## 7671 Reputation

17 years, 240 days

## A bug in applyrule?...

Maple 2021

This looks like a bug to me but please correct me if it is not.

 > restart;
 > kernelopts(version);

 > half_angle_rule := [         sin(x::name) = 2*sin(x/2)*cos(x/2),         cos(x::name) = 1 - 2*sin(x/2)^2 ];

In this example, Maple applies the rule to the first element only.
It should apply to both.

 > A := < sin(u), sin(u) >; applyrule~(half_angle_rule, A);

In this example, Maple applies the rule to the second element only.
It should apply to both.

 > B := < cos(u), cos(u) >; applyrule~(half_angle_rule, B);

## Almost a rational solution to a transcen...

Maple

While solving an exercise in class, I ran into the following interesting solution of a transcendental equation.  It was not intentionally designed to be like this.

```restart;
eq := 2*exp(-2*t) + 4*t = 127:
fsolve(eq, t=0..infinity);
31.75000000```

The solution looks like a rational number while it was expected to be transcendental.  Let's increase the number of digits:

```Digits := 20:
fsolve(eq, t=0..infinity);
Digits := 10:
31.750000000000000000```

Let's make it even more accurate:

```Digits := 29:
fsolve(eq, t=0..infinity);
Digits := 10:
31.750000000000000000000000000
```

And even more:

```Digits := 40:
fsolve(eq, t=0..infinity);
Digits := 10:
31.74999999999999999999999999986778814353```

Is there a deep reason why the solution is so close to being a rational or is it just a coincidence?

Maple 2021

I may be misunderstanding the documentation of implicitplot.  Can someone set me straight?

This is extracted from the implicitplot's help page:

`implicitplot(-x^2 + y, x = 0 .. 2, y = 0 .. x);`

The plotting range is limited to y ≤ x, as intended.  Let us verify that it does the right thing:

```display(
implicitplot(-x^2 + y, x = 0 .. 2, y = 0 .. x, color=red),
plot(x, x=0..2, color=blue)
);
```

Yes, indeed it does.

Now let us try limiting the plotting range to y ≤ 1 − x2. Here is what we get:

```display(
implicitplot(-x^2 + y, x = 0 .. 2, y = 0 .. 1-x^2, color=red),
plot(1-x^2, x=0..2, color=blue)
);
```

I expected the red curve to lie entirely below the blue curve but it doesn't. Am I misunderstanding implicitplot?

## How to evaluate this integral?...

Maple 2021

I need to calculate dozens of piecewise-defined  (but elementary) definite integrals of the following kind. Maple returns them unevaluated. Is there a trick to force evaluation?

```restart;
u := piecewise(0 <= x and 0 <= y and y <= x, x-1, 0 <= x and 0 <= y and x <= y, x, 0);
v := piecewise(0 <= x and 0 <= y and y <= x, x-1, y <= 0 and 0 <= x and -x <= y, x-1, 0);
plot3d(u*v, x=-1..1, y=-1..1);
# integrating over (0,1)x(0,1) works
int(u*v, x=0..1, y=0..1);
# but integrating over (-1,1)x(-1,1) returns unevaluated.
# How to force evaluation on (-1,1)x(-1,1)?
int(u*v, x=-1..1, y=-1..1);
```

integration-problem.mw

## How to suppress axes labels...

Maple 2021

I want to plot a 2D graph without labels. The labes=["",""] option does half of the job—it prints the empty string for labels (that's good) but it reserves room for them (that's bad).  In the following code I use a large size labelfont in order to exaggerate the effect:

```restart;
plots:-setoptions(labelfont=[TIMES,64]);  # large labelfont selected on purpose
p1 := plot([[0,0],[1,1]], labels=["", ""]);```

Note the large blank space at the bottom reserved for the the non-existent label.

I know one way to eliminate the label altogether:

`p2 := subs(AXESLABELS=NULL, p1);`

This does the right job but is there a more orthodox way of doing that?

Afterthought:  It would be good if the labels option to the plot command  accepted none as argument, as in labels=[none, none].

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