Unit 5: Analytical Applications
of Differentiation第 5 单元：导数的分析应用
Move beyond computation to formal reasoning. This unit develops the analytical tools — theorems, tests, and justifications — needed to draw rigorous conclusions about function behavior from derivatives.超越单纯的计算，迈入形式化推理。本单元构建分析工具——定理、判别法与论证——以便从导数严谨地推断函数行为。
Using the Mean Value Theorem运用中值定理
The Mean Value Theorem (MVT) is one of calculus's most powerful existence theorems. It bridges the gap between average behavior and instantaneous behavior, guaranteeing that somewhere on an interval, a function's instantaneous rate of change matches its average rate of change.中值定理（Mean Value Theorem，MVT）是微积分中最强大的存在性定理之一。它在平均行为与瞬时（instantaneous）行为之间架起桥梁，保证在某个区间内，函数的瞬时变化率（rate of change）等于平均变化率。
If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c$ in $(a, b)$ such that:若 $f$ 在 $[a, b]$ 上连续（continuous）且在 $(a, b)$ 上可导（differentiable），则至少存在一个 $c \in (a, b)$，使得：
In other words, the theorem guarantees the existence of a point where the tangent line is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$.换言之，该定理保证存在某点处，切线（tangent line）与连接 $(a, f(a))$ 和 $(b, f(b))$ 的割线（secant line）平行。
Checking Hypotheses检查前提条件
Before applying the MVT, you must verify both conditions: continuity on the closed interval $[a, b]$ and differentiability on the open interval $(a, b)$. If either condition fails, the conclusion is not guaranteed. On the AP Exam, always state explicitly that these conditions are met before drawing a conclusion.应用 MVT 之前，必须验证两个条件：在闭区间 $[a, b]$ 上连续，并在开区间 $(a, b)$ 上可导。若任一条件不成立，结论不成立。在 AP 考试中，作结论前应当明确说明这些条件已满足。
A strong MVT justification names the function, confirms continuity and differentiability, computes the average rate of change, and then concludes that there exists a value $c$ with $f'(c)$ equal to that average. Use the theorem's name by name.一份完整的 MVT 论证要点名函数、确认连续性与可导性、计算平均变化率，最后得出结论：存在某 $c$ 使 $f'(c)$ 等于该平均值。务必直接称呼定理名称。
| Learning Objective学习目标 | Essential Knowledge核心知识 |
|---|---|
FUN-1.B Justify conclusions about functions by applying the Mean Value Theorem over an interval.通过在某区间上应用中值定理，论证关于函数的结论。 |
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then MVT guarantees a point in the open interval where the instantaneous rate of change equals the average rate of change over the interval.若 $f$ 在 $[a,b]$ 上连续且在 $(a,b)$ 上可导，则 MVT 保证开区间内存在一点，使瞬时变化率等于该区间上的平均变化率。 |
slope）等于割线斜率。Mean Value Theorem Explorer中值定理交互演示
Move the interval endpoints on a fixed graph of $f(x)=x^3-3x$. The maroon line is the secant over $[a,b]$. The purple tangent is drawn at a point $c$ where $f'(c)$ matches the secant slope.在 $f(x)=x^3-3x$ 的固定图像上拖动区间端点。栗色线是 $[a,b]$ 上的割线，紫色切线绘制在某点 $c$ 处，此处 $f'(c)$ 与割线斜率相等。
Extreme Value Theorem, Global vs. Local Extrema, and Critical Points极值定理、全局与局部极值、临界点
The Extreme Value Theorem (EVT) provides another existence guarantee: a continuous function on a closed interval must attain both an absolute maximum and an absolute minimum. This simple-sounding result is the foundation for all optimization analysis.极值定理（Extreme Value Theorem，EVT）给出了另一个存在性保证：闭区间上的连续函数必能取到至少一个绝对最大值（absolute maximum）和至少一个绝对最小值（absolute minimum）。这一看似简单的结论是所有最优化（optimization）分析的基础。
