Prerequisite Math Knowledge for AP Calculus BC | 微积分BC所需高中数学基础知识点归纳

📚 Prerequisite Math Knowledge for AP Calculus BC | 微积分BC所需高中数学基础知识点归纳

Success in AP Calculus BC depends heavily on a strong command of foundational high school mathematics. Before tackling limits, derivatives, integrals, and series, students must be fluent in algebra, functions, trigonometry, and analytic geometry. This guide consolidates the essential precalculus topics that form the backbone of calculus. Mastering these will build confidence and reduce common mistakes when more advanced concepts are introduced.

要在AP微积分BC中取得成功，扎实的高中数学基础至关重要。在学习极限、导数、积分和级数之前，学生必须熟练掌握代数、函数、三角学和解析几何。本文梳理了构成微积分核心的必备预微积分知识点。掌握这些内容将建立信心，并减少在引入更高级概念时常犯的错误。

1. Algebraic Manipulation and Exponents | 代数运算与指数

Expanding polynomials: use the distributive property and FOIL to multiply binomials and polynomials accurately.

(x + a)(x + b) = x² + (a + b)x + ab

多项式展开：运用分配律和FOIL方法准确计算二项式与多项式的乘积。

Factoring: master common factor extraction, difference of squares, perfect square trinomials, sum and difference of cubes, and grouping.

a² − b² = (a − b)(a + b), a³ ± b³ = (a ± b)(a² ∓ ab + b²)

因式分解：熟练掌握提取公因式、平方差、完全平方三项式、立方和与立方差以及分组分解法。

Laws of exponents: apply product, quotient, power of a power, zero, and negative exponent rules fluently.

aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ (a ≠ 0)

指数定律：熟练运用乘法法则、除法法则、幂的幂、零指数和负指数规则。

Radicals and rational exponents: convert between radical form and fractional exponents and simplify expressions.

ⁿ√a = a^(1/n), ⁿ√(aᵐ) = a^(m/n)

根式与有理指数：掌握根式与分数指数的互化，并能化简含根式的表达式。

Rationalizing denominators: eliminate radicals from denominators using conjugates or appropriate multiplication.

分母有理化：利用共轭式或恰当的乘法消除分母中的根号。

---

2. Solving Equations and Inequalities | 方程与不等式求解

Linear equations and inequalities: isolate variables and handle sign changes when multiplying or dividing by a negative number.

线性方程与不等式：正确移项，并注意乘除负数时不等号方向的改变。

Quadratic equations: solve by factoring, completing the square, and the quadratic formula.

ax² + bx + c = 0 ⇒ x = [−b ± √(b² − 4ac)] / (2a)

二次方程：会用因式分解、配方法和求根公式求解。

Polynomial and rational equations: use factoring, common denominators, and the zero-product property to find solutions.

多项式方程与分式方程：通过因式分解、通分以及零乘积性质求根，并注意排除使分母为零的值。

Systems of equations: solve linear and non-linear systems using substitution and elimination methods.

方程组：掌握代入法和消元法求解线性方程组及含二次方程的非线性方程组。

Inequalities and sign charts: solve polynomial and rational inequalities by testing intervals on a number line.

不等式与符号表：在数轴上通过区间测试法解多项式不等式和分式不等式。

Absolute value equations and inequalities: interpret absolute value as distance and split into cases.

绝对值方程与不等式：将绝对值理解为数轴上的距离，并正确分情况拆解。

---

3. Functions: Definitions and Properties | 函数：定义与性质

Function definition: a relation where each input has exactly one output, represented as f(x).

函数定义：每个输入值对应唯一输出值的对应关系，通常记作 f(x)。

Domain and range: identify allowable x-values (domain) and possible y-values (range) from equations and graphs.

定义域与值域：能根据解析式和图像确定自变量x的取值范围（定义域）和因变量y的取值范围（值域）。

Composite functions: understand (f ∘ g)(x) = f(g(x)) and evaluate step by step.

(f ∘ g)(x) = f(g(x))

复合函数：理解 (f ∘ g)(x) = f(g(x)) 的含义，并能逐步代入求值。

Inverse functions: a function that ‘undoes’ the original; graph is a reflection over y = x; check with f(f⁻¹(x)) = x.

