Many analysis applications in areas such as finance and energy monitoring require the repeated execution of expensive modeling functions over streams of rapidly changing data. These applications can often be expressed declaratively, but current continuous query processing technology does not provide adequate performance in the presence of these expensive functions. For important classes of real-valued functions evaluated in predicates, we use Taylor approximations to determine ranges of stream inputs for which the system knows the outcome of the predicates for potential query results. We show how to integrate this technique into a prototype continuous query processor. We then report on experiments for a financial application using real bond market data. The experiments show that our techniques significantly reduce the number of function calls compared to traditional memoization.