From Spivak’s famous book Calculus. Chapter 14 from the book Calculus by Michael Spivak is called “The Fundamental Theorem of Calculus”. I’m not sure why it's written in singular like that since there are two fundamental theorems. In fact, this chapter presents us with the first fundamental theorem of calculus…

Simple yet so deep, complex, and nuanced. When I initially decided to tackle solving all problems from Spivak’s famous book Calculus, the first milestone I set was completing the chapter on the fundamental theorems of calculus. It had occurred to me while studying physics that though I could use the…

Irrational numbers, four functions, and a very, very important number. I’ve always found the logarithm and exponential functions to be in the category of concepts that seem simple enough on the surface, but that can be quite tricky at times when you try to reason about them. Recently I went…

Sometimes You Need Creativity to Discover a Function One of the (many) interesting aspects of problems in Spivak’s famous book Calculus is that you are introduced to and asked to find (“discover” somehow seems like the more apt word here) functions which (to me at least) seem quite esoteric. I’d…

A Rabbit Hole Filled With Supremums and Infimums — Usually, when you take calculus for the first time you become aware that there are theoretical underpinnings which are omitted from the material covered in your textbook or your lectures. Such courses focus on introducing concepts (in particular those of a derivative and an integral) and on computations involving such…

Let me show you what a nuanced solution to such a problem looks like for me. In Spivak’s famous book Calculus, some of the problems are marked with either one or two asterisks, and these indicate a relatively higher level of difficulty. …

An intuitive theoretical explanation of why mathematical induction works Mathematical induction is typically used to prove statements of the form

A tricky but reasonable affair! I’d like to show you basic usage of two plotting commands in Maple by solving a relatively difficult problem to reason about and visualize (at least for some of us!). The commands are implicitplot and plot3d. Each of these Maple commands (ie functions) has multiple…

376 hours, 293 problems, 335 pages written In life, many times memories of challenging times are sweet, or at least bittersweet. …