反函数：从一个函数逆向得出输入的函数；图像关于 y=x 对称；验证 f(f⁻¹(x)) = x。

Even and odd functions: even if f(−x) = f(x) (symmetric about y-axis), odd if f(−x) = −f(x) (symmetric about origin).

奇偶性：若 f(−x)=f(x) 则为偶函数（关于y轴对称）；若 f(−x)=−f(x) 则为奇函数（关于原点对称）。

Transformations of graphs: translate, reflect, stretch, and compress functions using f(x−h)+k, −f(x), af(x), etc.

图像变换：掌握平移 f(x−h)+k、对称 −f(x)、伸缩 af(x) 等函数图像变换规律。

---

4. Polynomial Functions | 多项式函数

Degree and leading coefficient: determine end behavior of polynomial graphs from the sign and parity of the leading term.

次数与首项系数：根据多项式首项的次数奇偶性和系数符号判断图像两端走势。

Zeros and factors: if c is a zero, then (x−c) is a factor; use the Factor Theorem to link roots and factors.

零点与因式：若 c 为零点，则 (x−c) 为因式；运用因式定理建立根与因式的联系。

Remainder Theorem: when dividing f(x) by (x−c), the remainder is f(c).

f(x) ÷ (x−c) ⇒ remainder = f(c)

余式定理：多项式 f(x) 除以 (x−c) 的余式等于 f(c)。

Polynomial long division and synthetic division: divide polynomials to factor and find slant asymptotes later.

多项式长除法与综合除法：用除法分解多项式，为后续寻找斜渐近线打下基础。

Multiplicity of zeros: a zero of multiplicity m makes the graph touch or cross the x-axis differently.

零点重数：重数为m的零点会使图像在x轴上只触及（偶数重）或穿过（奇数重）。

Rational Root Theorem: possible rational roots of a polynomial are of the form ±(factors of constant term)/(factors of leading coefficient).

有理根定理：整系数多项式可能的有理根为 ±(常数项因数)/(首项系数因数)。

---

5. Rational Functions and Asymptotes | 有理函数与渐近线

Vertical asymptotes: occur where the denominator is zero and the numerator is non-zero after simplification.

垂直渐近线：化简后分母为零且分子非零的x值处产生垂直渐近线。

Horizontal asymptotes: determined by comparing degrees of numerator and denominator.

水平渐近线：通过比较分子与分母多项式的次数确定，分三种情况（分子次数小于、等于或大于分母）。

Slant (oblique) asymptotes: when the numerator’s degree is exactly one more than the denominator’s; found by polynomial long division.

斜渐近线：当分子次数比分母恰好高一次时出现，通过多项式长除法求得。

Holes (removable discontinuities): occur at a common factor in numerator and denominator; the function is undefined at that point.

可去间断点（空洞）：分子与分母有公因式时，在该x值处函数无定义，形成图像上的“空洞”。

Partial fraction decomposition: express a rational function as a sum of simpler fractions, often used in integration later.

部分分式：将一个复杂分式拆分为几个简单分式之和，后续积分中经常用到。

Graphing rational functions: combine intercepts, asymptotes, holes, and sign charts to sketch the curve.

有理函数作图：通过截距、渐近线、空洞及符号表综合绘制图像。

---

6. Exponential and Logarithmic Functions | 指数与对数函数

Exponential functions: f(x) = a·bˣ, where b > 0, b ≠ 1; understand growth (b>1) and decay (0

f(x) = a · bˣ, b > 0, b ≠ 1

指数函数：形式为 f(x)=a·bˣ (b>0, b≠1)；区分增长 (b>1) 与衰减 (0

Logarithmic functions: y = log_b(x) means b^y = x; domain x>0, range all reals.

对数函数：y = log_b(x) 等价于 b^y = x；定义域 x>0，值域为全体实数。

Key properties: log_b(MN) = log_b(M)+log_b(N), log_b(M/N)=log_b(M)−log_b(N), log_b(M^p)=p·log_b(M).

更多咨询请联系16621398022（同微信）