Swimming bacteria detect chemical gradients by performing temporal comparisons of recent measurements of chemical concentration. These comparisons are described quantitatively by the chemotactic response function, which we expect to optimize chemotactic behavioral performance. We identify two independent chemotactic performance criteria: In the short run, a favorable response function should move bacteria up chemoattractant gradients; in the long run, bacteria should aggregate at peaks of chemoattractant concentration. Surprisingly, these two criteria conflict, so that when one performance criterion is most favorable, the other is unfavorable. Because both types of behavior are biologically relevant, we include both behaviors in a composite optimization that yields a response function that closely resembles experimental measurements. Our work suggests that the bacterial chemotactic response function can be derived from simple behavioral considerations and sheds light on how the response function contributes to chemotactic performance.