I occasionally pull out a calculus book for mental exercise myself (you can only do so much sudoko).
But calculus isn’t the only way to improve a child’s thinking skills. Most people I know who studied calculus way back when couldn’t even tell you what a derivative is anymore. So unless the kid is definitely bound for engineering or some physical science, the time spent learning calculus might be better spent learning something more applicable to his future avocation.
Calculus is a tool of thinking most effective when young minds are learning stuff. Calculus and a foreign language or two, preferably a very different foreign Language like Mandarin or Korean. All of these things give a young mind different pathways, different approaches to perception and aid in thinking about, and problem solving in, all subjects.