Mathematics 153
Notes from May 27, 2020, lecture session

What was covered:
The material from Unit 12, more or less through a comparison of the contributions by Newton and Leibniz.   The discussion includes numerous similarities and differences in their independent discoveries.  At the beginning of the unit many earlier discoveries during the century before Newton and Leibniz are discussed.  These include progress on various key problems in four diretions which involved forerunners of the work by Newton and Leibniz:   Area and volume computations, infinite series and product expansions for a wide range of functions and constants, maximiation and minimization problems, and the definition of tangent lines for a class of curves which is much larger than the family of conic sections.  One particularly significant advance during this time, due to Wallis, was the reformulation of numerous geometrical problems of the preceding types in terms of purely algebraic problems.   The unit ends with listings of results due to Newton and Leibniz in other areas of mathematics; both lists are extensive.  
Relevant directory files for Unit 12:
Here is a listing of files from Unit 12 which are related to the May 27 lecture, with comments on their significance.  These links are all clickable. 
http://math.ucr.edu/~res/math153-2020/week9/unit12/history12.pdf  This is the main file for the unit and should ultimately be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week9/unit12/cavalieri.png   This is a corrected version of the whiteboard which summarizes Cavalieri's Principle.
https://www.youtube.com/watch?v=Ow6QqpeMYQ4   A YouTube video with further discussion and illustrations of Cavalieri's Principle.
http://math.ucr.edu/~res/math153-2020/week9/unit12/zzz-3D%20Printing%20Time%20Lapse%20Photography%20-%20Yoda.mp4    A video which shows a 3D printer build a 3-dimensional figure from horizontal slices, like the conceptual idea behind Cavalieri's Principle.
http://math.ucr.edu/~res/math153-2020/week9/unit12/history12a.pdf   A discussion of the proof for Cavalieri's Principle.  This involves graduate level mathematics and will not be tested.
https://mathworld.wolfram.com/Cycloid.html   An animated description of the cylcoid curve, which was first mentioned in the 15th century and was analyzed in detail during the next two centuries, mainly before the development of calculus.
http://math.ucr.edu/~res/math153-2020/week9/unit12/tangent-line.png   This is an edited version of the whiteboard which summarizes the construction of tangent lines due to Fermat and Barrow.
https://www.youtube.com/watch?v=1YtQEQ-HpqI  A YouTube video with further discussion and illustrations of the construction for tangent lines.
http://math.ucr.edu/~res/math153-2020/week9/unit12/history12b.pdf    AND
http://math.ucr.edu/~res/math153-2020/week9/unit12/history12e.pdf   Derivation of the Newton Binomial Series formula for (1+x)a  where a is not a nonnegative integer.  
https://www.cafepress.com/+fundamental_theorem_of_calculus_mousepad,93746919   A sumary of the Fundamental Theorem of Calculus which was independently discovered by Newton and Leibniz.
http://math.ucr.edu/~res/math153-2020/week9/unit12/logistic-equation.pdf   AND
http://math.ucr.edu/~res/math153-2020/week9/unit12/logistic2.pdf  Two files which illustrate the usefulness of formal integration techniques for analyzing population growth, including its early near-exponential increase and its ultimate leveling off too a stable value (which might be zero).