On an 18-hour flight from California to Singapore a few years ago, Timothy D. Cook, Apple’s chief operating officer, had little time for small talk with a colleague. Glued to his business class seat, Mr. Cook had his nose in spreadsheets, preparing for a thorough review of Apple’s Asian operations.

The two landed at 6 a.m., took time to shower and headed into a meeting with Apple’s local executives. Twelve hours later, and well past dinnertime, the local executives were ready to call it quits.

“They were absolutely exhausted,” said Michael Janes, the Apple executive who accompanied Mr. Cook. “Tim was not. He was ready to jump to the next slide and the next slide after that. He is absolutely relentless.”

That relentlessness could be indispensable in the months ahead, because Mr. Cook may be tested as never before. He has been charged with running Apple’s day-to-day operations while his boss, Steven P. Jobs, the company’s visionary chief executive, is on medical leave.