If $f$ is continuous on $[a, b]$, then $f$ attains at least one minimum value and at least one maximum value on $[a, b]$.若 $f$ 在 $[a, b]$ 上连续，则 $f$ 在 $[a, b]$ 上至少取得一个最小值和一个最大值。
Critical Points临界点
A critical point of $f$ is a point $x = c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist. Critical points are the only candidates for local extrema—but not every critical point produces one. A critical point where the derivative changes sign yields a local extremum; one where it does not (such as an inflection point with a horizontal tangent) does not.$f$ 的临界点（critical point）是其定义域中满足 $f'(c) = 0$ 或 $f'(c)$ 不存在的点 $x = c$。临界点是局部极值（local extremum）的唯一候选——但并非每个临界点都产生极值。导数在该点变号才会产生局部极值；若不变号（如带水平切线的拐点（inflection point）），则不构成极值。
Not every critical point is an extremum. For example, $f(x) = x^3$ has $f'(0) = 0$, but $x = 0$ is neither a local max nor a local min. Always apply a test—first or second derivative—before concluding.并非每个临界点都是极值。例如 $f(x) = x^3$ 满足 $f'(0) = 0$，但 $x = 0$ 既不是局部最大值也不是局部最小值。下结论前务必应用判别法——一阶导数判别法（first derivative test）或二阶导数判别法（second derivative test）。
| Learning Objective学习目标 | Essential Knowledge核心知识 |
|---|---|
FUN-1.C Justify conclusions about functions by applying the Extreme Value Theorem.通过应用极值定理论证关于函数的结论。 |
EVT guarantees a min and max on $[a,b]$ if $f$ is continuous. Critical points occur where $f'=0$ or $f'$ DNE. All local extrema occur at critical points, but not all critical points are local extrema.若 $f$ 连续，EVT 保证在 $[a,b]$ 上取到最小值和最大值。临界点出现在 $f'=0$ 或 $f'$ 不存在的位置。所有局部极值都出现在临界点，但并非所有临界点都是局部极值。 |
Determining Intervals on Which a Function Is Increasing or Decreasing判定函数的递增与递减区间
The first derivative is the primary tool for analyzing where a function rises and falls. The sign of $f'(x)$ on an interval directly tells you the behavior of $f$ on that interval.一阶导数（first derivative）是分析函数上升与下降的主要工具。$f'(x)$ 在某区间上的符号直接表征 $f$ 在该区间的行为。
If $f'(x) > 0$ for all $x$ in an interval, then $f$ is increasing on that interval. If $f'(x) < 0$ for all $x$ in an interval, then $f$ is decreasing on that interval.若区间内所有 $x$ 满足 $f'(x) > 0$，则 $f$ 在该区间上递增（increasing）。若区间内所有 $x$ 满足 $f'(x) < 0$，则 $f$ 在该区间上递减（decreasing）。
Process流程
To determine intervals of increase or decrease: find all critical points of $f$ (where $f' = 0$ or $f'$ does not exist), then test the sign of $f'$ in each interval between consecutive critical points. Use a sign chart or number line as a workspace, but remember that on the AP Exam, sign charts alone are not sufficient justification—you must express your reasoning in sentences.判定递增或递减区间的流程：找出 $f$ 的所有临界点（$f' = 0$ 或 $f'$ 不存在之处），然后检验 $f'$ 在相邻临界点之间各区间上的符号。可用符号表（sign chart）或数轴作为草稿，但在 AP 考试中，仅凭符号表不足以作为论证——必须用完整句子表达推理。
When asked whether $f$ is increasing or decreasing at a point, frame your answer in terms of the derivative: "$f$ is increasing on $(a, b)$ because $f'(x) > 0$ for all $x$ in $(a, b)$." Never say "it's going up" without referencing the derivative.被问及 $f$ 在某点附近递增或递减时，应以导数措辞作答："$f$ 在 $(a, b)$ 上递增，因为对所有 $x \in (a, b)$ 有 $f'(x) > 0$"。切勿仅说"图像向上"而不引用导数。
