What is the author's tone in The Last Lecture?

Background on The Last Lecture: 

The Last Lecture, cowritten by Randy Pausch, a computer science professor, and Jeffrey Zaslow, a reporter, is actually based on an inspirational lecture given by Randy Pausch.

Diagnosed with terminal pancreatic cancer, Randy Pausch dedicated his "last lecture" to a reflection on what really matters in life. Pausch delivered his last lecture, "Really Achieving Your Childhood Dreams," to his friends, students, family, and co-workers at Carnegie Mellon University in 2007. His lecture received attention and praise from the media, and the book was published in 2008.

Tying in tone:

According to literarydevices.net, tone, in written composition, is the writer's attitude toward a subject or an audience. Generally, an author conveys tone through their choice of words. 

The book is written with humor and wisdom and outlines the points Pausch made in his "last lecture" at Carnegie Melon. Because of this, the tone can be characterized as reflective. In a reflection of his own childhood experiences and dreams, Pausch emphasizes the importance of childhood dreams in everyone's lives, and also comments on ways to achieve those dreams as we grow into adults.

I have attached a video of Pausch's lecture.

In The Last Lecture, what was the author's mood during his presentation?

Randy's positive mood is reflective of his lecture's topic.

Randy knew that people might perceive his "last lecture" as focusing on death and dying.  However, Randy challenged this preconception through his choice of topic. Randy decided to make the lecture about dreams and the power of transformative hope:

Sitting there, I knew that despite the cancer, I truly believed I was a lucky man because I had lived out these dreams. And I had lived out my dreams, in great measure, because of things I was taught by all sorts of extraordinary people along the way. If I was able to tell my story with the passion I felt, my lecture might help others find a path to fulfilling their own dreams.

Randy's last lecture is not about his passing.  Rather, he seized it as an opportunity to talk about the beauty of dreams.  

This topic contributed to Randy's very upbeat mood. Randy was convinced that his entire life was a testament to dreams being realized.  It is for this reason he apologizes to the audience for not being "as depressed or morose as I should be."  Randy feels that the arc of his life has been a glorious one because he accomplished meaningful dreams.  As a result, his mood while giving the lecture is positive.