IB Math Review Set D: Functions and Calculus | IB数学复习题集D：函数与微积分

📚 IB Math Review Set D: Functions and Calculus | IB数学复习题集D：函数与微积分

This review set brings together the essential topics in functions and introductory calculus that frequently appear in IB Mathematics: Analysis and Approaches. The problems in Review Set D are designed to test your conceptual understanding, algebraic manipulation, and application of derivatives and integrals. Whether you are preparing for a unit test or the final examination, a solid grasp of these core ideas will sharpen your problem‑solving skills and boost your confidence. Each section below corresponds to a key skill area that you will encounter in the set.

本复习题集汇聚了IB数学分析与方法课程中函数与微积分入门部分的常考内容。Review Set D 的题目旨在检验你的概念理解、代数运算能力以及导数和积分的应用。无论你是在准备单元测验还是大考，扎实掌握这些核心知识都将提升你的解题能力和自信心。下面每一节对应你在习题集中将会遇到的关键技能领域。

---

1. Domain and Range | 定义域与值域

Understanding the domain of a function means identifying all real x‑values for which the function is defined. For rational functions, exclude values that make the denominator zero; for square root functions, ensure the radicand is non‑negative. The range is the set of all possible output values, which can often be found by considering the behaviour of the graph or by solving for x in terms of y.

理解函数的定义域就是确定所有使函数有意义的实数 x。对于有理函数，要排除使分母为零的值；对于平方根函数，要确保被开方数非负。值域是所有可能的输出值的集合，通常可以通过观察图像变化趋势或用 y 表示 x 来求解。

For example, given f(x) = √x + 4 / (x − 1), the domain requires x + 4 ≥ 0 and x − 1 ≠ 0, giving x ∈ [−4, 1) ∪ (1, ∞).

例如，给定 f(x) = √x + 4 / (x − 1)，定义域要求 x + 4 ≥ 0 且 x − 1 ≠ 0，得到 x ∈ [−4, 1) ∪ (1, ∞)。

• Domain restrictions: denominators, even roots, logarithms.
• 定义域限制：分母、偶次根、对数。
• Range analysis: sketch graph, use inverse, or apply transformations.
• 值域分析：画图、利用反函数或运用变换。

---

2. Composite and Inverse Functions | 复合函数与反函数

The composite function (g ◦ f)(x) = g(f(x)) applies f first, then g. The domain of g ◦ the domain of g. Inverse functions reverse the effect of the original function: f⁻¹(y) = x ⇔ f(x) = y. A function must be one‑to‑one on its domain to have an inverse; restricting the domain can help.

复合函数 (g ◦ f)(x) = g(f(x)) 先作用 f，再作用 g。g ◦ f 的定义域由 f 的定义域中那些使得 f(x) 落在 g 的定义域内的 x 组成。反函数逆反原函数的作用：f⁻¹(y) = x ⇔ f(x) = y。函数在其定义域上必须是一一映射才具有反函数；通过限制定义域可以解决这个问题。

y = f(x) and then solve for y. The graphs of f and f⁻¹ are symmetric about the line y = x.

代数上求反函数时，将 y = f(x) 中的 x 与 y 互换，然后解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。

Common exam problems: given f and g in symbolic form, find (f ◦ g)(x), state its domain, and verify that (f⁻¹ ◦ f)(x) = x.

常见考题：给出符号形式的 f 和 g，求 (f ◦ g)(x)，说明其定义域，并验证 (f⁻¹ ◦ f)(x) = x。

---

3. Graph Transformations | 图像变换

= a f(x) stretches vertically by factor a; y = f(b x) compresses horizontally by factor 1/|b|. Reflections occur when a or b is negative.

变换可以让你平移、拉伸、压缩或翻折函数的基本图像。主要规则是：y = f(x) + d 垂直平移 d；y = f(x + c) 水平平移 −c；y = a f(x) 垂直拉伸 a 倍；y = f(b x) 水平压缩到 1/|b|。当 a 或 b 为负时发生翻折。

tart with the horizontal shift, then horizontal stretch/compression, then vertical stretch/compression, and finally the vertical shift.

组合变换需要注意顺序。通常先进行水平变换（函数内部的运算），再进行垂直变换。若函数写为 y = a f(b(x + c)) + d，顺序为水平平移、水平伸缩、垂直伸缩，最后是垂直平移。

shifting right by π/2, compressing horizontally by factor 1/3, stretching vertically by factor 2, and shifting up by 1.

例如，y = 2 sin(3(x − π/2)) + 1 的图像可由 y = sin x 先右移 π/2，水平压缩 1/3，垂直拉伸 2 倍，再上移 1 而得到。

---

4. Quadratic Functions and the Discriminant | 二次函数与判别式

roots; Δ = 0 gives one repeated real root; Δ < 0 yields no real roots (two complex roots).

