Sections 1 to 7: critical points, the EVT, the MVT, Rolle, monotonicity, the first and second derivative tests, concavity and inflection, curve sketching, the closed-interval method, applied optimizationCALC I第 1 至 7 节：极值点、极值定理、均值定理、罗尔定理、单调性、一阶与二阶导数判别法、凹凸性与拐点、曲线描绘、闭区间方法、应用最优化CALC I
Name:姓名：Date:日期：
PART I · CORE TECHNIQUES第一部分 · 核心技能Computational fluency · 28 marks计算熟练度 · 28分
Derivatives, Sign Charts, and Curve Analysis导数、符号图与曲线分析
Show all differentiation and factoring. Build a complete sign chart for $f'$ (and for $f''$ where required) before stating any conclusion. Name the test you apply at each critical point.展示全部求导与因式分解过程。在得出任何结论之前，先构建 $f'$（以及需要时 $f''$）的完整符号图。在每个极值点处说明所使用的判别法。
Q1MEDIUMCOREcritical points and the first derivative test极值点与一阶导数判别法[8 marks]
Let $f(x) = x^{4} - 4x^{3} + 4x^{2}$.
(a)Find $f'(x)$ and factor it completely. State all critical points of $f$.求 $f'(x)$ 并完全因式分解。列出 $f$ 的所有极值点。[3]
(b)Construct a sign chart for $f'$ and hence determine the intervals on which $f$ is increasing and the intervals on which $f$ is decreasing.构建 $f'$ 的符号图，并据此确定 $f$ 的单调递增区间和单调递减区间。[3]
(c)Apply the First Derivative Test to each critical point. Classify each as a local maximum, a local minimum, or neither, and justify your answer by reference to the sign change of $f'$.对每个极值点应用一阶导数判别法。将其分类为局部极大值、局部极小值或两者均不是，并以 $f'$ 的符号变化为依据作出说明。[2]
Q2MEDIUMCOREconcavity, inflection points, and the second derivative test凹凸性、拐点与二阶导数判别法[8 marks]
Let $g(x) = x e^{-x^{2}/2}$.
(a)Compute $g'(x)$ and find all critical points of $g$. Apply the Second Derivative Test to classify each one.计算 $g'(x)$，求出 $g$ 的所有极值点。应用二阶导数判别法对每个极值点进行分类。[4]
(b)Compute $g''(x)$ and find all inflection points of $g$. You must verify that $g''$ changes sign at each candidate.计算 $g''(x)$，求出 $g$ 的所有拐点。必须验证 $g''$ 在每个候选点处确实改变符号。[4]
Q3HARDCOREclosed-interval method and the Extreme Value Theorem闭区间方法与极值定理[6 marks]
Let $h(x) = 2x^{3} - 9x^{2} + 12x - 4$ on the closed interval $[-\tfrac{1}{2},\, 4]$.