Using the First Derivative Test to Determine Relative (Local) Extrema运用一阶导数判别法确定相对（局部）极值
The First Derivative Test uses sign changes of $f'$ around a critical point to classify it as a local maximum, local minimum, or neither.一阶导数判别法（first derivative test）通过 $f'$ 在临界点附近的变号情况，将该临界点判定为局部最大值、局部最小值或两者皆非。
If $c$ is a critical point of $f$ and $f'$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$. If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$. If $f'$ does not change sign, then $c$ is neither.若 $c$ 是 $f$ 的临界点，且 $f'$ 在 $c$ 处由正变负，则 $f$ 在 $c$ 取得局部最大值。若 $f'$ 在 $c$ 处由负变正，则 $f$ 在 $c$ 取得局部最小值。若 $f'$ 不变号，则 $c$ 既非最大值也非最小值。
Writing a Complete Justification书写完整论证
A full justification requires three elements: (1) identify $c$ as a critical point, (2) describe the sign change of $f'$ around $c$, and (3) state the conclusion using the name of the test. For instance: "$f'(c) = 0$, and $f'$ changes from positive to negative at $x = c$, so by the First Derivative Test, $f$ has a local maximum at $x = c$."完整论证包含三要素：(1) 指出 $c$ 是临界点；(2) 描述 $f'$ 在 $c$ 附近的变号；(3) 引用判别法名称给出结论。例如："$f'(c) = 0$，且 $f'$ 在 $x = c$ 处由正变负，因此由一阶导数判别法，$f$ 在 $x = c$ 取得局部最大值。"
Apply an appropriate mathematical definition, theorem, or test. When justifying extrema, choose between the First Derivative Test and Second Derivative Test based on what information is available.应用恰当的数学定义、定理或判别法。论证极值时，根据已有信息在一阶导数判别法与二阶导数判别法之间作出选择。
Using the Candidates Test to Determine Absolute (Global) Extrema运用候选点法确定绝对（全局）极值
When a function is continuous on a closed interval $[a, b]$, the Extreme Value Theorem guarantees absolute extrema exist. The Candidates Test is the procedure for finding them.当函数在闭区间 $[a, b]$ 上连续时，极值定理保证全局极值（absolute extremum）存在。候选点法（candidates test）就是寻找它们的标准流程。
On a closed interval $[a, b]$: (1) Find all critical points of $f$ in $(a, b)$. (2) Evaluate $f$ at each critical point and at both endpoints $a$ and $b$. (3) The largest value is the absolute maximum; the smallest is the absolute minimum.在闭区间 $[a, b]$ 上：(1) 找出 $f$ 在 $(a, b)$ 内的所有临界点；(2) 在每个临界点和两端点 $a$、$b$ 处计算 $f$ 的值；(3) 最大值即为绝对最大值，最小值即为绝对最小值。
This method works because absolute extrema on a closed interval can only occur at critical points or endpoints—there are no other possibilities. The test is straightforward but requires careful computation; do not forget to evaluate the function at the endpoints.此方法之所以有效，是因为闭区间上的绝对极值只能出现在临界点或端点处——别无他处。流程虽直接，但计算需谨慎；切勿忘记在端点处计算函数值。
Students sometimes forget to check the endpoints. On a closed interval, the absolute max or min may occur at an endpoint rather than at a critical point. Always evaluate $f(a)$ and $f(b)$.学生常忘记检查端点。在闭区间上，绝对最大值或最小值可能出现在端点而非临界点处。务必计算 $f(a)$ 和 $f(b)$。
Determining Concavity of Functions over Their Domains在定义域上判定函数的凹凸性
Concavity describes the curvature of a graph. While the first derivative tells us direction (increasing/decreasing), the second derivative tells us how the rate of change itself is changing—the shape of the curve.凹凸性（concavity）描述图像的弯曲方式。一阶导数告诉我们方向（递增/递减），二阶导数（second derivative）则告诉我们变化率本身如何变化——即曲线的形状。
