Hi, there!
As is often the case, telecommuting has become popular even among
my online students since the COVID-19 occurred. It was the apparent
result, for they had no choice but to keep social distancing. This comes
about as a corollary regardless of their favorite commuting style.
In Osaka, a two-hour commute from a bedroom community to a down-
town office is taken for granted. People used to stand the whole way on
jam-packed trains; however, due to or thanks to the pandemic, they end-
ed up being able to work at home.
I don’t care about commuting so much that one hour to two on a train is
nothing. I doze off in my seat, except when I remember some math word
problems I’ve wanted to solve.
Today the acronym “MIT” came into my mind in commuting, and I want
to rechallenge its math exam to review a basic derivative.
Derive the formula \(\frac{d}{dx}a^{x} = M(a)a^{x}\) directly from the definition of the
derivative, and identify \(M(a)\) as a limit. (Source: Massachusetts Insti-
tute of Technology exam)
Here is my simple solution to the problem.
\(\frac{d}{dx}(a^{x}) = \displaystyle\lim_{h\rightarrow 0}(\frac{a^{x + h} – a^{x}}{h})\)
\(= \displaystyle\lim_{h\rightarrow 0}(\frac{a^{x}a^{h} – a^{x}}{h})\)
\(= \displaystyle\lim_{h\rightarrow 0}(\frac{a^{x}(a^{h} – 1)}{h})\)
\(= a^{x}\displaystyle\lim_{h\rightarrow 0}(\frac{a^{h} – 1}{h})\)
To avoid a precarious situation in my financial standing, I might want
to focus on a derivative word problem rather than being deeply com-
mitted to a financial derivative market.
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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