一条曲线的方程满足 $\frac{\mathrm{d}y}{\mathrm{d}x}=3(4x-7)^{\frac12}-4x^{-\frac12}$。已知曲线经过点 $\big(4,\frac52\big)$。

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。

对括号幂积分时要除以括号内的斜率。再用给定点确定积分常数。

Integrating the two terms gives

At $x=4$, the first two terms sum to $27/2-16=-5/2$. The given height is $5/2$, so $C=5$. Hence $y=\tfrac12(4x-7)^{3/2}-8\sqrt x+5$.