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

标签：

• Kyoto-University
• Implicit-Differentiation
• Limit
• Parametric-Differentiation
• Arc-Length
• Definite-Integral

Author

祭音Myyura (co-authored with GPT 5.6 SOL)

Description

1. Find the angles of intersection of the curves

   $$2x^2+y^2=20,\qquad 4y^2-x^2=8.$$

2. Evaluate

   $$\lim_{x\to1}x^{1/(x-1)}.$$

3. A circle of radius $1$ rolls once, without slipping, around the outside of a fixed circle of radius $4$. Initially, the marked point $p$ is the outermost point of the small circle.

   1. Sketch the curve traced by $p$.
   2. Find its length.
   3. Find the area enclosed by it.

题目描述

1. 求曲线 $2x^2+y^2=20$ 与 $4y^2-x^2=8$ 的交角。
2. 计算 $\displaystyle\lim_{x\to1}x^{1/(x-1)}$。
3. 半径为 $1$ 的小圆在半径为 $4$ 的静止大圆外侧无滑动地滚行一周。初始时小圆上的定点 $p$ 位于最外侧。
   1. 画出 $p$ 的轨迹；
   2. 求轨迹长度；
   3. 求轨迹所围成的面积。

Kai

Q.1

Solving the two equations gives

$$y^2=4,\qquad x^2=8,$$

so the intersections are $(\pm2\sqrt2,\pm2)$. Implicit differentiation gives the tangent slopes

$$m_1=-\frac{2x}{y},\qquad m_2=\frac{x}{4y}.$$

At every intersection,

$$m_1m_2=-\frac{x^2}{2y^2}=-1.$$

Thus the curves meet orthogonally, and every angle of intersection is

$$\boxed{\frac{\pi}{2}}.$$

Q.2

Let the limit be $L$. Then

$$\log L =\lim_{x\to1}\frac{\log x}{x-1} =1.$$

Therefore

$$\boxed{L=e}.$$

Q.3

Let $t$ be the polar angle of the small circle's center. Its center is

$$C(t)=(5\cos t,5\sin t).$$

The no-slip condition makes the small circle rotate through angle $5t$. Since $p$ initially points outward, its trajectory is the four-cusped epicycloid

$$\boxed{ x(t)=5\cos t+\cos5t,\qquad y(t)=5\sin t+\sin5t,\quad 0\le t\le2\pi }.$$

(1)

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The cusps are at radius $4$ and polar angles $\pi/4,3\pi/4,5\pi/4,7\pi/4$; the initial point is $p=(6,0)$.

(2)

Differentiating,

$$\|\gamma'(t)\| =5\sqrt{2+2\cos4t} =10|\cos2t|.$$

Hence

$$L=\int_0^{2\pi}10|\cos2t|\,dt =\boxed{40}.$$

(3)

Since

$$xy'-yx'=30(1+\cos4t),$$

Green's theorem yields

$$\begin{aligned} S&=\frac12\int_0^{2\pi}(xy'-yx')\,dt\\ &=15\int_0^{2\pi}(1+\cos4t)\,dt =\boxed{30\pi}. \end{aligned}$$