一条曲线的方程为 $y=\tan^{-1}(4x)$。

(a) 求该曲线的斜率为 $\frac14$ 时 $x$ 的精确值。

(b) 求 $\displaystyle\int_0^{0.25}y\,\mathrm{d}x$ 的精确值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

反正切求导后把斜率设为四分之一，解平方方程时保留正负两个根。

Since $y=\tan^{-1}(4x)$,

Setting this equal to $1/4$,

Hence

把反正切关系改写成正切后隐式求导，也可求出逆导数。

From $\tan y=4x$, differentiating with respect to $y$ gives

Using $\tan y=4x$,

When $\mathrm{d}y/\mathrm{d}x=1/4$, the reciprocal derivative is $4$. Hence

分部积分让反正切被求导。剩下的一次分子与分母导数成比例，产生自然对数。

Take $u=\tan^{-1}(4x)$ and $\mathrm{d}v=\mathrm{d}x$. Then $\mathrm{d}u=4\,\mathrm{d}x/(1+16x^2)$ and $v=x$. Integration by parts gives

The logarithmic term uses $w=1+16x^2$, so $\mathrm{d}w=32x\,\mathrm{d}x$ and the remaining integral is $\frac18\int w^{-1}\,\mathrm{d}w$. The lower-limit contribution is zero. At $x=1/4$, $\tan^{-1}(4x)=\pi/4$, giving

用 $4x=\tan u$ 把积分改成 $u\sec^2u$，再分部积分。

Let $4x=\tan u$. Then

The limits change from $x=0$ and $x=1/4$ to $u=0$ and $u=\pi/4$, respectively. On this interval, $\cos u>0$, and $\int\tan u\,\mathrm{d}u=-\ln(\cos u)+C$.

Thus