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.请将所有辅助步骤写在草稿纸上。AP 考试第一部分分为不可使用计算器和可使用计算器两个部分，每道题已标注对应类型。
Oil is pumped into a tank so that $V(t)=10t-t^{2}/4$ gallons at time $t$ minutes. At $t=4$, oil enters the tank at石油被泵入储罐，使得 $t$ 分钟时罐内油量为 $V(t)=10t-t^{2}/4$ 加仑。在 $t=4$ 时，石油注入储罐的速率为
Q5MEDIUM4.4 Related Rates (Parametric)4.4 相关变化率（参数形式）No Calculator
A spherical balloon is inflated so that its volume increases at a constant rate of $k$ cm³/s. At the instant when its radius is $a$ cm, where $a>0$, which expression gives $\dfrac{dr}{dt}$?一个球形气球被充气，其体积以恒定速率 $k$ cm³/s 增大。在半径为 $a$ cm 的瞬间（$a>0$），下列哪个表达式表示 $\dfrac{dr}{dt}$？
Q6HARD4.4 Related Rates (Shadow)4.4 相关变化率（影子）Calculator
A $6$-ft-tall person walks away from a $15$-ft lamppost at $4$ ft/s. The tip of their shadow moves at一个身高 $6$ 英尺的人以 $4$ 英尺/秒的速率远离一根高 $15$ 英尺的路灯。其影子尖端移动的速率为
Let $f$ be twice-differentiable with $f''<0$ on an open interval around $x=a$. The tangent-line approximation to $f$ near $x=a$ is设 $f$ 在 $x=a$ 的某开区间上二阶可导且 $f''<0$。$f$ 在 $x=a$ 附近的切线近似是
(A)an overestimate高估值
(B)an underestimate低估值
(C)exact精确值
(D)not determinable from the given information无法根据已知信息确定
Q12HARD4.4 Related Rates (Angle)4.4 相关变化率（角度）Calculator
A spotlight on the ground is $20$ m from a wall. A $2$-m-tall figure walks from the light toward the wall at $1$ m/s. When the figure is $4$ m from the wall, the length of the shadow on the wall changes at a rate closest to地面上有一盏聚光灯，距墙 $20$ 米。一个身高 $2$ 米的人以 $1$ 米/秒的速率从灯处走向墙壁。当此人距墙 $4$ 米时，墙上投影长度的变化速率最接近
A company's cost in dollars for producing $x$ items is $C(x)=0.01x^{2}+20x+500$. The marginal cost at $x=100$ is某公司生产 $x$ 件产品的成本（美元）为 $C(x)=0.01x^{2}+20x+500$。在 $x=100$ 时的边际成本为
Q14MEDIUM5.1 MVT5.1 微分中值定理Preview of Unit 5No Calculator
Note: The Mean Value Theorem is formally CED Topic 5.1 (Unit 5). Included here as a preview because tabular-rate FRQs in Unit 4 routinely cite it.注：微分中值定理正式属于 CED 考点 5.1（第五单元）。此处作为预习内容纳入，因为第四单元的表格式变化率自由回答题中常常会引用该定理。
Which hypothesis is required for the Mean Value Theorem on $[a,b]$?微分中值定理在 $[a,b]$ 上成立所需的条件是什么？
(A)$f$ differentiable on $(a,b)$ and continuous on $[a,b]$.$f$ 在 $(a,b)$ 上可导且在 $[a,b]$ 上连续。
A particle moves along a horizontal line. At time $t=2$ seconds its velocity is $v(2)=-3$ m/s and its acceleration is $a(2)=5$ m/s$^{2}$. Consider the three statements below.一个质点沿水平直线运动。在 $t=2$ 秒时，其速度为 $v(2)=-3$ m/s，加速度为 $a(2)=5$ m/s$^{2}$。考虑以下三个命题。
I.The particle is moving toward the left at $t=2$.在 $t=2$ 时质点向左运动。
II.The particle is speeding up at $t=2$.在 $t=2$ 时质点在加速。
III.The particle's speed is decreasing at $t=2$.在 $t=2$ 时质点的速率在减小。
The height $H$ of a seedling, in centimetres, is measured at time $t$ hours. Selected values are given.某幼苗的高度 $H$（单位：厘米）在时刻 $t$（单位：小时）测得，部分数据如下。
$t$ (h)（小时）
0
2
4
6
$H(t)$ (cm)（厘米）
20
26
30
32
The tangent line to the graph of $H$ at $t=4$ is used to approximate $H(5)$. Based on the data, this approximation is most likely利用 $H$ 的图像在 $t=4$ 处的切线来近似 $H(5)$。根据上述数据，该近似值最可能是
(A)an overestimate, because the data suggest $H$ is concave down偏大，因为数据表明 $H$ 的图像向下凹（concave down）
(B)an underestimate, because the data suggest $H$ is concave down偏小，因为数据表明 $H$ 的图像向下凹（concave down）
(C)an overestimate, because the data suggest $H$ is concave up偏大，因为数据表明 $H$ 的图像向上凹（concave up）
(D)an underestimate, because the data suggest $H$ is concave up偏小，因为数据表明 $H$ 的图像向上凹（concave up）
Q22MEDIUM4.4 Related Rates (Parametric Reverse)4.4 相关变化率（参数反向）No Calculator
At an instant when a spherical balloon has radius $a$ cm, where $a>0$, its radius is increasing at a constant rate of $k$ cm/s. Which expression gives $\dfrac{dV}{dt}$ at that instant?在某一瞬间，球形气球的半径为 $a$ cm（$a>0$），且半径以恒定速率 $k$ cm/s 增大。下列哪个表达式表示该瞬间的 $\dfrac{dV}{dt}$？