(a)State why the Extreme Value Theorem guarantees that $h$ attains both an absolute maximum and an absolute minimum on $[-\tfrac{1}{2}, 4]$.说明极值定理为何保证 $h$ 在 $[-\tfrac{1}{2}, 4]$ 上同时取到绝对最大值和绝对最小值。[1]
(b)Find all critical points of $h$ inside $(-\tfrac{1}{2}, 4)$.求 $h$ 在 $(-\tfrac{1}{2}, 4)$ 内的所有极值点。[2]
(c)Evaluate $h$ at each critical point and at each endpoint, and hence state the absolute maximum value and the absolute minimum value of $h$ on $[-\tfrac{1}{2}, 4]$. State where each is attained.在每个极值点和每个端点处计算 $h$ 的值，由此写出 $h$ 在 $[-\tfrac{1}{2}, 4]$ 上的绝对最大值与绝对最小值。注明各自在何处取到。[3]
Q4HARDCOREfull curve sketch of a rational function有理函数的完整曲线描绘[6 marks]
Let $\displaystyle f(x) = \frac{x^{2} - 4}{x^{2} - 1}$. Carry out a complete analysis of $f$ in the order below.设 $\displaystyle f(x) = \frac{x^{2} - 4}{x^{2} - 1}$。按以下顺序对 $f$ 进行完整分析。
(a)State the domain of $f$. Find all intercepts and determine the symmetry of the graph. State all vertical and horizontal asymptotes, with supporting limits.写出 $f$ 的定义域。求所有截距并确定图形的对称性。写出所有竖直渐近线和水平渐近线，并附上相应的极限说明。[2]
(b)Compute $f'(x)$ and construct a complete sign chart for $f'$ (include the asymptote locations as partition points). State the intervals of increase and decrease and classify any local extrema.计算 $f'(x)$，构建 $f'$ 的完整符号图（将渐近线位置作为分割点）。写出单调递增和递减区间，并对所有局部极值进行分类。[2]
(c)Compute $f''(x)$ and determine the concavity of each branch. State the $x$-coordinates of any inflection points, or explain why none exist. Using all of the above, describe the key features of the graph in words and coordinates; do not draw the graph but list: the $y$-intercept, any local extrema with coordinates, each asymptote, and the concavity of each branch.计算 $f''(x)$，确定每段曲线的凹凸性。写出所有拐点的 $x$ 坐标，或说明不存在拐点的原因。综合以上结果，用文字和坐标描述图形的关键特征；不需要作图，但须列出：$y$ 轴截距、所有局部极值及其坐标、各条渐近线，以及每段曲线的凹凸性。[2]
PART II · DEFINITIONS AND PROOF第二部分 · 定义与证明Rigorous arguments · 26 marks严格论证 · 26分
Rolle, the Mean Value Theorem, and Derivative Estimates罗尔定理、均值定理与导数估计
State every hypothesis of the theorem before invoking its conclusion. In inequality proofs, identify the function you apply the MVT to and exhibit the guaranteed $c$ explicitly.在引用定理结论之前，逐一陈述所有假设条件。在不等式证明中，明确说明对哪个函数应用均值定理，并显式给出所保证的 $c$ 值。
Q5HARDPROOFRolle's Theorem: hypotheses, conclusion, and finding $c$罗尔定理：假设条件、结论与求 $c$ 值[8 marks]
Throughout this question, identify $[a, b]$ explicitly and verify each hypothesis.在本题全程，明确指出区间 $[a, b]$，并逐一验证每个假设条件。
(a)State Rolle's Theorem in full, including all three hypotheses.完整陈述罗尔定理，包括全部三个假设条件。[2]
(b)Apply Rolle's Theorem to $p(x) = x^{3} - x$ on the interval $[-1, 1]$. Verify all three hypotheses, and then find every value $c \in (-1, 1)$ guaranteed by the theorem.将罗尔定理应用于 $p(x) = x^{3} - x$ 在区间 $[-1, 1]$ 上。验证全部三个假设条件，然后求定理所保证的所有 $c \in (-1, 1)$ 值。[3]
(c)Use Rolle's Theorem to show that the polynomial $q(x) = x^{5} + 3x + 1$ has at most one real root. (Hint: assume it has two distinct roots $a < b$ and derive a contradiction.)利用罗尔定理证明多项式 $q(x) = x^{5} + 3x + 1$ 至多有一个实数根。（提示：假设它有两个不同的实数根 $a < b$，并推导出矛盾。）[3]
