Have you seen my latest blunder? Here it is:
To magnify my shame, here is the video version:
The goal was to solve this limit:
I received an email the other day — a response to the antiderivative challenge. Enjoy it. And ask yourself: what does maths do for you?
What is it that drives folks … to delight in taking five lines to solve a maths question that we could solve in one line?
Then read further.
Is it possible to solve ∫(1/t)dt using the reverse power law?
We can write the problem as follows:
The Power Rule for derivatives is one of the first tricks we learn in Calculus I. It’s such a refreshing alternative to using the limit definition.
Let’s start with a question. For the graph of xⁿ, what is the slope at a given point?
For example, f(x) =x². We can represent f(x) as the area of a square with sides, x. We increase each side by an infinitesimal, dx.
Yesterday (October 30, 2020), YouTuber Steven Chow (blackpenredpen) posted a video with this goal: Integrate x² without the power rule. Before seeing Steve’s solution, I posted my own video. (I didn’t want to be unduly influenced.)
Have a crack at it yourself. Then see how I solved it (below).
You want to find the roots of a quadratic polynomial. Good news! The method of completing the square always works. What makes that so, and what other information can we glean from this method?
Let’s find out with an example. We’ll find the roots of:
This derivation of the power rule does not use limits. Its inspiration is Abraham Robinson’s non-standard analysis and hyper-real numbers.
If you learned a derivation of the power rule, it probably started something with a definition of the derivative: