📚 Edexcel IAL Pure Mathematics 3 (P3) Complete Revision Guide | 爱德思IAL纯数学3（P3）完整复习指南
Edexcel International A Level Pure Mathematics 3 (P3) is the third pure mathematics unit in the IAL Mathematics qualification. It extends the algebra, trigonometry, and calculus introduced in P1 and P2 and introduces more advanced techniques such as partial fractions, implicit differentiation, integration by parts, numerical methods, and three-dimensional vectors. Success in P3 requires fluency in symbolic manipulation, careful attention to domain and range, and a disciplined approach to multi-step problems.
爱德思国际A Level纯数学3（P3）是IAL数学资格中第三个纯数学单元。它扩展了P1和P2中引入的代数、三角和微积分，并引入了更高级的技巧，例如部分分式、隐函数微分、分部积分、数值方法和三维向量。要在P3中取得成功，需要熟练的符号运算能力、对定义域和值域的仔细关注，以及处理多步骤问题的严谨方法。
1. Algebra and Functions | 代数与函数
) + B/(x-b), then solve for A and B by comparing coefficients or substituting convenient x-values.
P3的代数重点是分解有理函数为部分分式、处理绝对值函数以及熟练应用图像变换。对于具有不同线性因子的有理式，可写成 f(x)/((x-a)(x-b)) = A/(x-a) + B/(x-b)，然后通过比较系数或代入方便的x值来求解A和B。
The modulus function |x| is defined as x when x ≥ 0 and -x when x < 0. Equations such as |2x-3| = 5 split into 2x-3 = 5 or 2x-3 = -5. Inequalities like |x+1| < 4 are rewritten as -4 < x+1 < 4.
绝对值函数|x|定义为当x≥0时为x，当x<0时为-x。例如 |2x-3| = 5 可拆分为 2x-3 = 5 或 2x-3 = -5。形如 |x+1| < 4 的不等式可改写为 -4 < x+1 < 4。
1/a; y = af(x) is a vertical stretch by scale factor a; y = -f(x) reflects in the x-axis.
关键图像变换包括平移、伸缩和对称。y = f(x+a) 是水平平移 -a；y = f(ax) 是水平伸缩，比例为 1/a；y = af(x) 是垂直伸缩，比例为 a；y = -f(x) 关于 x 轴对称。
| Transformation | 变换 | Effect on graph of y = f(x) | 对 y = f(x) 图像的影响 |
|---|---|
| y = f(x) + c | Vertical translation by c | 垂直平移 c |
| y = f(x + c) | Horizontal translation by -c | 水平平移 -c |
| y = a f(x) | Vertical stretch by scale factor a | 垂直伸缩 a |
| y = f(ax) | Horizontal stretch by scale factor 1/a | 水平伸缩 1/a |
| y = -f(x) | Reflection in the x-axis | 关于 x 轴对称 |
2. Trigonometric Functions and Identities | 三角函数与恒等式
eir definitions are sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = cos θ/sin θ = 1/tan θ.
P3引入了倒数三角函数 sec、cosec 和 cot，并加强了解方程和微积分所需的恒等式熟练度。其定义为 sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = cos θ/sin θ = 1/tan θ。
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
θ, 2cos²θ - 1, and 1 - 2sin²θ. sin 2θ = 2 sin θ cos θ, and tan 2θ = 2 tan θ / (1 - tan²θ).
二倍角公式对于改写乘积和幂次至关重要。cos 2θ 的三种形式为 cos²θ - sin²θ、2cos²θ - 1 和 1 - 2sin²θ。sin 2θ = 2 sin θ cos θ，tan 2θ = 2 tan θ / (1 - tan²θ)。
An expression of the form a sin θ + b cos θ can be written as R sin(θ + α), where R = √a² + b² α = b/a. This is very useful for solving equations and finding maximum and minimum values.
形如 a sin
Published by TutorHao | Exam Prep Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图