グローバルビジネスで役立つ数学（20）極限値の問題（英語版）

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
【レッスン】私のオンライン英語レッスンをご希望の方はこちらをご覧ください。
【コンテント】当サイトで提供する情報はその正確性と最新性の確保に努めていま
　すが完全さを保証するものではありません。当サイトの内容に関するいかなる間
　違いについても一切の責任を負うものではありません。