(t), t=0..Too, size=[1200,800]);

 
> 

Y(t);

piecewise(t < 3*sqrt(2), sqrt(2)*(1/2)^floor(-ln(-(1/8)*t*sqrt(2)+3/4)/ln(2))*(t-2*sqrt(2)*(3/2-2*(1/2)^floor(-ln(-(1/8)*t*sqrt(2)+3/4)/ln(2))))-(1/2)*(t-2*sqrt(2)*(3/2-2*(1/2)^floor(-ln(-(1/8)*t*sqrt(2)+3/4)/ln(2))))^2, 0)

 (1)
> 

Too;

3*2^(1/2)

 (2)
> 

 

Sudoku...

vv 13992
April 19 2018
0 0

See:

https://www.mapleprimes.com/posts/207910-A-Compact-Sudoku-Solver-Via-Groebner

color, linestyle, ......

vv 13992
April 17 2018
0 2

Probably the designers considered that the color, linestyle, thickness are enough for the legend (which I also do).

Anyway, the symbols seem to be useless here because their exact position is not known.

Physics...

vv 13992
April 17 2018
1 1

You have the Physics package loaded. It redefines many things, including diff.
The conversion works as expected if you don't use the package (which is not needed in the worksheet).

polar...

vv 13992
April 16 2018
2 9

If you have the curve in polar coordinates, then DirectSearch or fsolve is not needed.
Something like this (for a modified cardioid):

r:=t->2*(1-cos(t))+1:
plots:-animate(plot, 
[[[r(t-u)*cos(t),r(t-u)*sin(t),t=0..2*Pi] ,
[[0,r(Pi/2-u)*sin(Pi/2)],[0,2+r(Pi/2-u)*sin(Pi/2)]]   ],thickness=[1,6]],
u=0..2*Pi, scaling=constrained);

view...

vv 13992
April 16 2018
1 0

You cannot use plot3d(f(x,y)=0.1, x=0..1,y=0..1);
Probably you want plots[implicitplot](f(x,y)=0.1, x=0..1,y=0..1);

For z=0..1  just use the view option.
Compare:

plot3d(x^2+y^2, x=-1..1, y=-1..1);
plot3d(x^2+y^2, x=-1..1, y=-1..1, view=0..1);


 

student...

vv 13992
April 15 2018
1 5

In my opinion the student should be taught to use a single simple line of code

s := n -> s(n-1)^2 - 2;   s(1) := 4;

defining his own sequence:
s(4);
      37634
 seq(s(n), n=1..5);  
      4, 14, 194, 37634, 1416317954

0^0...

vv 13992
April 13 2018
0 0

See a full explanation here: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

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