If $f''(x) > 0$ on an interval, then $f$ is concave up on that interval, so the graph bends upward like a cup. If $f''(x) < 0$ on an interval, then $f$ is concave down on that interval, so the graph bends downward like a frown.若 $f''(x) > 0$ 在某区间成立，则 $f$ 在该区间上凹（concave up），图像如杯状向上弯曲。若 $f''(x) < 0$ 在某区间成立，则 $f$ 在该区间下凹（concave down），图像如倒置的杯状向下弯曲。
Equivalently, $f$ is concave up on an interval where $f'$ is increasing, and concave down where $f'$ is decreasing. This connection between $f'$ behavior and $f$ concavity is essential for graph analysis.等价地，$f$ 在 $f'$ 递增的区间上上凹，在 $f'$ 递减的区间上下凹。$f'$ 行为与 $f$ 凹凸性之间的这一联系是图像分析的核心。
Points of Inflection拐点
A point of inflection is a point where the concavity of $f$ changes. At an inflection point, $f''$ must either equal zero or fail to exist—but as with critical points, $f'' = 0$ alone does not guarantee an inflection point. You must confirm that concavity actually changes by checking the sign of $f''$ on either side.拐点（inflection point）是 $f$ 凹凸性发生变化的位置。拐点处 $f''$ 必须等于零或不存在——但与临界点类似，仅 $f'' = 0$ 并不能保证拐点存在。必须通过检验 $f''$ 在两侧的符号确认凹凸性确实发生变化。
Take $f(x) = x^4$. Then $f'(x) = 4x^3$ and $f''(x) = 12x^2$.取 $f(x) = x^4$，则 $f'(x) = 4x^3$，$f''(x) = 12x^2$。
- $f''(0) = 0$ — so $x = 0$ is a candidate for an inflection point.$f''(0) = 0$ —— 所以 $x = 0$ 是拐点候选。
- But $f''(x) = 12x^2 \ge 0$ for every real $x$. The sign of $f''$ never changes.然而对所有实数 $x$ 都有 $f''(x) = 12x^2 \ge 0$。$f''$ 的符号从不变化。
- Therefore $f$ is concave up everywhere, including at $x = 0$ — there is no inflection point at $x = 0$.因此 $f$ 处处上凹，$x = 0$ 处也不例外——该点不是拐点。
Takeaway:要点： "$f''(c) = 0$" is a necessary condition for an inflection point, but it is not sufficient. You must verify a sign change of $f''$ across $c$. The same caveat applies to $f''(c)$ failing to exist."$f''(c) = 0$" 是拐点的必要条件，但不充分。必须验证 $f''$ 在 $c$ 两侧变号。$f''(c)$ 不存在的情形也有同样的注意事项。
When justifying concavity, always reference $f''$. Say "$f$ is concave up on $(a, b)$ because $f''(x) > 0$ on $(a, b)$," not just "the graph is concave up." Precision in naming $f$, $f'$, and $f''$ is critical.论证凹凸性时必须引用 $f''$。要写"$f$ 在 $(a, b)$ 上上凹，因为 $f''(x) > 0$ 在 $(a, b)$ 上成立"，而不仅是"图像上凹"。精确称呼 $f$、$f'$、$f''$ 至关重要。
Using the Second Derivative Test to Determine Extrema运用二阶导数判别法确定极值
The Second Derivative Test provides an alternative to the First Derivative Test for classifying critical points. It is often more efficient when the second derivative is easy to evaluate.二阶导数判别法（second derivative test）为临界点分类提供了一阶导数判别法之外的另一种途径。当二阶导数易于计算时，它通常更高效。
If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $x = c$. If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $x = c$. If $f''(c) = 0$, the test is inconclusive so you should switch to the First Derivative Test.若 $f'(c) = 0$ 且 $f''(c) > 0$，则 $f$ 在 $x = c$ 取得局部最小值。若 $f'(c) = 0$ 且 $f''(c) < 0$，则 $f$ 在 $x = c$ 取得局部最大值。若 $f''(c) = 0$，则该判别法无法判定，应改用一阶导数判别法。
One Critical Point on an Interval区间上仅有一个临界点
When a continuous function has exactly one critical point on an interval and that critical point is a local extremum, it is also the absolute extremum on that interval. This is a powerful shortcut in optimization problems.当连续函数在某区间上恰有一个临界点且该临界点是局部极值时，它也是该区间上的绝对极值。这在最优化问题中是一条强有力的捷径。
