May 2024 – Page 3 – The Renewal Equation

How To?

Olly / May 6, 2024May 6, 2024 / Updates

Today I read part of the next section of my calculus text, having finished the exercises from the previous section on Saturday. (I would have liked to read more, but the day was very busy in other ways. Sigh.)

I’ve been thinking quite a bit about the best way to organize my calculus review. Continuing to do all of the exercises in every section would be excessive, I think, and it would also take a very long time. I’m not sure of the best alternative. One option is to do all of the odd-numbered exercises in each section. That would be an improvement, but I think it might still be too much. Another idea is to read all of the sections in a chapter, then do the chapter review exercises provided at the end, perhaps going back to the section exercises for more practice on certain concepts. I think this may be what I will try first. I’m also open to suggestions, however.

Day 100

Olly / May 4, 2024May 5, 2024 / Updates

This is my 100th daily post. Hooray!

Looking back, I have made both less and more progress than I anticipated. Almost all of the calculus review remains to do, and it’s clear that part of the project is going to be a long haul. On the other hand, my work on calculus prerequisites and logic has been fruitful, and quite a bit of geometry knowledge has returned to me without deliberate review. More importantly, my ability to think mathematically and my joy in doing so have both resurged far beyond my expectations. Overall, I think the project has been a success so far.

I plan to keep going as I have been, for the most part. I’ve decided to begin taking Sundays off from blogging, however, and relaxing my study goals for that day. I think a weekly rest will help me sustain the project long term.

So see you on Monday, faithful readers. Thank you for all your support during these first 100 days.

Pro Forma

Olly / May 3, 2024May 3, 2024 / Updates

I did more calculus exercises today despite also sleeping quite a bit. I’m afraid I don’t have anything fun to share, however.

Graph of the Day

Olly / May 2, 2024May 2, 2024 / Math Posts

Today I spent two hours on calculus exercises. Among other things, this involved drawing thirteen graphs.¹ That part was a bit tiresome, but I did learn from it. My favorite was the graph of $$f(x) = \frac{3x+|x|}{x}\text{,}$$ which didn’t look at all the way I had imagined before thinking it through. Click here to see the graph.

To do in comments. ↩︎

Sphere versus Cube

Olly / May 1, 2024May 2, 2024 / Math Posts

Today I was able to spend more than an hour on calculus exercises. (Yay!) Now I am pondering the following:

The volume of a sphere of radius $r$ is given by $V=\frac{4}{3}\pi r^3$. The surface area of the same sphere is given by $A=4\pi r^2$, the derivative of the formula for the volume. This makes sense, as a small change in volume involves adding a thin layer to the surface of the sphere. In the limit, that layer will be identical to the sphere’s surface.

On the other hand, the surface area of a cube of side length $x$ is given by $A=6x^2$. This is not the derivative of $V=x^3$, the formula for the cube’s volume. Why?¹

To do. ↩︎