Mathematics 153
Notes from May 29, 2020, lecture session

What was covered:
The material from the remainder of Unit 12 and the first part of Unit 14.  The former includes further comments on the approaches of Newton and Leibniz along with examples for which each approach is more effective.   It also mentions a few of the numerous other substantial contributions to mathematics which Newton and Leibniz did besides set up calculus; in all of these cases there was no overlap between the work of these two scholars.  The discussion includes numerous similarities and differences in their independent discoveries.  In nearly all these cases one can see a major difference between the interests of Newton and Leibniz.  As a physicist, Newton's work was strongly motivated by interest in solving problems from physics.  In contrast, Leibniz' motivations were largely philsophical and conceptual, and his work generally tended towards issues of the latter type; it should be noted that there are notable exceptions in both of their collected works.  
Material in Unit 14 was covered through the discussion of solid analytic geometry and the legacy of Bernoulli family.
Finally, it seems worthwhile to address a point in Leibniz' philosophy which received a great deal of ridicule in Candide, which was written by Voltaire (François-Marie Arouet, 1694 – 1778).   One character in this work was constantly saying that this is the best of all possible worlds.  This is a distortion of Leibniz' view that the world was created so that there would be a minimum amount of evil.  Of course, one can argue against this viewpoint, but doing so is a more complicated task.
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/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 and modeling population growth, including its early near-exponential increase and its ultimate leveling off.   For this application, expressing the solution in finite terms (Leibniz' preference) is more effective than writing things in terms of infinite series (Newton's preference).  
http://math.ucr.edu/~res/math153-2020/week9/unit12/history14c.pdf   AND
http://math.ucr.edu/~res/math153-2020/week9/unit12/comments-on-14c.png   Discussion of functions which are easily expressible in terms of elementary functions from first year calculus but have indefinite integeals which are not expressible in such terms.  For many of these examples such as exp(–x2), it is fairly straightforward to express the indefinite integrals as convergent power series.   For this application, writing things in terms of infinite series (Newton's preference) is more effective than expressing the solution in finite terms (Leibniz' preference).  
http://math.ucr.edu/~res/math153-2020/week9/unit12/history14d.pdf   AND
http://math.ucr.edu/~res/math153-2020/week9/unit12/comments-on-14d.png   AND
http://math.ucr.edu/~res/math153-2020/week9/unit12/bump-functions.png  Construction and examples of infinitely differentiable functions defined near a point a such that the value of the function at a, and the values for the derivatives of all orders at a, are equao to zero.  Such a function cannot have a convergent power series expansion centered at a, and therefore in these cases expressing the solution in finite terms (Leibniz' preference) is more effective than writing things in terms of infinite series (Newton's preference).
One important conclusion from the preceding five files is that there are situations in which Newton's preferred way of looking at functions is better than Liebniz` way,  and vice versa.
Relevant directory files for Unit 14:
Here is a listing of files from Unit 14 which are related to the May 29 lecture, with comments on their significance.  These links are all clickable. 
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14.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/week8/unit11/catenary+parabola.png   This file was previously included, but it was mentioned here in connection with the work of the Bernoullis
https://www.youtube.com/watch?v=LHttKi-QIW8    A YouTube video on the various members of the Bernoulli family, who made numerous contributions to mathematics and physics during the late 17th century and most of the 18th century.    Material from this video may appear on the final exam.
http://math.ucr.edu/~res/math153-2020/weekTEN/unit14/history14e.pdf   This file discusses some of the applications of calculus to questions in algebra and number theory.  The first item is the sequence of Bernoulli numbers, which is named after the first John Bernoulli.