Use the First Derivative Test when you already have sign information about $f'$ or when $f''$ is difficult to compute. Use the Second Derivative Test when $f''$ is readily available and you simply need to check its sign at a critical point.已知 $f'$ 的符号信息或 $f''$ 难以计算时，使用一阶导数判别法。$f''$ 现成可用且只需在临界点处检查其符号时，使用二阶导数判别法。
Sketching Graphs of Functions and Their Derivatives绘制函数及其导数的图像
This topic ties together everything you've learned about derivatives and graph behavior. You should be able to move fluidly between the graph of $f$, the graph of $f'$, and the graph of $f''$.本主题串联了关于导数与图像行为的所有内容。应当能在 $f$、$f'$、$f''$ 的图像之间自如切换。
Key features of a function's graph—intervals of increase/decrease, local extrema, concavity, and inflection points—can all be identified from $f'$ and $f''$. Conversely, given a graph of $f$, you should be able to sketch $f'$.函数图像的关键特征——递增/递减区间、局部极值、凹凸性、拐点——都可由 $f'$ 和 $f''$ 识别。反之，给定 $f$ 的图像，应能绘出 $f'$ 的图像。
Reading $f$ from $f'$从 $f'$ 读出 $f$
Where $f' > 0$, $f$ is increasing. Where $f' < 0$, $f$ is decreasing. Where $f'$ changes sign, $f$ has a local extremum. Where $f'$ is increasing, $f$ is concave up. Where $f'$ is decreasing, $f$ is concave down. Where $f'$ has an extremum, $f$ has an inflection point.$f' > 0$ 处，$f$ 递增；$f' < 0$ 处，$f$ 递减；$f'$ 变号处，$f$ 取得局部极值；$f'$ 递增处，$f$ 上凹；$f'$ 递减处，$f$ 下凹；$f'$ 取得极值处，$f$ 有拐点。
Do not confuse properties of $f'$ with properties of $f$. When looking at the graph of $f'$, a positive $y$-value means $f$ is increasing—not that $f$ is positive. Always be explicit: "Because $f'(x) > 0$…" rather than "because it is positive…"切勿混淆 $f'$ 与 $f$ 的性质。看 $f'$ 的图像时，正的 $y$ 值意味着 $f$ 递增——而非 $f$ 为正。务必表述清晰："由于 $f'(x) > 0$……"，而非"由于它为正……"。
Connecting a Function, Its First Derivative, and Its Second Derivative连接函数、一阶导数与二阶导数
This topic consolidates the relationships among $f$, $f'$, and $f''$ into a unified analytical framework. Being able to translate information between these three levels is arguably the most important skill in this unit.本主题将 $f$、$f'$、$f''$ 之间的关系整合为统一的分析框架。能在这三个层级之间转换信息，可以说是本单元最重要的技能。
| Feature of $f$$f$ 的特征 | Indicated by $f'$由 $f'$ 表征 | Indicated by $f''$由 $f''$ 表征 |
|---|---|---|
| Increasing递增 | $f' > 0$ | — |
| Decreasing递减 | $f' < 0$ | — |
| Local max局部最大值 | $f'$ changes + to −$f'$ 由 + 变 − | $f''(c) < 0$ at critical point临界点处 $f''(c) < 0$ |
| Local min局部最小值 | $f'$ changes − to +$f'$ 由 − 变 + | $f''(c) > 0$ at critical point临界点处 $f''(c) > 0$ |
| Concave up上凹 | $f'$ increasing$f'$ 递增 | $f'' > 0$ |
| Concave down下凹 | $f'$ decreasing$f'$ 递减 | $f'' < 0$ |
| Inflection point拐点 | $f'$ has extremum$f'$ 取得极值 | $f''$ changes sign$f''$ 变号 |
Identify how mathematical characteristics or properties of functions are related in different representations. Practice translating between graphs, tables, and analytic expressions of $f$, $f'$, and $f''$.识别函数的数学特征或性质在不同表示间的关联。练习在 $f$、$f'$、$f''$ 的图像、表格与解析表达式之间相互转换。
Connecting $f$, $f'$, and $f''$ on One Fixed Window在同一固定窗口中连接 $f$、$f'$、$f''$
