京都大学 情報学研究科 通信情報システム専攻 2022年8月実施 専門基礎A [A-1]

Author

SUN

Description

Answer all the following questions.

(1)

Find all the local maxima and minima, and corresponding and with respect to function . Let and be real numbers.

(2)

Let be a domain bounded by and , where . Compute the following integral .

(3)

Find the length of the curve given as follows.

(4)

Find the eigenvectors of matrix , and show the conditions on which they become orthogonal to each other. Let be a real number.

Kai

(1)

To find critical points, set partial derivatives to zero:

From , we have or .

• If , then . Points: .
• If , then . Points: .

Second-order partial derivatives:

Hessian determinant :

• For (Saddle point)
• For (Saddle point)
• For . Since , it is a local minimum:

• For . Since , it is a local maximum:

(2)

The domain is bounded by and , which intersect at and . For , the range of is .

Using integration by parts:

Evaluating from 0 to 1:

(3)

The arc length for from to :

Let , then . When ; when .

(4)

Characteristic equation :

Eigenvectors for :

Eigenvectors for :

For and to be orthogonal: