…he had only to put the corpse where he had left the new baby at the last quarter of the moon, and the hyenas would do the rest. (1.9)
This is a rather grim start to the novel, with dead man-apes everywhere. Life in the dim dawn of man was "nasty, brutish, and short" (to quote the philosopher Thomas Hobbes). But—things are going to get better. You get the bad stuff so you can see how much better the future is going to be, with its slabs and aliens and human advancement. Progress is climbing away from the mortal grave.
Part 3, Chapter 18
…though even the smallest could completely destroy the ship if it slammed into it at tens of thousands of miles an hour, the chance of this happening was negligible. (18.3)
The danger from asteroids emphasizes how vulnerable human beings are…though the scientific assessment of chances is a comfort. Morality looms, but in 2001 you've always got some technobabble to make things seem better.
Part 4, Chapter 25
Yet still he called stupidly, as if an incantation could bring back the dead: "Hello Frank…Hello Frank…Can you read me?" (25.30)
Frank Poole's fate is probably the most traumatic, or unpleasant, death in the novel. Bowman calling out helplessly into the vast indifferent cosmos links the big universe all around to death. It's like he's floating alone in a big sea of mortality (though of course the aliens are out there to rescue him eventually).
Part 4, Chapter 26
It was impossible to tell that the sleeping man was not dead; there was not the slightest visible sign of vital activity. (26.43)
The hibernating folks seem dead. Blurring the line between living and dead emphasizes the extent to which Bowman and all humans are just living on borrowed time. Could an alien tell the difference between a live human and a dead one?
He had thought that a hibernating man showed no sign of life, but now he knew that this was wrong. Though it was impossible to define it, there was a difference between hibernation and death. (26.22)
Even with all their technology, humans can't bridge that gap between death and life. Yet.
"Unless you obey my instructions, I shall be forced to disconnect you." (26.35)
The best evidence you get that Hal is alive is that he's afraid to die. Death defines living—and where does that leave the immortal aliens?
Part 4, Chapter 27
To Hal, this was the equivalent of Death. For he had never slept, and therefore he did not know that one could wake again….(27.9)
Again, you feel sorry for poor Hal…and again, he often seems more human than Bowman, who never seems all that worried about death. (There's a long tradition of artificial monsters being more human than their creators—see Frankenstein, for example.)
Part 4, Chapter 29
Bowman could bear no more. He jerked out the last unit, and Hal was silent forever. (29.59)
Bowman kills Hal—a sort of echo of the novel's other murder, when Moon-Watcher killed One-Ear. Both are part of humans' evolutionary push out to the stars. Transcendence requires a bunch of dead bodies (flesh or metal), apparently.
Part 5, Chapter 32
They would replace their natural bodies as they wore out—or perhaps even before that—by constructions of metal and plastic and would thus achieve immortality. (32.15)
Humans speculate how aliens might become immortal. Of course, this could be Clarke himself speculating, since it's exactly how he makes his aliens immortal. Mortal humans like to think about immortality, in gods or aliens.
Part 5, Chapter 39
He could not possibly survive until Discovery II made its rendezvous with Japetus, four or five years hence. (39.4)
Bowman realizes that he's going to die alone in space. He's remarkably calm about it. He's something of a cold fish.