こんにちは。数III頻出の減衰曲線についてまとめてみました。ここでは, についてですが,
,
,
についても同様ですので, 以下を参考にしてみてください。グラフは実際の関数
をそのまま書いたのでは分かりにくくなるため, 変数を調整して書いています。実際のテストではここに書いてあるグラフのように, 特徴を強調して書いても大丈夫です。最後に面積についての問題を載せています。どうぞ楽しんでください。
として,
の範囲でグラフを描いてみる。
第2次導関数まで求めてみると,
ここで, より,
となる
の値を考えると,
より,
となる
の値は,
である。(もちろん三角関数の合成から求めてもよい)
となる
の値を考えると,
より,
となる
の値は,
である。
これらをもとにグラフの概形を書くと, 以下のようになる。
極値に関して
![Rendered by QuickLaTeX.com x=\dfrac{\pi}{4}, \dfrac{9}{4}\pi, \dfrac{17}{4}\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a5d8b76e1e5bbf244c6a89a001a0ffea_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \dfrac{\pi}{4}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-96d0ea0dd5de99724c2fb375f21df81f_l3.png)
![Rendered by QuickLaTeX.com 2\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-782e019267fd6fd3c4330c15db8f7e68_l3.png)
極大値は左から順に,
![Rendered by QuickLaTeX.com \dfrac{1}{\sqrt2}e^{-\frac{\pi}{4}}, \dfrac{1}{\sqrt2}e^{-\frac{9}{4}\pi}, \dfrac{1}{\sqrt2}e^{-\frac{17}{4}\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b94e563b6966620466ffea91307f0b1d_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com \dfrac{1}{\sqrt2}e^{-\frac{\pi}{4}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d1998dff07fd115acf93f15f30c81261_l3.png)
![Rendered by QuickLaTeX.com e^{-2\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-06736bb6b1b0d319dfa50b15b28ef496_l3.png)
![Rendered by QuickLaTeX.com x=\dfrac{5}{4}\pi, \dfrac{13}{4}\pi, \dfrac{21}{4}\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5493e959da68b2833f031e159b8f9ece_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \dfrac{5}{4}\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c40c471ae3e4273cd9addde79b0be830_l3.png)
![Rendered by QuickLaTeX.com 2\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-782e019267fd6fd3c4330c15db8f7e68_l3.png)
極小値は左から順に,
![Rendered by QuickLaTeX.com -\dfrac{1}{\sqrt2}e^{-\frac{5}{4}\pi}, -\dfrac{1}{\sqrt2}e^{-\frac{13}{4}\pi}, -\dfrac{1}{\sqrt2}e^{-\frac{21}{4}\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1199c09fdb73e770f3667b505c3e378e_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com -\dfrac{1}{\sqrt2}e^{-\frac{5}{4}\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-7ba32e588d1fb3cb4cc3567ef4bca251_l3.png)
![Rendered by QuickLaTeX.com e^{-2\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-06736bb6b1b0d319dfa50b15b28ef496_l3.png)
変曲点の
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x=\dfrac{\pi}{2}, \dfrac{3}{2}\pi, \dfrac52\pi, \dfrac72\pi, \dfrac92\pi, \dfrac{11}{2}\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-afd91c499277746e71dea3c5ccf86086_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com x=\dfrac{\pi}{2}+(n-1)\pi\ (n=1, 2, 3, 4, 5, 6)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1b3f20461e710871ad63ad7cfb9b6b6b_l3.png)
座標は左から順に
![Rendered by QuickLaTeX.com \left(\dfrac{\pi}{2}, e^{-\frac{\pi}{2}}\right), \left(\dfrac32\pi, -e^{-\frac32\pi}\right), \left(\dfrac52\pi, e^{\frac52\pi}\right), \left(\dfrac72\pi, -e^{-\frac72\pi}\right), \left(\dfrac92\pi, e^{-\frac92\pi}\right), \left(\dfrac{11}{2}\pi, -e^{-\frac{11}{2}\pi}\right)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b76fbf0b1e4c22e193b5332c4fd33489_l3.png)
![Rendered by QuickLaTeX.com \bullet\,](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e24e17238759f68d54354dd3ed64127d_l3.png)
![Rendered by QuickLaTeX.com \left(\dfrac{\pi}{2}+(n-1)\pi, (-1)^{n-1}\cdot e^{\frac{\pi}{2}+(n-1)\pi}\right)\ (n=1, 2, 3,\cdots)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fa99878951ba54383ba8b99f6132fe86_l3.png)
減衰のしかたは, 曲線
![Rendered by QuickLaTeX.com y=e^{-x}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fb4009e9690e0590275911c39b7c197b_l3.png)
![Rendered by QuickLaTeX.com y=-e^{-x}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6ba0719dbeedab78221887638ad4d838_l3.png)
以上の特徴は暗記しておくと, のちのち便利です。
曲線と
軸との交点を原点Oから正の方向に順に
,
,
,
とする。
(1) この曲線と線分とで囲まれた部分の面積
を求めよ。
(2) を求めよ。
【東京女子大】
求める面積は
