こんにちは。数IIIやってると頻繁に出てくるグラフってあるんです。ですから, ある程度記憶しておくと便利かなと思って書いておきます。今回はその対数関数編です。
のグラフ
とおくと,
とすると,
である。
これらをもとにグラフを描くと, 以下のようになる。
極値に関して
![Rendered by QuickLaTeX.com x=\dfrac1e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ca97213da0a18d13f19548efc5e3d9d1_l3.png)
![Rendered by QuickLaTeX.com -\dfrac1e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1119f5c970910cb3e141bd12ed4015d7_l3.png)
極大値なし
変曲点なし
漸近線なし
のグラフ
とおくと,
なので,
とすると,
,
とすると,
である。
これらをもとにグラフを描くと, 以下のようになる。
極値に関して
![Rendered by QuickLaTeX.com x=e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cfac656029e5f37d5a3136139ca282e5_l3.png)
![Rendered by QuickLaTeX.com \dfrac1e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-702249f607145a7aa682337cb6d6a53a_l3.png)
極小値なし
変曲点
![Rendered by QuickLaTeX.com \left(e^{\frac32}, \dfrac32e^{-\frac32}\right)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-04ea6d6ca77d689fada53e6fcaf78b6b_l3.png)
漸近線
![Rendered by QuickLaTeX.com x=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4c2c95e89d8f5c6e0e66407c3fe20028_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to+0}\dfrac{\log x}{x}=-\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-65ab5683067cf71e680185578ec2caeb_l3.png)
![Rendered by QuickLaTeX.com y=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f6f465e0f12abc1eee8e266b91926f96_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to\infty}\dfrac{\log x}{x}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2d546f0c8f57638692966af36f8813f4_l3.png)
のグラフ
とおくと,
なので,
とすると,
とすると,
これらをもとにグラフを描くと, 以下のようになる。(描画の都合上極値などが示せていない)
極値に関して
![Rendered by QuickLaTeX.com x=e^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-263ddeb2d22585565601014f8a8289dd_l3.png)
![Rendered by QuickLaTeX.com \dfrac2e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-aac52a76f981bdf69a017d23003eb910_l3.png)
極小値なし
変曲点
![Rendered by QuickLaTeX.com \left(e^{\frac83}, \dfrac83e^{-\frac43}\right)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a9dc3e051b673f8a6a730309474f9c3d_l3.png)
漸近線
![Rendered by QuickLaTeX.com x=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4c2c95e89d8f5c6e0e66407c3fe20028_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to+0}\dfrac{\log x}{\sqrt{x}}=-\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-27c4849427108123318b0b83892d8a97_l3.png)
![Rendered by QuickLaTeX.com y=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f6f465e0f12abc1eee8e266b91926f96_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to\infty}\dfrac{\log x}{\sqrt{x}}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0d75f75563e567e7f2e1bcecf73b8094_l3.png)
のグラフ
とおくと,
なので,
は略
とすると,
これらをもとにグラフを描くと, 以下のようになる。(描画の都合上極大値が示せていない)
極値に関して
![Rendered by QuickLaTeX.com x=e^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-263ddeb2d22585565601014f8a8289dd_l3.png)
![Rendered by QuickLaTeX.com \dfrac{4}{e^2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5bab17a4a27b25895e23ecae5332d1be_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
変曲点はあるが特に覚えなくてよい。
漸近線
![Rendered by QuickLaTeX.com x=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4c2c95e89d8f5c6e0e66407c3fe20028_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to+0}\dfrac{\left(\log x\right)^2}{x}=\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c3ac0c8e061e4c144eaccfe34cba677b_l3.png)
![Rendered by QuickLaTeX.com y=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f6f465e0f12abc1eee8e266b91926f96_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to\infty}\dfrac{\left(\log x\right)^2}{x}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-38889687c5db9b22ceb97e0d04763412_l3.png)
のグラフ
とおくと,
なので,
は略
とすると,
これらをもとにグラフを描くと, 以下のようになる。(描画の都合上極大値が示せていない)
![Rendered by QuickLaTeX.com x=e^4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-32739602b3bc291c979f03fca2f108e7_l3.png)
![Rendered by QuickLaTeX.com \dfrac{16}{e^2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1f0c27ffa6c654d5ac9adff3cdd8fc51_l3.png)
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
変曲点はあるが特に覚えなくてよい。
漸近線
![Rendered by QuickLaTeX.com x=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4c2c95e89d8f5c6e0e66407c3fe20028_l3.png)
![Rendered by QuickLaTeX.com y](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d76ceac31cb52dd9eb4431a14c502dc_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to+0}\dfrac{\left(\log x\right)^2}{\sqrt{x}}=\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cd7a82fc5ce85313a59290c29588a1e5_l3.png)
![Rendered by QuickLaTeX.com y=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f6f465e0f12abc1eee8e266b91926f96_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to\infty}\dfrac{\left(\log x\right)^2}{\sqrt{x}}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-58288ce0b999564f04937f10edfeadb7_l3.png)
のグラフ
とおくと,
なので,
とすると,
とすると,
これらをもとにグラフを描くと, 以下のようになる。
極値に関して
![Rendered by QuickLaTeX.com x=e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cfac656029e5f37d5a3136139ca282e5_l3.png)
![Rendered by QuickLaTeX.com e](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c55cd1118b31908836c0ba5f222b44f6_l3.png)
極大値はなし。
変曲点
![Rendered by QuickLaTeX.com \left(e^2, \dfrac{e^2}{2}\right)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d42b5029ed47d6f094e3649929ed6253_l3.png)
漸近線
![Rendered by QuickLaTeX.com x=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8a4b1fabb9dd2ab5aece4bfdff8f5021_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to1+0}\dfrac{x}{\log{x}}=\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5e357f6955ff7cd79a3f81dbcc981617_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to1-0}\dfrac{x}{\log{x}}=-\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-72c58fd7fbfdd92d5502bc57b0dd6a93_l3.png)
ちなみに
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to\infty}\dfrac{x}{\log{x}}=\infty](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ff9f7d99de7cb52d02f28aef81ca346e_l3.png)
![Rendered by QuickLaTeX.com \displaystyle \lim_{x\to+0}\dfrac{x}{\log{x}}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-754c3e598e62d5dfd63e87137e9406a1_l3.png)