二次函数 f(x) = ax² + bx + c 的图像为抛物线。判别式 Δ = b² − 4ac 揭示了根的性质：Δ > 0 有两个不同的实根；Δ = 0 有一个重根；Δ < 0 没有实根（两个复数根）。

the square converts standard form to vertex form. The axis of symmetry is x = h = −b/(2a).

顶点式 f(x) = a(x − h)² + k 能直观显示顶点 (h, k)。通过配方法可以将一般式化为顶点式。对称轴为 x = h = −b/(2a)。

Intersection of a quadratic and a line can be analysed by setting them equal and examining the discriminant of the resulting quadratic equation. This is a favourite IB problem style.

通过将二次函数与直线联立并考察所得二次方程的判别式，可以分析两者的交点情况。这是IB中常见的题型。

---

5. Limits and Continuity | 极限与连续性

The limit of a function f(x) as x approaches a, written lim_x→a f(x), describes the value that f(x) approaches. For a limit to exist, the left‑hand and right‑hand limits must be equal. A function is continuous at x = a if lim_x→a f(x) = f(a).

函数 f(x) 当 x 趋近于 a 时的极限，记作 lim_x→a f(x)，描述的是 f(x) 趋近的值。极限存在的条件是左极限和右极限相等。若 lim_x→a f(x) = f(a)，则函数在 x = a 处连续。

Algebraic techniques such as factoring, rationalising, or using the squeeze theorem help evaluate indeterminate forms like 0/0. For rational functions, if substitution gives a 0/0 form, it often indicates a common factor that can be cancelled.

代数技巧如因式分解、有理化或使用夹逼定理可以帮助求未定式 0/0 的极限。对于有理函数，若代入后得到 0/0，通常表明存在可以约去的公因式。

Understanding continuity is essential for justifying when a derivative exists; differentiability implies continuity, but continuity does not guarantee differentiability.

理解连续性是论证导数存在的基础；可导必连续，但连续不一定可导。

---

6. Introduction to Differentiation | 导数入门

The derivative f′(x) is defined as the limit: f′(x) = lim_h→0 . Extensions include constant multiples, sums, and the chain rule for composite functions.

导数 f′(x) 定义为极限：f′(x) = lim_h→0 [f(x+h) − f(x)] / h。它表示瞬时变化率，也是切线的斜率。幂函数求导法则：若 f(x) = xⁿ，则 f′(x) = n xⁿ⁻¹。扩展包括常数倍、和差公式以及复合函数的链式法则。

Common derivatives:
d/dx (sin x) = cos x, d/dx (cos x) = −sin x,
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x.

常用导数：
d/dx (sin x) = cos x, d/dx (cos x) = −sin x,
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x。

v + u v′. Quotient rule: (u/v)′ = (u′ v − u v′) / v². Chain rule: dy/dx = (dy/du)×(du/dx).

积的求导法则：(u v)′ = u′ v + u v′。商的求导法则：(u/v)′ = (u′ v − u v′) / v²。链式法则：dy/dx = (dy/du)×(du/dx)。

---

广告 / Sponsored

7. Tangents and Normals | 切线与法线

a). The normal line is perpendicular to the tangent, so its slope is −1/f′(a) provided f′(a) ≠ 0. Equations of normals follow the point‑slope form similarly.

曲线在点 (a, f(a)) 处的切线斜率为 f′(a)。切线方程为 y − f(a) = f′(a)(x − a)。法线与切线垂直，因此其斜率为 −1/f′(a)（当 f′(a) ≠ 0）。法线方程同样使用点斜式。

) or where it is parallel to a given line. You may also be asked to show that a line is tangent to a curve by equating the function and line and their respective slopes at the point of contact.

很多IB题目要求你找出切线水平的点（f′(x) = 0）或与已知直线平行的切线。也可能会要求你通过同时让函数值相等且导数相等来证明某条直线与曲线相切。

---

8. Stationary Points and Optimization | 驻点与最优化

if f″(x) = 0, the test is inconclusive and the first derivative sign table should be used.

驻点出现在 f′(x) = 0 的位置。它们可以分为局部极大值、局部极小值或拐点。二阶导数判别法使用 f″(x)：若在驻点处 f″(x) > 0，则为局部极小值；若 f″(x) < 0，则为局部极大值；若 f″(x) = 0，则此法无效，需使用一阶导数符号表。

Optimization problems involve finding the maximum or minimum value of a quantity described by a function, often with a constraint. Steps: write the quantity to be optimised as a function of one variable, differentiate, find stationary points, and justify the nature using a sign diagram or second derivative.

最优化问题要求你找出由函数表示的某个量的最大值或最小值，通常带有一个约束条件。步骤为：将待优化量表示为单变量函数，求导，找出驻点，并用符号图或二阶导数说明其性质。

Typical IB example: a box with square base and volume V has surface area A. Express A in terms of the base side length, differentiate, and find the dimensions that minimise material used.

