# Calculus Fun Facts [B]

**Jeff Eldridge • Edmonds Community College**

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.

**References**

**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)

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