Variable Speed

Is it possible within the audacity program, to create an envelope within which one can change the speed so that for example, it starts off at normal speed but by the time it gets to the end of the envelope, the speed has doubled but this is performed gradually over the length of the envelope?

Could try “sliding time scale / pitch shift” effect in Audacity 1.3 “Effect”…
'sliding time scale -- pitch shift' effect in Audacity 1,3.png
or this “warping” plug-in here …

The “sliding time scale / pitch shift” can change speed whilst preserving original pitch, (but introduces conspicuous artifacts ).

The “warping” plug-in does not preserve pitch when the tempo increases: pitch increaes with speed, (but less artifacts than the other effect though).

Audacity 1.3.x
Provided that the rate of change is fairly slow you can also use a “Time Track”
A limitation of this method is that rapid changes in speed are likely to cause jumps in speed rather than a smooth transition, but for gradual speed changes this is a convenient, quick and high quality way of doing it.

To “fix” the effect to the track (apply it to the data), use “Mix and Render” from the Tracks menu and then remove the time track.