京都大学 情報学研究科 知能情報学専攻 2024年8月実施 情報学基礎 F1-2

Author

itsuitsuki

Description (English)

In the questions below, denotes the natural logarithm of , and denotes Napier's constant (the base of the natural logarithm).

Q.1

Answer the following questions. Derivations must be clearly shown.

(1) Let be a positive integer. Compute the -th derivative of .

(2) Compute the following limit.

Q.2

Compute the volume common to a sphere and a cylinder , . Derivation must be clearly shown.

Q.3

Answer the following questions.

(1) Show that the inequality

holds for any positive integer .

(2) Show that is an irrational number using the inequality in (1).

Kai

Q.1

(1)

Since

let the -th derivative be

where

Then

and therefore

so

Hence

(2)

詳しく書いてくれる方が居って頂ければ幸いです。自分が時間不足なので暫く略します。 とする。

Q.2

詳しく書いてくれる方が居って頂ければ幸いです。自分が時間不足なので暫く略します。

cylinder coordinates and

Q.3

(1)

Since ,

we have .

And since

and since

we can conclude

(2)

Assume is rational, setting it where and .

Then

where we set , and multiply these with ,

with should be integer but it is in by the inequality chain above, thus causing contradiction.

Hence no such exists, making irrational.