Tangent Line
Core Sentence / 核心句
A tangent line gives the local linear direction of a curve at a point.
中文理解：切线不是随便贴一下的线，而是函数在某一点附近最好的线性近似。
Formula / 公式或规则
When To Use / 什么时候用
看到 tangent slope、local linear approximation、derivative at a point，就想到 tangent line。
Watch Out / 易错点
切线斜率是 instantaneous，不是 secant interval average。
Search and Source Mentions / 搜索与出现位置
Mentioned In / 出现位置
- 微积分BC第5次正课 · Tangent-Line Approximation
- the tangent line becomes the nearby first-order model L(x)
- 微积分BC第4次正课 · Tangent-Line Equation
- derivative slope and the point (a,f(a)) determine the local line
- Chapter 6 · Implicit Curve and Tangent
- zooming in turns an implicit curve into its tangent line, whose slope is
- 02 - The Paradox of the Derivative · Knowledge Point 6: Secant Line Becomes Tangent Line
Knowledge Point 6: Secant Line Becomes Tangent Line
- 02 - The Paradox of the Derivative · 4. The derivative is the slope of the tangent line.
4. The derivative is the slope of the tangent line.
- 03 - Derivative Formulas through Geometry · