题目

Problem

The curve $C$ has parametric equations

x=\frac{2}{3}t^{\frac{3}{2}}-2t^{\frac{1}{2}},\qquad y=2t+5,\qquad 0<t\le3.

(a) Find the exact length of $C$ .

[5]

(b) Find the set of values of $t$ for which $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}>0$ .

[5]

题目中文翻译

曲线 $C$ 的参数方程为

x=\frac{2}{3}t^{\frac{3}{2}}-2t^{\frac{1}{2}},\qquad y=2t+5,\qquad 0<t\le3

(a) 求曲线 $C$ 的精确长度。

(b) 求满足 $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}>0$ 的 $t$ 值集合。

CIE 9231 2023 November Paper 23 Q5