Once at an airport, I noticed the Big 747 Jumbo Jets.
They of course had the big jet engines under the wings.
However, in the tail, I noticed a tiny jet exhaust.
Wheres the intake for this mini-engine?
Without having access to the technical specs, I believe you may have seen a vent from the plane’s interior.
The small jet you saw aft is the auxiliary power unit (APU). It provides electrical power and bleed air for ground operations. I don’t know where exactly the intakes are, but they should be on the aft fuselage, possibly with doors that open when it’s running.
Yep. I’m a Boeing engineer in real life. The APU is common to all our Boeing jets, the APU exhaust on the 747 is the most visually prominent being a round exhaust. Others, like the 777 and 767 have a slotted exhaust instead of a round exhaust, and aren’t as visually obvious.
It is a common misconception that the APU provides extra thrust, it doesn’t. It is a small gas turbine (about the physical size of an automobile engine) that provides electrical power for systems and hydraulics on board while the main engines aren’t spinning to provide power. When on an airplane, you can notice a distinctive dimming and re-brightening of the lights in the cabin when the airplane transfers from the APU to main power before takeoff and after landing.
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Does Boeing need someone with TSO/JCL (mainframe) experience in Seattle/Everett? 
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Good to have you aboard, Garg, but you didn’t quite answer the question. Where is the air intake for the APU?
Here’s a picture of a 747 that shows the APU intake: http://www.airliners.net/open.file?id=99910
It’s the boxy opening next to the vertical stabilizer.
(and here’s one that shows the exhaust)