TAG-Heuer doesn't have much information on the vintage watches indeed, and unfortunately Heuer's record keeping wasn't quite as meticulous as Omega's and they don't have a serial and reference number database as a result. It makes it more difficult to find information on a watch, but on the other hand it makes the digging around more interesting for us! You've come to the right place anyway :)

And your request is quite an easy one. You've helped us a lot by already providing the 7721 number (yes, it is a model reference). Here's your watch in a Heuer catalogue from 1970:

So it actually fits into the period around the end of production for the first Carreras (1964 on) and commencement of the first automatic Carreras (1969 on), which in fact overlap with each other for a couple of years of production. Another telltale for that dating is the movement - Valjoux didn't produce the 7733 until 1969 and Heuer look only to have started using it in 1970, so it can't be earlier than that date. A 113xxx serial number could be a bit early - I wonder if that second number could be a 7 instead? It can be hard to be certain, even under fairly high magnification.

We don't frown on providing a price indication, but it is helpful to know whether it's for a sale, or for insurance purposes etc. Although nice watches in their own right, these were introduced as an "economy" range, a way into Heuer ownership that was less prohibitive than one of the named series watches. That still reflects on the price today, in as much as the series watches (Carrera, Autavia, Monaco etc) are more pursued by collectors - at the bottom end, a 7721 is below the $1000 mark though some dealers are pricing much more ambitiously than this, apparently on the basis that any vintage Heuer has considerable value.