Contextual FRQs require labeled units, clear statement of what the derivative means in context, and justification using signs of $v$ and $a$ for motion problems. For related-rates, include a diagram, equation relating quantities, and differentiation with respect to time.情境类自由回答题要求标注单位、明确说明导数在情境中的含义，并在运动问题中用 $v$ 和 $a$ 的符号进行论证。对于相关变化率问题，需画图、建立各量之间的方程，并对时间求导。
FRQ 1MEDIUM4.2 Motion4.2 运动No Calculator
A particle moves along the $x$-axis with velocity $v(t)=3t^{2}-12t+9$ for $t\ge 0$ (in m/s).一个质点沿 $x$ 轴运动，速度函数为 $v(t)=3t^{2}-12t+9$（单位：m/s），$t\ge 0$。
(a)Find all times $t\ge 0$ when the particle is at rest.求所有 $t\ge 0$ 时质点静止的时刻。
(b)On which interval(s) is the particle moving right?质点在哪个（些）区间上向右运动？
(c)At $t=2$, is the particle speeding up or slowing down? Justify using the signs of $v$ and $a$.在 $t=2$ 时，质点是在加速还是减速？用 $v$ 和 $a$ 的符号进行论证。
A water trough is $10$ ft long with cross-section an isosceles triangle ($2$ ft wide at top, $2$ ft deep). Water fills the trough at $3$ ft³/min.一个水槽长 $10$ 英尺，横截面为等腰三角形（顶部宽 $2$ 英尺，深 $2$ 英尺）。水以 $3$ ft³/min 的速率注入水槽。
(a)Express $V$, the water volume, in terms of the depth $h$. Justify using similar triangles.将水的体积 $V$ 用水深 $h$ 表示，并用相似三角形进行推导。
(b)At the instant $h=1$ ft, find $\dfrac{dh}{dt}$. Include units.在 $h=1$ 英尺的瞬间，求 $\dfrac{dh}{dt}$，并标注单位。
(c)Is the water surface width increasing faster or slower when $h=1$ ft vs $h=1.5$ ft? Justify.当 $h=1$ 英尺与 $h=1.5$ 英尺时，水面宽度的增大速率哪个更快？请说明理由。
A tank holds $G(t)$ gallons of water at time $t$ minutes. Selected values of $G$:一个储罐在 $t$ 分钟时储有 $G(t)$ 加仑水。$G$ 的部分取值如下：
$t$ (min)
$0$
$2$
$4$
$6$
$8$
$G(t)$ (gal)
$120$
$108$
$88$
$60$
$24$
(a)Estimate $G'(4)$ using a symmetric difference quotient. Include units and interpret.用对称差商估算 $G'(4)$，标注单位并解释其含义。
(b)Must there be a time $t\in(0,8)$ for which $G'(t)=-12$? Justify using the MVT.在 $(0,8)$ 内是否必然存在某时刻 $t$ 使得 $G'(t)=-12$？用微分中值定理进行论证。(MVT is formally Unit 5, Topic 5.1: see Q14/Q15 preview above.)（微分中值定理正式属于第五单元考点 5.1，参见上方 Q14/Q15 预习内容。）
(c)Use the linearization of $G$ at $t=4$ to estimate $G(4.5)$. Is the estimate likely an overestimate or underestimate, based on the concavity suggested by the table? Justify.用 $G$ 在 $t=4$ 处的线性化估算 $G(4.5)$。根据表格所提示的凹凸性，该估算是高估还是低估？请说明理由。
Evaluate each limit. For each, state the indeterminate form before applying L'Hôpital's Rule, and justify each step.求下列各极限。对每一题，在应用洛必达法则前先说明不定式的类型，并对每一步进行论证。
(d)Explain why L'Hôpital's Rule fails to evaluate $\displaystyle\lim_{x\to \infty}\dfrac{x+\sin x}{x}$ even though it is an $\tfrac{\infty}{\infty}$ form, then compute the limit directly.说明为何尽管 $\displaystyle\lim_{x\to \infty}\dfrac{x+\sin x}{x}$ 为 $\tfrac{\infty}{\infty}$ 型，洛必达法则却无法用于求该极限，然后直接计算该极限。
FRQ 6HARDFRQ LEVEL4.2 Motion from a Velocity Graph4.2 由速度图像分析运动No Calculator
A particle moves along a horizontal line for $0\le t\le 8$ seconds. The graph of its velocity $v$, in metres per second, consists of three line segments and is shown below.一个质点在 $0\le t\le 8$ 秒内沿水平直线运动。其速度 $v$（单位：米每秒）的图像由三条线段组成，如下图所示。
(a)Find all times $t$ in $(0,8)$ at which the particle changes direction. Justify your answer.求所有使质点改变运动方向的时刻 $t\in(0,8)$，并说明理由。
(b)Find all intervals on $(0,8)$ during which the particle is speeding up. Justify your answer using the signs of $v$ and $a$.求质点在 $(0,8)$ 上加速的所有区间，并利用 $v$ 与 $a$ 的符号加以论证。
(c)Find the acceleration of the particle at $t=5$. Include units, and explain what this value means in the context of the particle's motion.求质点在 $t=5$ 时的加速度，写明单位，并结合质点运动的情境解释该值的含义。
(d)A student claims that the particle attains its maximum speed only at $t=4$. Evaluate this claim, with justification.某学生声称质点仅在 $t=4$ 时达到最大速率。请评价该说法并给出论证。