Q6HARDPROOFMean Value Theorem: finding $c$ and proving an inequality均值定理：求 $c$ 值与证明不等式[10 marks]
State which function and which interval you apply the MVT to in each part.在每个小题中，说明对哪个函数和哪个区间应用均值定理。
(a)State the Mean Value Theorem in full.完整陈述均值定理。[2]
(b)For $f(x) = \ln x$ on the interval $[1, e]$, verify the hypotheses of the MVT and find every $c \in (1, e)$ guaranteed by the theorem.对 $f(x) = \ln x$ 在区间 $[1, e]$ 上，验证均值定理的假设条件，并求定理所保证的所有 $c \in (1, e)$ 值。[3]
(c)Prove that $|\sin b - \sin a| \le |b - a|$ for all real numbers $a$ and $b$. State the key property of $\cos$ that you use.证明对所有实数 $a$ 和 $b$ 有 $|\sin b - \sin a| \le |b - a|$。说明你所使用的 $\cos$ 的关键性质。[3]
(d)Suppose $f$ is differentiable on $\mathbb{R}$ with $f(2) = 5$ and $f'(x) \le 3$ for every $x$. Use the MVT to find the best possible upper bound for $f(7)$.设 $f$ 在 $\mathbb{R}$ 上可微，且 $f(2) = 5$，$f'(x) \le 3$ 对所有 $x$ 成立。利用均值定理求 $f(7)$ 的最优上界。[2]
Q7HARDPROOFMVT corollary: zero derivative implies constant均值定理推论：导数为零意味着函数为常数[8 marks]
This question develops the corollary that powers antiderivative uniqueness.本题推导支撑原函数唯一性的推论。
(a)Prove: if $f'(x) = 0$ for every $x$ in an open interval $I$, then $f$ is constant on $I$. Your proof must apply the MVT to two arbitrary points of $I$ and deduce that $f$ takes the same value at both.证明：若 $f'(x) = 0$ 对开区间 $I$ 中的每一个 $x$ 成立，则 $f$ 在 $I$ 上为常数函数。证明中须对 $I$ 中的两个任意点应用均值定理，并推出 $f$ 在两点处取相同的值。[4]
(b)Using part (a), prove: if $f'(x) = g'(x)$ for every $x \in I$, then $f(x) = g(x) + C$ for some constant $C$.利用 (a) 的结论证明：若 $f'(x) = g'(x)$ 对所有 $x \in I$ 成立，则存在常数 $C$ 使得 $f(x) = g(x) + C$。[2]
(c)Use the inequality $\ln x \le x - 1$ for $x > 0$ (which you may assume) to show by the MVT that $e^{x} \ge 1 + x$ for all $x \ge 0$. (Apply the MVT to $e^{t}$ on $[0, x]$.)利用不等式 $\ln x \le x - 1$（$x > 0$ 时，可直接使用此结论），通过均值定理证明 $e^{x} \ge 1 + x$ 对所有 $x \ge 0$ 成立。（对 $[0, x]$ 上的 $e^{t}$ 应用均值定理。）[2]
PART III · APPLICATIONS AND SYNTHESIS第三部分 · 应用与综合Extended optimization problems · 28 marks综合最优化问题 · 28分
Applied Optimization应用最优化
In every problem: name the quantity to optimize, write the objective function in one variable by using the constraint, state the feasible domain, find the critical point(s), and confirm the nature of the optimum by a test. Do not stop at the critical value of an intermediate variable; answer the question that was asked.在每道题中：明确要优化的量，利用约束条件将目标函数写成单变量形式，写出可行域，求极值点，并通过判别法确认极值的性质。不要停在中间变量的极值处；要回答题目所要求的问题。
Q8HARDAPPLIEDmaximum enclosed area with a fixed perimeter固定周长下的最大围合面积[8 marks]
A farmer wants to enclose a rectangular plot of land and then subdivide it into three equal pens by placing two internal fences parallel to one pair of sides. The total length of fencing available (perimeter plus two internal dividers) is $120$ metres.一位农民想围住一块矩形土地，并通过在一对平行边方向放置两道内部围栏，将其分成三个相等的区域。可用围栏的总长度（周长加两道内部隔栏）为 $120$ 米。
