1) To emphasize the symmetries of the graphs, the center of the composition take the point (-1,1).

2) For greater clarity, the range along y-axis is reduced.

3) In order to estimate "by eye" the slopes of the graphs, selected the same scales on axes.

plot([seq(1+2*(1+x)^n, n = 1 .. 4)], x = -3 .. 1, -2 .. 4, color = [red, blue, yellow, green], thickness = 2, legend = [n = 1, n = 2, n = 3, n = 4], scaling = constrained);

restart;

a:=Vector([2,3,4,5]): l:=[]:

for k to ArrayNumElems(a) do

if a[k]>=3 then l:=[op(l), k] fi:

od:

l;

                        [2, 3, 4]

Try first all equations lead to a uniform form.

Example:

A := y - x = 0:

k := lcoeff(lhs(A), [x, y]);

A/k;

                k := -1                

               x - y = 0

Carl, you are right. I was inattentive. New procedure   SpiralMatrix1  solves the original problem:

restart;

SpiralMatrix1:=proc(A)

local n, A1, SpiralMatrix, a;

uses LinearAlgebra;

n:=RowDimension(A);

SpiralMatrix:=proc(A::Matrix)

local n, Turn, L, C, k;

uses LinearAlgebra;

n:=RowDimension(A);

Turn:=proc(B::Matrix)

local n;

uses LinearAlgebra;

n:=RowDimension(B);

[[seq(B[1,j],j=1..n),seq(B[i,n],i=2..n),seq(B[n,j],j=n-1..1,-1),seq(B[i,1],i=n-1..2,-1)], Matrix([seq([seq(B[i,j],j=2..n-1)],i=2..n-1)])];

end proc;

L:=[]; C:=A;

for k to floor(n/2) do

L:=[op(L),op(Turn(C)[1])]; C:=Turn(C)[2];

od;

if type(n,even) then Matrix(n,n,L) else Matrix(n,n,[op(L),C[1,1]]) fi;

end proc; 

A1:=SpiralMatrix(Matrix(n,n,symbol=a));

assign(seq(seq(A1[i,j]=A[i,j],j=1..n),i=1..n));

Matrix([seq([seq(a[i,j],j=1..n)],i=1..n)]);

end proc;

Example:

A:=Matrix([seq([i $ 5], i=1..5)]);

SpiralMatrix1(A);

SpiralMatrix:=proc(A::Matrix)

local n, Turn, L, C, k;

uses LinearAlgebra;

n:=RowDimension(A);

Turn:=proc(B::Matrix)

local n;

uses LinearAlgebra;

n:=RowDimension(B);

[[seq(B[1,j],j=1..n),seq(B[i,n],i=2..n),seq(B[n,j],j=n-1..1,-1),seq(B[i,1],i=n-1..2,-1)], Matrix([seq([seq(B[i,j],j=2..n-1)],i=2..n-1)])];

end proc;

L:=[]; C:=A;

for k to floor(n/2) do

L:=[op(L),op(Turn(C)[1])]; C:=Turn(C)[2];

od;

if type(n,even) then Matrix(n,n,L) else Matrix(n,n,[op(L),C[1,1]]) fi;

end proc;

Example:

A:=Matrix([seq([i $ 5], i=1..5)]);

SpiralMatrix(A);

restart;

f(x) = 2.25*f(x-1)-0.5*f(x-2);

convert(%, rational);

f := unapply(rsolve({%, f(1) = 1/3, f(2) = 1/12}, f(x)), x);

plots[display](<plot(f, 1 .. 40, color = red) | plots[logplot](f, 1 .. 40, color = red)>);

Left - the usual plot, right - logplot.

Here is the simplest code, generating the first  N  rows of Pascal's Triangle:

PascalTriangle:=proc(N)

local n;

for n from 0 to N-1 do print(seq(binomial(n,k), k=0..n)) od;

end: 

Example:

PascalTriangle(10);

Line:=plot(x^2-2*x+2, x=-0.5..2.5, -0.5..3.5, thickness=2):

Point:=plottools[disk]([1,1],0.04, color=blue):

plots[display](Line, Point, scaling=constrained);

For such a simple task we don't need in Maple (and it will not help if, for example, instead of 1000 we take 10 ^ 10). Instead the range 1 .. 1000 consider the range 0 .. 999, adding zeros in front if the number of digits is less than 3:  000, 001, 002, etc. Total used 3 * 1000 = 3000 digits and because of the equality of all digits  the answer will be 3000/10 = 300

The procedure  MultiAdd  solves your problem:

MultiAdd:=proc(f::procedure, L::list(range))

local M, T, m, S;

uses combinat;

M:=map(i->[$ i], L);

T:=cartprod(M);  S:=0;

while not T[finished] do

m:=T[nextvalue]();  S:=S+f(op(m))

end do;

S;

end proc:

Example:

MultiAdd((i,j,k)->1/(i*j*k), [1..5, 2..8, 3..7]);

                              3360747/784000

Replace  Diff  by  diff :

restart: with(DEtools):

DEexp1:=2*y(x)*(y(x)-4):

DE:=diff(y(x),x)=DEexp1: 

DEplot(DE, y(x), x=-1..1, y=-2..7.4);

You can not substract a vector from a number. You can use the formula:

theta:=Vector(m, i->90) - lambda;  # m is the numbers of rows

Example:

m:=4:  theta:=Vector(m, i->90) - lambda:

lambda:=<10, 90, 0, -60>:

theta;

                   <80, 0, 90, 150>

For longitude all is similar only use  `if`  command.

RealRange(0, 2*Pi)  minus  `union`(solve({x >= 0, cos(x) = 0, x <= 2*Pi}, AllSolutions, explicit));

 is already implemented in Maple (version 16). See  ?worksheet,plotinterface,zoom

Addition: in older versions zooming can be done by scaling along axes.

Example of zooming of the plot  y=x*sin(7*x)  in 10 times:

plots[animate](plot, [(x/k)*sin(7*x/k)*k, x = -Pi .. Pi,thickness=2, tickmarks=[0,0]], k=1..10, frames=200);

p1:=a*n1+b/n2+c*(n1+n2)^2:

subs(n1+n2=1, p1);

                       a*n1+b/n2+c