Scared of bugs?

My daughter is almost 21 months old. over the last few weeks she has become increasingly frightened of bugs and spiders. We had a GIANT spider in the house Sunday and her daddy killed it and scooped it up and tossed it outside. She has been screaming about spiders all day long every day since then and starts shaking. She does this over the vacuum anytime we turn it on, but this is a new development. She wont go to sleep without us showing her the spider is not in the kitchen anymore. She cries at night when it is time to go to sleep and even talks about spiders in her sleep. I guess my question is this. Is it normal for kids to develop fears out of no where like this? And is it normal for them to be so bad like this? To the point of screaming and shaking?