Calculus Fun Facts [B]

When defining a Riemann sum, does it suffice to consider only regular partitions? If so, is there any reason to use non-regular partitions? Is there a way to develop MacLaurin polynomials (with error bounds!) without knowing anything about infinite or alternating series? Why do we use the ∂ symbol to denote the boundary of a region as well as a partial derivative? And more!

Linked above is a PDF of the Beamer slides from the presentation in handout mode. See below for direct links to the Desmos graphs.

Solving calculus problems without calculus:

• Desmos graph of unit circle
• How to find tangent lines without calculus (James Tanton May 2018 Curriculum Essay)
• Computing `int_(-pi)^(pi) sin^2(x) \ dx ` without calculus
• Why `int_0^pi sin(x) \ dx = 2` 

Regular partitions suffice:

• Do regular partitions suffice for Riemann integrability? (math.stackexchange thread)
• Elements of Real Analysis by Charles Denlinger (pp. 380–381)
• "Partitions of the Interval in the Definition of Riemann's Integral," Jinteng Tong, Journal of Math. Educ. in Sc. and Tech. 32 (2001), pp. 788–793

A useful irregular partition:

• Sums of Powers of Positive Integers (MAA)
• Sum of Squares Formula (James Tanton video)
• How Fermat computed `int_0^a x^p \ dx ` (from A History of Mathematics by Carl Boyer, 1968, pp. 384–385)
• Desmos graph of a useful irregular partition
• The p = -1 case (math.stackexchange thread)

Making `x^a ` meaningful:

• Visualizing `L(x) ` using a slope field

Why we use that `partial ` symbol for boundaries:

• Terry Tao comment on Math Overflow

Other cool stuff not mentioned in the talk:

• "A Tour of the Calculus Gallery" by William Dunham (MAA Monthly, January 2005)