Warmup What’s a $dx$?
Reflection p. 77 “Thanks to the real numbers, we can measure the length of any line, even irrational lines such as the diagonal of the unit square.”
Why do we need permission to measure something? How do the real numbers provide it? Are they needed?
How do you measure the area of a rectangle, triangle, generic polyhedron? How about a circle? Why is how to measure things a hard question?
Arbitrarily Small Pieces
This concept as you use it in Calculus class isn’t about a set of things. You don’t grab hold of them or use them. Instead, when invoking them a scenario is described.
Warmup With a partner, take turns assigning real numbers and picking smaller ones than the assigned size.
Draw a triangular prism cut from a rectangular parallelepiped.
Impossible warmup Try to find the volume of a tetrahedron using geometry and algebra. Try with calculus.
Recall Find the volume of a tetrahedron using the geometric series.
Recall Find and prove the sum of the general geometric series $a+ar+ar^2 +\ldots$.
Why does Stillwell say it is more cautious to use the limit of partial sums instead of the algebraic trick directly on the infinite series?*
Knowing the Unknowable
What is $\pi$ and how do we know it? Using the analogy of the three blind men and the elephant, we know something about $\pi$ every time we have a different way to see it. What different ways of seeing it are there?
Archimedes not only approximated $\pi$ (though no one called it that then. When did people start calling it by its modern name? How did this happen?) he established a common ratio to the measurement of area and circumference of the circle. Stillwell recounts this and hints at the proof.
Problem Follow Archimedes and Stillwell. Use Eudoxus’ method of exhaustion to show that the area of a circle is the same as a triangle with height equal to the radius and base equal to the circumference of the circle. Hint: Check JTG Chapter 4.
Read the entry for [$\pi$ on Wikipedia][], especially the history. Take some notes and ask some questions.
![Progress in computing $\pi$ (from Wikipedia)][computing pi]
This progress looks like it gets quick around 1950. Then realize the graph is logarithmic.
Discussion What is the point of all this?
Research How many digits does one need for various practical purposes? How many are in use in various places?
Research What characterizes computational advances, particularly in the computer age beyond “Duh, computers are fast”?
Research What does your calculator do? Wolfram alpha?
Reflect How do you check the answer to a calculation of $\pi$ to a quadrillion decimal places?
Research What is $\tau$ in relation to $\pi$ and why does it matter?
Warmups
Use differentials to compute tangents to curves other than the parabola.
Use the game of suggesting arbitrarily small numbers to see how close $2x+dx$ is to $2x$.
Discuss Why are the Marquis L’Hôpital’s arguments “indefensible” as suggested on p. 91?
Research Examine Fermat’s concept of adequality from a perspective of scientific or technological change. It is an interesting example because we would today consider it a better approach than what persisted, Fermat was not unknown, and yet adequality was not adopted then.
Warmup State briefly the fundamental theorem of calculus.
Problem Unpack Stillwell’s argument about the arc length of the circle starting on p. 96. What is an arctan? Where are $y$ and $x$?
Computing $\pi$
We have seen a number of ideas for methods of computing $\pi$. But we have not actually used them to compute $\pi$.
Problem Implement these by hand without using a calculator. Then use SAGE to implement them.
Reflect Compare them as technologies. Consider particularly efficiency by hand and computer, ease of implementation from concept, and elegance.
Reflect Find a a solid body with finite volume and infinite surface area. What other impossible objects can you find/create using calculus?
Further Topics About Analysis
Research Stillwell claims on p. 99 that “Berkeley’s criticism stung, and mathematicians tried to answer it, though for a long time without much success.” Find mathematicians’ responses to Berkeley.
Research Find out more about nonstandard analysis.
Research Find out more about using spaces of functions as mathematical objects.