Adjust the cubic coefficient $k$ in $f(x)=x^3-3kx$. The graph windows stay fixed so you can focus on how zeros, extrema, and concavity relationships shift together.调整 $f(x)=x^3-3kx$ 中的三次项系数 $k$。图像窗口保持固定，以便集中观察零点、极值与凹凸性关系如何同步变化。
Function $f$函数 $f$
First derivative $f'$一阶导数 $f'$
Introduction to Optimization Problems最优化问题简介
Optimization problems ask you to find the maximum or minimum value of a quantity subject to given constraints. These are among the most applied and most tested topics in AP Calculus. The key challenge is translating a word problem into a single-variable function that you can analyze with calculus.最优化（optimization）问题要求在给定约束（constraint）下，求某量的最大值或最小值。这是 AP 微积分中最具应用性、考查频率最高的主题之一。关键难点在于将文字题转化为单变量函数，再用微积分分析。
(1) Identify the quantity to be optimized. (2) Write it as a function of one or more variables. (3) Use a constraint equation to reduce to a single variable. (4) Find critical points. (5) Determine which critical point yields the desired extremum. (6) Answer the question that was asked.(1) 确定待优化的量。(2) 将其写成一个或多个变量的函数。(3) 利用约束方程化为单变量。(4) 求临界点。(5) 判断哪个临界点对应所求极值。(6) 回答题目所问的问题。
Common optimization contexts include maximizing area or volume given a constraint on perimeter or material, minimizing distance or cost, and finding dimensions that optimize a geometric quantity.常见的最优化情境包括：在周长或材料约束下最大化面积或体积、最小化距离或成本、求使某几何量最优的尺寸。
Solving Optimization Problems求解最优化问题
This topic extends the setup from Topic 5.10 into full solutions with interpretation. On the AP Exam, you are expected not just to find the optimal value, but to explain what it means in context.本主题将主题 5.10 的建模扩展为带有情境解释的完整解答。在 AP 考试中，不仅要求出最优值，还要解释其在情境中的含义。
Minimum and maximum values of a function take on specific meanings in applied contexts. After computing, always circle back to the original question and state your answer in terms of the scenario described.函数的最小值与最大值在应用情境中具有具体含义。计算完成后，务必回到原题，用情境的语言陈述结论。
A complete optimization solution must justify why the critical value is indeed a maximum or minimum. Use the First or Second Derivative Test, or the Candidates Test on a closed interval. Simply finding $f'(c) = 0$ is not sufficient—you must confirm the nature of the extremum.完整的最优化解答必须论证临界值确实为最大或最小。可使用一阶或二阶导数判别法，或在闭区间上使用候选点法。仅得到 $f'(c) = 0$ 不够——必须确认极值的类型。
Exploring Behaviors of Implicit Relations探究隐式关系的行为
The analytical techniques developed throughout this unit—finding critical points, determining increasing/decreasing behavior, analyzing concavity—extend naturally to implicitly defined functions. The key difference is that derivatives obtained via implicit differentiation are typically expressed in both $x$ and $y$.本单元发展的分析技巧——求临界点、判定递增/递减、分析凹凸性——可自然推广到隐式定义的函数。主要差异在于：通过隐式求导（implicit differentiation）得到的导数通常同时含 $x$ 与 $y$。
For an implicitly defined function, a critical point occurs where $\frac{dy}{dx} = 0$ or $\frac{dy}{dx}$ does not exist. Since $\frac{dy}{dx}$ is usually a fraction involving $x$ and $y$, set the numerator equal to zero (for $\frac{dy}{dx} = 0$) or analyze where the denominator equals zero (for $\frac{dy}{dx}$ DNE).对于隐式函数，临界点出现在 $\frac{dy}{dx} = 0$ 或 $\frac{dy}{dx}$ 不存在的位置。由于 $\frac{dy}{dx}$ 通常是含 $x$、$y$ 的分式，令分子为零（对应 $\frac{dy}{dx} = 0$），或分析分母为零的位置（对应 $\frac{dy}{dx}$ 不存在）。
