No, writing in cursive is not typical in preschool. In fact, printing isn't typical in preschools, either, other than just trying to teach kids to recognize their A,B,C's and perhaps how to spell their first names. Many children enter kindergarten without having good handwriting skills, and by that, I mean even having difficulty holding a pencil properly, recognizing all their letters, etc.
I'm wondering if the school your daughter is at is teaching the D'Nealian style of printing, which isn't really cursive writing, but it has a more "scroll-like" appearance, with little "hooks" or "tails" on some of the letters, so that the transition to actual cursive writing later on, where all the letters connect to form words, comes easier.