Try using splines for this (CurveFitting:-Spline).

You can also use CurveFitting:-PolynomialInterpolation  command.  But  if you want your plot to look like the plot in the picture ((in particular, the part of the graph below Ox-axis does not have inflection points), then you need to reduce the values of your function at the points  x=3  and  x=4 .

An example:

restart;
XY:=[[-4,3],[-3.5,1.8],[-3,1],[-2,1],[-1,1],[0,0],[1,-1],[2,-1.62],[3,-2],[4,-2],[5,1]]:
f:=unapply(CurveFitting:-PolynomialInterpolation(XY, x), x);
plot(f(x), x = -4 .. 5.2, thickness=2, gridlines, size=[600,500]);

Edit.

In a for-loop:

for i from 11 to 20 do
i^2;
od;

Directly (elementwise):

[$11..20]^~2;

If you want to create a matrix consisting of matrices, then here is an example of how to do it:

Download matrix_of_matrices.mw

In Maple everything look a little better.
 

Download plot3d.mw

Download convolution_new.mw

We use polar coordinates for this. One focus of the conic is at the origin. The eccentricity  e  varies from 0 to  2  in increments of  0.02. When e = 1 we get a parabola. Different types of conics are made in different colors:

Download conics_animation.mw

Contoures lines for y in red:

Download Contour_Plot_MaplePrimes_new.mw

I think that from a pedagogical point of view, a beginner should first of all master the  dsolve  command with  numeric option. This is the basic command for numerically solving differential equations. It allows you to find the values of dependent variables at individual points and also to build graphs of solutions using  plots:-odeplot  command. In the example below, we see that the graph is a spiral that rotates very quickly with an ever increasing speed.

dsolve.mw
 

All plottings are made in Cartesian coordinates, as plots:-odeplot command does not support  coords=polar  option.

Download dsolve.mw

I think you are simply misinterpreting the Gauss-Bonnet formula in your examples. What are you writing about "...area of upper cone's spherical cap plus length of dual cone's intersection with unit sphere..."  will be true only if the vertices of both cones coincide with the center of the sphere and the proof for this case is trivial (without Gauss-Bonnet formula)  2*Pi*(1-h)+2*Pi*h=2*Pi  .  Consider the extreme case: the radius of the base of the upper cone tends to  0 , and its height to  2 . Then the radius of the base of the lower dual cone will be tending to  0 , and your whole sum will be tending to  0 .

I could not express analytically the relationship between the variables  x, y, z , but it is easy to do numerically. By changing  u  and  v  with a certain step (for example, in a double loop), you easily get the corresponding triples  [x, y, z] . All these triples lie on an interesting surface:

restart;
alpha:=1:
eq1 := x(u,v) = (sqrt(2)*cos(v)^2*cos(2*u) + cos(u)*sin(2*v))/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
eq2 := y(u,v) = (sqrt(2)*cos(v)^2*sin(2*u) - sin(u)*sin(2*v))/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
eq3 := z(u,v) = 3*cos(v)^2/(2 - alpha*sqrt(2)*sin(3*u)*sin(2*v));
plot3d(rhs~([eq1,eq2,eq3]), u=-Pi/2..Pi/2, v=0..Pi, axes=normal, labels=[x,y,z], grid=[200,200], orientation=[-10,70]);

plot(2, phi=0..Pi/2, color="LightGreen", coords=polar, filled);

If you want to highlight the border with the desired color, then do this:

A:=plot(2, phi=0..Pi/2, color="LightGreen", coords=polar, filled);
B:=plot([[t,0,t=0..2],[2*cos(t),2*sin(t),t=0..Pi/2],[0,t,t=0..2]], color="Green", thickness=3):
plots:-display(A,B);

Edit.

You can re-use the  series  command to an already existing series to obtain the desired simplification.