Second Derivatives of Implicit Relations隐式关系的二阶导数
When computing $\frac{d^2y}{dx^2}$ for an implicit relation, the result typically involves $x$, $y$, and $\frac{dy}{dx}$. You may substitute the expression for $\frac{dy}{dx}$ to simplify. The second derivative can then be used to determine concavity and apply the Second Derivative Test, just as with explicit functions.对隐式关系计算 $\frac{d^2y}{dx^2}$ 时，结果通常含 $x$、$y$ 和 $\frac{dy}{dx}$。可将 $\frac{dy}{dx}$ 的表达式代入以化简。所得二阶导数可像处理显式函数那样，用于判定凹凸性并应用二阶导数判别法。
Apply appropriate mathematical rules or procedures, and provide reasons or rationales for solutions and conclusions. With implicit relations, precision in algebraic manipulation and careful chain rule application are essential.应用恰当的数学规则或步骤，并为解答与结论提供理由或依据。处理隐式关系时，精确的代数运算与谨慎应用链式法则（chain rule）至关重要。
Exam Strategy考试策略
Always refer to $f$, $f'$, and $f''$ by name. Never write "it" or "the function" when referencing a derivative. Ambiguity costs points. If the question names the function $g$, use $g$, $g'$, and $g''$ consistently.始终以名称指代 $f$、$f'$、$f''$。引用导数时切勿写"它"或"该函数"。歧义会扣分。若题目将函数记为 $g$，则始终一致地使用 $g$、$g'$、$g''$。
If a question asks "Does $g$ have a relative minimum, maximum, or neither at $x = 10$? Justify your answer," begin your response by repeating the framework: "$g$ has a relative minimum at $x = 10$ because…"若题目问"$g$ 在 $x = 10$ 处是相对最小值、相对最大值还是两者皆非？说明理由"，作答应当沿用其句式开头："$g$ 在 $x = 10$ 处取得相对最小值，因为……"
When given a graph of $f'$, make arguments about $f$ based directly on the graph of the derivative. For example: "$f$ is concave up on $a < x < b$ because the graph of $f'$ is increasing on $a < x < b$." This is more reliable than trying to read the graph of $f$ itself.给定 $f'$ 的图像时，应直接基于导数的图像论证 $f$。例如："$f$ 在 $a < x < b$ 上上凹，因为 $f'$ 的图像在 $a < x < b$ 上递增。"这比试图读出 $f$ 自身的图像更可靠。
Before applying any theorem (MVT, EVT), explicitly confirm its hypotheses are satisfied. State continuity and differentiability conditions before drawing conclusions.应用任何定理（MVT、EVT）之前，明确确认其前提满足。在下结论前先陈述连续性与可导性条件。
Finding a critical point is not enough. You must justify that it yields a maximum or minimum using a derivative test or the Candidates Test. A complete answer includes the value and its context interpretation.仅找到临界点不够。必须用导数判别法或候选点法论证它对应最大或最小。完整解答还包括该值及其情境解释。
Common Mistakes常见错误
The most prevalent error. Students read a graph of $f'$ and describe features as if they are features of $f$. For example, seeing that $f'$ is positive and concluding "$f$ is positive" instead of "$f$ is increasing."最常见的错误。学生读 $f'$ 的图像，却把特征当作 $f$ 的特征。例如，看到 $f'$ 为正就得出"$f$ 为正"而非"$f$ 递增"。
Vague language like "it is increasing" can be interpreted as referring to $f$ or $f'$. Always name the function explicitly."它递增"这类含糊措辞既可指 $f$ 也可指 $f'$。务必明确点名所指函数。
A critical point is only an extremum if the derivative changes sign (First Derivative Test) or if the second derivative test confirms it. $f'(c) = 0$ alone is insufficient.临界点只有在导数变号（一阶导数判别法）或二阶导数判别法确认的情形下才是极值。仅有 $f'(c) = 0$ 不充分。
