一条曲线满足 $\dfrac{\mathrm{d}y}{\mathrm{d}x}=12\big(\tfrac{1}{2}x-1\big)^{-4}$。已知曲线经过点 $P(6,4)$。

(a) 求曲线在 $P$ 点处的切线方程。

(b) 求曲线的方程。

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。

把切点的横坐标代入导数求斜率，再用点斜式。

At $x=6$, the gradient is $12(3-1)^{-4}=3/4$. The tangent through $(6,4)$ is $y-4=\tfrac34(x-6)$, or $y=\tfrac34x-\tfrac12$.

积分时把括号当作整体，并除以括号内的斜率。再用 $P$ 确定常数。

On the branch through $P$, integration gives

since the derivative of the power contributes the factor $(-3)(1/2)$. At $x=6$ this gives $4=-1+C$, so the equation on this branch is $y=5-8(\tfrac12x-1)^{-3}$.