典型的IB例题：一个正方形底、体积为 V 的盒子，表面积为 A。用底面边长表示 A，求导，找出用料最省的尺寸。

---

9. Basic Integration | 基本积分

or n ≠ −1. Standard integrals include ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ eˣ dx = eˣ + C, and ∫ 1/x dx = ln|x| + C.

积分是微分的逆过程。不定积分 ∫ f(x) dx = F(x) + C，其中 F′(x) = f(x)。对于幂函数，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。标准积分包括 ∫ sin x dx = −cos x + C、∫ cos x dx = sin x + C、∫ eˣ dx = eˣ + C 以及 ∫ 1/x dx = ln|x| + C。

Integration rules: constant multiple, sum rule. There is no simple product or quotient rule for integration; you often need to simplify expressions first. For composite functions, consider using the reverse chain rule or substitution.

积分法则：常数倍、和差法则。积分没有简单的积或商法则；通常需要先对表达式进行化简。对于复合函数，可考虑使用逆向链式法则或换元法。

tion dy/dx = f(x) involves integrating and then using an initial condition to solve for C.

求解微分方程 dy/dx = f(x) 的特解需要先积分，然后利用初始条件来确定常数 C。

---

10. Definite Integrals and Area | 定积分与面积

The definite integral ∫ₐᵇ f(x) dx represents the net area bounded by y = f(x), the x‑ amental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a) where F is any antiderivative of f.

定积分 ∫ₐᵇ f(x) dx 表示由 y = f(x)、x 轴以及垂直线 x = a 和 x = b 围成的净面积。使用微积分基本定理计算：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的任一原函数。

(x) − g(x)] dx. When the curves cross, you must split the interval into subintervals where the upper and lower functions swap.

曲线之间的面积：若在 [a, b] 上 f(x) ≥ g(x)，则两者之间的面积为 ∫ₐᵇ [f(x) − g(x)] dx。当曲线相交时，必须将区间分为几段，分别确定上下函数。

Property: ∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx. Also, area below the x‑ inite integral, so absolute values or separate integrals are used to calculate total area.

性质：∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx。此外，x 轴下方的面积在定积分中会取负值，因此计算总面积时需使用绝对值或分段积分。

---

11. Kinematics and Rate of Change | 运动学与变化率

y integrating velocity: s(t) = ∫ v(t) dt, using an initial condition to find the constant.

在运动学中，若位移 s(t) 是时间的函数，则速度 v(t) = s′(t)，加速度 a(t) = v′(t) = s″(t)。反之，可通过积分速度得到位移：s(t) = ∫ v(t) dt，并利用初始条件求出常数。

Total distance travelled differs from displacement. If velocity changes sign, integrate |v(t)| or sum the areas of the velocity‑time graph. Many IB problems connect differentiation and integration in a real‑world motion context.

总路程与位移不同。若速度改变符号，需对 |v(t)| 积分或将速度‑时间图下的面积相加。许多IB题目在现实运动的语境中将微分与积分联系起来。

Rate of change questions often ask: "Find the rate at which ... is increasing when ...". Set up a relationship equation, differentiate with respect to time (implicitly), and substitute known rates.

变化率问题常问：“当...时，求...的增长率”。先建立关系式，对时间求导（隐函数求导），再代入已知速率。

---

12. Exam Technique and Common Pitfalls | 考试技巧与常见误区

Success in Review Set D relies on more than memorising formulas. Always pay attention to domain restrictions when composing functions, and check that your inverse function passes the horizontal line test on the restricted domain. When differentiating, watch for chain rule applications, especially inside trigonometric, exponential, and logarithmic expressions.

在 Review Set D 中获得成功不仅靠记忆公式。复合函数时务必注意定义域限制，并检查你的反函数在限制域上是否通过水平线检验。求导时，要留意链式法则的应用，特别是三角函数、指数和对数内部的复合部分。

In integration, do not forget the constant +C for indefinite integrals, and use parentheses carefully when evaluating definite integrals to avoid sign errors. Always sketch a rough graph to visualise the area being calculated.

积分时，不要忘记不定积分的常数 +C；计算定积分时括号要仔细使用以避免符号错误。总是画一个简图来直观感受所要计算的面积。

s: confusing f⁻¹(x) with 1/f(x), misapplying the quotient rule, ignoring the need to write "ln|x|" r ∫ 1/x dx, and forgetting to switch limits when using substitution in definite integrals.

常见误区：将 f⁻¹(x) 与 1/f(x) 混淆、错误地运用商法则、在 ∫ 1/x dx 中忘记写成 ln|x|、以及在定积分换元时忘记变换积分限。

Work systematically, show all steps clearly, and always interpret your numerical answers in the context of the problem. These habits will help you tackle the variety of challenges in Review Set D.

条理清晰地解题，展示完整步骤，并始终将数值结果放在问题背景中解释。这些习惯将帮助你从容应对 Review Set D 中的各种挑战。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）