(a)Let $x$ be the length of the side perpendicular to the internal dividers and $y$ be the length of the side parallel to them. Write down the constraint relating $x$ and $y$, and hence express the total area $A$ as a function of $x$ alone. State the feasible domain for $x$.设 $x$ 为垂直于内部隔栏方向的边长，$y$ 为平行于内部隔栏方向的边长。写出 $x$ 与 $y$ 之间的约束关系，并由此将总面积 $A$ 表示为仅关于 $x$ 的函数。写出 $x$ 的可行域。[3]
(b)Find $A'(x)$, locate the critical point, and verify it gives a maximum (not a minimum) by applying the Second Derivative Test.求 $A'(x)$，找出极值点，并通过二阶导数判别法验证该点给出最大值（而非最小值）。[3]
(c)State the dimensions $x$ and $y$ of the entire enclosure that maximise the area, and give the maximum area.写出使面积最大的围栏整体尺寸 $x$ 与 $y$，并给出最大面积。[2]
Q9HARDAPPLIEDminimum material for an open-top box无盖箱所用材料最少[10 marks]
A manufacturer must produce open-top rectangular boxes with a square base and a fixed volume of $32\,\text{cm}^{3}$. The base costs $\$3$ per $\text{cm}^{2}$ and each of the four sides costs $\$1$ per $\text{cm}^{2}$. The manufacturer wants to minimise the total material cost.一家制造商须生产正方形底面的无盖矩形箱，固定容积为 $32\,\text{cm}^{3}$。底面成本为每 $\text{cm}^{2}$ $\$3$，四个侧面各为每 $\text{cm}^{2}$ $\$1$。制造商希望最小化总材料成本。
(a)Let the side length of the square base be $x\,\text{cm}$ and the height be $h\,\text{cm}$. Write the volume constraint and hence express $h$ in terms of $x$. State the feasible domain for $x$.设正方形底面的边长为 $x\,\text{cm}$，高为 $h\,\text{cm}$。写出体积约束条件，并由此将 $h$ 表示为 $x$ 的函数。写出 $x$ 的可行域。[2]
(b)Write the total cost $C$ as a function of $x$ alone, combining the base cost and the four-side cost.将总成本 $C$ 表示为仅关于 $x$ 的函数，综合底面成本与四个侧面成本。[2]
(c)Find the critical point of $C$ on $(0, \infty)$. Use the Second Derivative Test (or analyse the sign of $C'$) to confirm it is a minimum.求 $C$ 在 $(0, \infty)$ 上的极值点。使用二阶导数判别法（或分析 $C'$ 的符号）确认该点为极小值点。[4]
(d)State the dimensions $x$ and $h$ that minimise cost, and give the minimum total cost to the nearest cent.写出使成本最小的尺寸 $x$ 与 $h$，并给出精确到分的最小总成本。[2]
Q10HARDAPPLIEDclosest point on a curve to an external point曲线上距外部点最近的点[10 marks]
Find the point(s) on the curve $y = \sqrt{x}$ (with $x \ge 0$) that are closest to the point $(2,\, 0)$.求曲线 $y = \sqrt{x}$（$x \ge 0$）上距点 $(2,\, 0)$ 最近的点。
(a)A point on the curve has coordinates $(x,\, \sqrt{x})$. Write the squared distance $D(x)$ from this point to $(2, 0)$. Explain why minimising $D(x)$ is equivalent to minimising the actual distance. State the feasible domain.曲线上一点的坐标为 $(x,\, \sqrt{x})$。写出该点到 $(2, 0)$ 的距离平方 $D(x)$。解释为何最小化 $D(x)$ 等价于最小化实际距离。写出可行域。[2]
(b)Compute $D'(x)$ and find all critical points of $D$ in the feasible domain. (After differentiating, multiply through by $2\sqrt{x}$ to obtain a polynomial equation.)计算 $D'(x)$，求 $D$ 在可行域内的所有极值点。（求导后，两边乘以 $2\sqrt{x}$ 以得到多项式方程。）[4]
(c)Check the value of $D$ at each critical point and at any relevant boundary. Confirm that the critical point is a minimum by the Second Derivative Test or by comparing values.计算 $D$ 在每个极值点及相关边界处的值。通过二阶导数判别法或比较函数值确认该极值点为极小值点。[2]
(d)State the coordinates of the closest point on $y = \sqrt{x}$ to $(2, 0)$ and give the minimum distance in exact form.写出 $y = \sqrt{x}$ 上距 $(2, 0)$ 最近点的坐标，并以精确形式给出最小距离。[2]