京都大学情報学研究科知能情報学専攻 2024年2月実施専門科目 F1-2

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Consider the following functions and .

(1) Evaluate whether and are differentiable, respectively. (2) Evaluate whether their derivatives and are continuous at , respectively.

Evaluate the directional differentiability and continuity of the following function.

Compute the volume in the -space for each of the following conditions.

(1) Under the surface and over the rectangle defined by on the -plane. (2) Under the surface of the paraboloid and over the -plane.

(1)

does not exist. Hence is not differentiable.

by Squeezing theorem. Hence is differentiable.

(2)

At , does not exist, so it is not continuous.

From (1), and does not exist, so is also not continuous at .

We have and . So is directionally differentiable at and axes.

and

limit of along different paths to is different, which means the limit does not exist. Hence is not continuous.

(1)

(2)

The partial surface to be integrated is

Let , and .