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Install the Digits app
The iPad mini is great for using as a calculator. The Digits app has all of the expected calculator functions but adds a useful "paper" tape to the mix. It makes doing a string of operations nice and lets you share the tape to provide a record. I use Digits all the time on my iPad mini.
Install the CalenMob Pro app
The iPad mini has a free calendar app but the CalenMob Pro app leaves it in the dust. It has several useful views and is a joy to use. It syncs nicely with Google Calendar and with multiple calendars enabled it is a very useful tool on the iPad mini.
Thanks to @jasonhiner for turning me onto CalenMob.
$6.99 (limited free version available)
Install NOAA Radar HD
The iPad mini screen is perfect for detailed weather tracking and the NOAA Radar app is great. It gives near real-time radar images of your immediate area and the big picture too. The app looks great and is extremely useful.