An example:

A:=series(ln(1+x), x);
series(A/x, x);

If you need to programmatically extract the exponent of  x  in the term  O(x^n) , then do so:

A:=series(ln(1+x), x):
op(-1,A);

                                             6

Here's an example of another way:

applyrule(O(x^n::posint)/x=O(x^(n-1)), O(x^11)/x);

Use  combinat:-choose  command for this. An example:

combinat:-choose({$1..10}, 3);

  {{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 2, 7}, {1, 2, 8}, {1, 2, 9}, {1, 2, 10}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 3, 7}, {1, 3, 8}, {1, 3, 9}, {1, 3, 10}, {1, 4, 5}, {1, 4, 6}, {1, 4, 7}, {1, 4, 8}, {1, 4, 9}, {1, 4, 10}, {1, 5, 6}, {1, 5, 7}, {1, 5, 8}, {1, 5, 9}, {1, 5, 10}, {1, 6, 7}, {1, 6, 8}, {1, 6, 9}, {1, 6, 10}, {1, 7, 8}, {1, 7, 9}, {1, 7, 10}, {1, 8, 9}, {1, 8, 10}, {1, 9, 10}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 3, 7}, {2, 3, 8}, {2, 3, 9}, {2, 3, 10}, {2, 4, 5}, {2, 4, 6}, {2, 4, 7}, {2, 4, 8}, {2, 4, 9}, {2, 4, 10}, {2, 5, 6}, {2, 5, 7}, {2, 5, 8}, {2, 5, 9}, {2, 5, 10}, {2, 6, 7}, {2, 6, 8}, {2, 6, 9}, {2, 6, 10}, {2, 7, 8}, {2, 7, 9}, {2, 7, 10}, {2, 8, 9}, {2, 8, 10}, {2, 9, 10}, {3, 4, 5}, {3, 4, 6}, {3, 4, 7}, {3, 4, 8}, {3, 4, 9}, {3, 4, 10}, {3, 5, 6}, {3, 5, 7}, {3, 5, 8}, {3, 5, 9}, {3, 5, 10}, {3, 6, 7}, {3, 6, 8}, {3, 6, 9}, {3, 6, 10}, {3, 7, 8}, {3, 7, 9}, {3, 7, 10}, {3, 8, 9}, {3, 8, 10}, {3, 9, 10}, {4, 5, 6}, {4, 5, 7}, {4, 5, 8}, {4, 5, 9}, {4, 5, 10}, {4, 6, 7}, {4, 6, 8}, {4, 6, 9}, {4, 6, 10}, {4, 7, 8}, {4, 7, 9}, {4, 7, 10}, {4, 8, 9}, {4, 8, 10}, {4, 9, 10}, {5, 6, 7}, {5, 6, 8}, {5, 6, 9}, {5, 6, 10}, {5, 7, 8}, {5, 7, 9}, {5, 7, 10}, {5, 8, 9}, {5, 8, 10}, {5, 9, 10}, {6, 7, 8}, {6, 7, 9}, {6, 7, 10}, {6, 8, 9}, {6, 8, 10}, {6, 9, 10}, {7, 8, 9}, {7, 8, 10}, {7, 9, 10}, {8, 9, 10}}

I think that all your questions can be answered only numerically with the specified parameter values. Here is an example:

restart;
A:=x->mue+(mun/gama)+(u0^2)-mud*x-mue*exp(x)-(mun/gama)*exp(gama*x)-(u0^2)*(1-2*x/u0^2)^(1/2):

mue:=1/(1+alpha+beta):
mun:=alpha/(1+alpha+beta):
mud:=beta/(1+alpha+beta):
u0:=(mue+mun*gama)^(-1/2):

A1:=eval(A(x),[alpha=0.2,beta=0.3,gama=17]);
r:=solve(op([1,1],indets(A1,sqrt))>=0); # The domain of A1
plot(A1, x=-0.4..op(2,r), size=[800,400]);
fsolve(A1,x=-0.1..0.1); # The first root
fsolve(A1,x=0.1..op(2,r));  # The second root
fsolve(diff(A1,x),x=0.1..op(2,r));  # The root of first derivative

To investigate the dependence of the results on the parameters, see the  Explore command in the help.

restart;
int(tanh(x)/sqrt(x^2+1), x=1..100, numeric);

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