京都大学 情報学研究科 知能情報学専攻 2023年8月実施 情報学基礎 F1-2

Author

Isidore, Casablanca, 祭音Myyura

Description

設問1

以下の積分を求めよ。計算過程も明示すること。

(1)

(2) としたときに、

設問2

以下の問いに答えよ。計算過程も明示すること。

(1) の小数第 位を四捨五入し、小数第 位まで求めよ。

(2) に対して、次の不等式が成り立つことを示せ。

設問3

の条件の下で、 の最大値と最小値を求めよ。

Kai

設問1

(1)

Let , we have . Then

(2)

Let , the Jacobian determinant

Then we have

設問2

(1)

Perform a Taylor series expansion of the function , then insert . Calculating until the 5th term could lead to a result , and then round it to

(2)

(solution by Isidore)

Perform the same expansion as (1) will directly prove

Let , then its derivative is

We use Newton-Raphson's method to calculate the root of .

Starting at , the value of can be calculated as following

Therefore, insert ,

Q.E.D

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(solution by Casablanca)

Easy to see that we only need to prove that:

Let . Then we have .

• (i) for (note: )
• (ii) for , , increases

and we consider the point on , .

Let

where

and we know that: , it's obvious that increases for , and .

Hence for .

Thus

And consider the intersection of and , we have

thus for , hence for ,

and from (i) and (ii), finally ,we know that

設問3

Perform the Lagrange multipliers method, we get

Calculate its derivative with the roots, the answer is