![Rendered by QuickLaTeX.com n\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-247e8f5c969525b056c6b7f4da04b0f0_l3.png)
![Rendered by QuickLaTeX.com (n+1)\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-ef427ef78860bfd37621fd4c3ee07111_l3.png)
したがって,
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_n&=&\left|\displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\sin x\, dx\right|\\&=&\displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\left|\sin x\right|\, dx\end{array}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b8b423ca6b4274a60f66538b90ccc2a8_l3.png)
![Rendered by QuickLaTeX.com x-n\pi=t](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1d2c0bf5948ac4b811752ad93fe5b24b_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{c|ccc} x &n\pi&\to&(n+1)\pi\\ \hline t &0&\to&\pi\end{array}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-196a758bec7163d53b38136e3fa37b87_l3.png)
また,
![Rendered by QuickLaTeX.com x=n\pi+t](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-bf0e039c8319bdeb8e156a431b413d47_l3.png)
![Rendered by QuickLaTeX.com dx=dt](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5d4f5ef0b2464816d39dce6a3975675d_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_n&=&\displaystyle\int_0^{\pi} e^{-n\pi-t}\left|\sin(n\pi+t)\right|\,dt\\&=&e^{-n\pi}\displaystyle\int_0^{\pi}e^{-t}\left|(-1)^n\sin t\right|\,dt\\&=&e^{-n\pi}\displaystyle\int_0^{\pi}e^{-t}\sin t\,dt\end{array}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-07f3f6587cdff666aec3c3bb614ef7ea_l3.png)
![Rendered by QuickLaTeX.com \sin x](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-83376fe5b9be200d8335f3ca7c2c294e_l3.png)
![Rendered by QuickLaTeX.com 0\leqq t\leqq\pi](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-7a0f89aa84eb15728a5d2d517efcbd80_l3.png)
![Rendered by QuickLaTeX.com \sin t\geqq0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-864a32a4a7a0a35016407a538739ab26_l3.png)
これは
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com e^{-\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5531a38a60838d1c78caef51120cc106_l3.png)
ここで,
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com S_0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-1a14941e5fd4cba60bc6c1f074271712_l3.png)
![Rendered by QuickLaTeX.com S_0&=&\displaystyle\int_0^{\pi}e^{-t}\sin t\,dt](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6c4504d0197e56427ec273f49caa5530_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_0&=&\displaystyle\int\left(-e^{-t}\right)'\sin t\, dt\\&=&\left[-e^{-t}\sin t\right]_0^{\pi}+\displaystyle\int_0^{\pi} e^{-t}\cos t\, dt\\&=&\displaystyle\int_0^{\pi} e^{-t}\cos t\, dt\\&=&\left[-e^{-t}\cos t\right]_0^{\pi}-\displaystyle\int_0^{\pi}e^{-t}\sin t\, dt\\&=&e^{-\pi}+1-S_0\end{array}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-083549ed53f2b7b8860ca9e6fa178231_l3.png)
これより,
![Rendered by QuickLaTeX.com S_0=\dfrac{e^{-\pi}+1}{2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5dfac092b1df471a381b7bf2db0442be_l3.png)
![Rendered by QuickLaTeX.com S_n=S_0e^{-n\pi}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-7826eae7224cb0c1f3c3f7408332ce93_l3.png)
![Rendered by QuickLaTeX.com S_n=\dfrac{e^{-\pi}+1}{2}e^{-n\pi}\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d8c5102d7cf7f114a077cf6aa79716d1_l3.png)
(2)
![Rendered by QuickLaTeX.com 0<e^{-\pi}<1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-0dcc5e279a5d81d2e8fe83e197f63a1b_l3.png)
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
よって,
![Rendered by QuickLaTeX.com \begin{array}{lll}\displaystyle\sum_{n=0}^{\infty}S_n&=&\dfrac{e^{-\pi}+1}{2}\cdot\dfrac{1}{1-e^{-\pi}}\\&=&\dfrac{1+e^{\pi}}{2(e^{\pi}-1)}\end{array}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-0d0d73d41caec80fe0beb2f476f1951d_l3.png)
![Rendered by QuickLaTeX.com \dfrac{1+e^{\pi}}{2(e^{\pi}-1)}\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-0c981c6b15c13058e0dd4b632902f75f_l3.png)
このように面積は等比数列になり, その無限級数和は収束します。