Final answer:
Propofol is the intravenous anesthetic agent that is extremely short-acting and can cause apnea if injected too quickly.
Step-by-step explanation:
The intravenous anesthetic agent known for being extremely short-acting and for the risk of causing apnea if administered too quickly is propofol. This injectable anesthetic is commonly used during the induction of anesthesia due to its rapid onset and short duration of action. Anesthetists must be careful to administer propofol at a controlled rate to prevent complications such as apnea, where a patient temporarily stops breathing.