On a closed interval, always evaluate $f$ at both endpoints. The absolute extremum may occur at an endpoint, not at an interior critical point.在闭区间上务必计算 $f$ 在两端点的值。绝对极值可能出现在端点而非内部临界点。
Saying "the graph goes up" is a description, not a justification. A justification uses calculus reasoning: "$f$ is increasing because $f'(x) > 0$."说"图像上升"只是描述，并非论证。论证须使用微积分推理："$f$ 递增，因为 $f'(x) > 0$"。
You must confirm continuity on $[a,b]$ and differentiability on $(a,b)$ before applying the Mean Value Theorem. Skipping this step means your argument is incomplete.应用中值定理前，必须确认 $[a,b]$ 上连续、$(a,b)$ 上可导。跳过这一步意味着论证不完整。
Flashcards闪卡
Click a card to reveal the answer.点击卡片查看答案。
Unit Quiz单元测验
Test your understanding. Select the best answer for each question.检验掌握情况。每题选择最佳答案。
1. The Mean Value Theorem guarantees the existence of a value $c$ in $(a,b)$ such that $f'(c)$ equals the average rate of change over $[a,b]$. Which conditions must hold?1. 中值定理保证 $(a,b)$ 内存在某 $c$ 使 $f'(c)$ 等于 $[a,b]$ 上的平均变化率。需要哪些条件？
2. If $f'(c) = 0$ and $f''(c) = 0$, what can you conclude about $f$ at $x = c$?2. 若 $f'(c) = 0$ 且 $f''(c) = 0$，关于 $f$ 在 $x = c$ 处可得出什么结论？
3. On the interval $(-2, 5)$, $f'(x) < 0$ for all $x$. Which statement is true?3. 在 $(-2, 5)$ 上对所有 $x$ 有 $f'(x) < 0$。哪一项陈述正确？
4. A continuous function $f$ on $[1, 7]$ has critical points at $x = 3$ and $x = 5$. To find the absolute maximum of $f$, which values must you compare?4. 连续函数 $f$ 在 $[1, 7]$ 上在 $x = 3$ 和 $x = 5$ 处有临界点。要求 $f$ 的绝对最大值，必须比较哪些值？
5. The graph of $f'$ is increasing on $(a, b)$. What does this tell you about $f$?5. $f'$ 的图像在 $(a, b)$ 上递增。这告诉你关于 $f$ 的什么？
Readiness Checklist备考清单
Click each item you've mastered. Aim for 100% before exam day.点击已掌握的条目。考试前争取达到 100%。
- State and apply the Mean Value Theorem with all three hypotheses陈述并应用中值定理，包含其全部前提
- State and apply the Extreme Value Theorem陈述并应用极值定理
- Identify critical points from $f'(x) = 0$ and $f'(x)$ undefined从 $f'(x) = 0$ 和 $f'(x)$ 未定义处识别临界点
- Determine intervals of increase / decrease from the sign of $f'$由 $f'$ 的符号判定递增 / 递减区间
- Apply the First Derivative Test to classify local extrema应用一阶导数判别法判定局部极值
- Use the Candidates Test to find absolute extrema on a closed interval用候选点法在闭区间上求绝对极值
- Determine concavity from the sign of $f''$由 $f''$ 的符号判定凹凸性
- Apply the Second Derivative Test and know when it is inconclusive应用二阶导数判别法，并清楚何时无法判定
- Locate and justify points of inflection定位并论证拐点
- Sketch $f$ from features of $f'$ and $f''$ (and vice versa)从 $f'$、$f''$ 的特征绘出 $f$（反之亦然）
- Read and interpret graphs of $f$, $f'$, $f''$ together同时阅读并解读 $f$、$f'$、$f''$ 的图像
- Set up an optimization problem from a word description根据文字描述建立最优化问题
- Solve optimization using an appropriate extremum test运用恰当的极值判别法求解最优化问题
- Apply extrema analysis to implicit relations将极值分析应用于隐式关系
- Write a full verbal justification referencing the relevant theorem/test写出完整的文字论证，引用相关定理或判别法
AP-Style Practice ProblemsAP 风格练习题
Exam-level practice for this unit — multiple-choice plus extended-response items modeled on the AP rubric. Built for top-score prep; go here after you've worked through the notes and the in-page quizzes above.本单元的考试级练习——多项选择题加按 AP 评分标准命制的简答题。为冲刺高分而设；建议完成上面的笔记与页内测验后再前往。