AP-Style Practice QuestionsAP 风格练习题
Topics主题 9.1 - 9.9BC
Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part, each question below is labeled accordingly.请在草稿纸上写出所有解题步骤。AP 考试第一部分分为不允许使用计算器和允许使用计算器两部分，每道题已注明。
A curve is defined by the parametric equations $x(t) = t^{3}$ and $y(t) = t^{2} + 5t$. What is $\dfrac{dy}{dx}$ at $t = 1$?曲线由参数方程 $x(t) = t^{3}$ 与 $y(t) = t^{2} + 5t$ 定义。求 $t = 1$ 时的 $\dfrac{dy}{dx}$。
A curve is given by $x(t) = t^{2} - 4t$ and $y(t) = t^{3} - 3t$. At which value(s) of $t$ does the curve have a horizontal tangent line?曲线由 $x(t) = t^{2} - 4t$ 与 $y(t) = t^{3} - 3t$ 给出。曲线在哪个（些）$t$ 值处有水平切线？
A curve is given by $x(t) = t^{3}$ and $y(t) = t^{2}$. Find $\dfrac{d^{2}y}{dx^{2}}$ at $t = 1$.曲线由 $x(t) = t^{3}$ 与 $y(t) = t^{2}$ 给出。求 $t = 1$ 时的 $\dfrac{d^{2}y}{dx^{2}}$。
A curve is defined by $x(t) = t^{2}+1$ and $y(t) = t^{3}-12t$. At $t = 2$, the curve is曲线由 $x(t) = t^{2}+1$ 与 $y(t) = t^{3}-12t$ 定义。在 $t = 2$ 处，曲线
Which integral gives the arc length of the curve $x(t) = t^{2}$, $y(t) = t^{3}$ for $0 \le t \le 1$?下列哪个积分表示曲线 $x(t) = t^{2}$、$y(t) = t^{3}$ 在 $0 \le t \le 1$ 上的弧长？
Find the length of the curve $x(t) = 3\cos t$, $y(t) = 3\sin t$ for $0 \le t \le \dfrac{\pi}{2}$.求曲线 $x(t) = 3\cos t$、$y(t) = 3\sin t$ 在 $0 \le t \le \dfrac{\pi}{2}$ 上的弧长。
$\mathbf{r}(t) = \langle t^{2}, \ln t \rangle$. Find $\mathbf{r}'(t)$.$\mathbf{r}(t) = \langle t^{2}, \ln t \rangle$。求 $\mathbf{r}'(t)$。
A particle's position is given by $\mathbf{r}(t)$, with $\mathbf{r}'(2) = \langle 0, 5 \rangle$. What does this tell you about the curve at $t = 2$?质点的位置为 $\mathbf{r}(t)$，且 $\mathbf{r}'(2) = \langle 0, 5 \rangle$。这说明曲线在 $t = 2$ 处具有怎样的性质？
$\mathbf{r}'(t) = \langle 4t, 2e^{2t}\rangle$ and $\mathbf{r}(0) = \langle -1, 3\rangle$. Find $x(1)$, the $x$-coordinate of $\mathbf{r}(1)$.已知 $\mathbf{r}'(t) = \langle 4t, 2e^{2t}\rangle$ 且 $\mathbf{r}(0) = \langle -1, 3\rangle$。求 $\mathbf{r}(1)$ 的 $x$ 坐标 $x(1)$。
$\displaystyle\int \langle 3t^{2}, \cos t \rangle \, dt =$
A particle moves with velocity $\mathbf{v}(t) = \langle 5, 12 \rangle$. What is the particle's speed?质点以速度 $\mathbf{v}(t) = \langle 5, 12 \rangle$ 运动，其速率是多少？
A particle has velocity $\mathbf{v}(t) = \langle 2, 3t^{2}-3 \rangle$ and $\mathbf{r}(0) = \langle 0, 0 \rangle$. Find the displacement vector of the particle on $[0,2]$.质点速度为 $\mathbf{v}(t) = \langle 2, 3t^{2}-3 \rangle$，且 $\mathbf{r}(0) = \langle 0, 0 \rangle$。求质点在 $[0,2]$ 上的位移向量。
A particle moves with velocity $\mathbf{v}(t) = \langle 3, t^{2}-4 \rangle$. Is the particle's speed increasing or decreasing at $t=1$?质点速度为 $\mathbf{v}(t) = \langle 3, t^{2}-4 \rangle$。质点在 $t=1$ 时速率是增加还是减小？
The polar curve $r(\theta) = 2 + 2\cos\theta$ has slope $\dfrac{dy}{dx}$ at $\theta = \dfrac{\pi}{2}$ equal to极坐标曲线 $r(\theta) = 2 + 2\cos\theta$ 在 $\theta = \dfrac{\pi}{2}$ 处的斜率 $\dfrac{dy}{dx}$ 等于
How many points on the cardioid $r = 1+\cos\theta$, for $0 \le \theta < 2\pi$, have a vertical tangent line?心脏线 $r = 1+\cos\theta$（$0 \le \theta < 2\pi$）上有多少个点具有竖直切线？
What is the area enclosed by the polar curve $r = 5$ for $0 \le \theta \le 2\pi$?极坐标曲线 $r = 5$ 在 $0 \le \theta \le 2\pi$ 上所围的面积是多少？
Find the area of one petal of the rose $r = \sin(3\theta)$.求玫瑰线 $r = \sin(3\theta)$ 一个花瓣的面积。
Which integral gives the area of the region inside $r = 3\sin\theta$ and outside $r = 1+\sin\theta$?下列哪个积分表示位于 $r = 3\sin\theta$ 内部且在 $r = 1+\sin\theta$ 外部的区域的面积？
Free-response answers must include setup, units, and interpretation in context where appropriate. A calculator is permitted unless marked otherwise.自由回答题须包含解题设置、单位，以及在适当时对结果的情境解释。除非特别注明，否则允许使用计算器。
A curve is defined by the parametric equations $x(t) = t^{3} - 3t$ and $y(t) = t^{2} - 4$.曲线由参数方程 $x(t) = t^{3} - 3t$ 与 $y(t) = t^{2} - 4$ 定义。
A particle moves in the plane with velocity vector $\mathbf{v}(t) = \langle 2t,\, 3e^{-t}\rangle$ for $t \ge 0$. At time $t=0$, the particle is at the point $(1,4)$.质点在平面内运动，速度向量为 $\mathbf{v}(t) = \langle 2t,\, 3e^{-t}\rangle$，$t \ge 0$。$t=0$ 时质点位于点 $(1,4)$。
A particle moves along a curve with velocity vector $\mathbf{v}(t) = \langle \cos t,\, \sin(2t)\rangle$ for $0 \le t \le \pi$.质点沿曲线运动，速度向量为 $\mathbf{v}(t) = \langle \cos t,\, \sin(2t)\rangle$，$0 \le t \le \pi$。
Consider the polar curve $r = 2 + 4\cos\theta$ for $0 \le \theta \le 2\pi$.考虑极坐标曲线 $r = 2 + 4\cos\theta$，$0 \le \theta \le 2\pi$。
Let $r_{1} = 4\cos\theta$ and $r_{2} = 2$ for $-\dfrac{\pi}{2} \le \theta \le \dfrac{\pi}{2}$.设 $r_{1} = 4\cos\theta$、$r_{2} = 2$，$-\dfrac{\pi}{2} \le \theta \le \dfrac{\pi}{2}$。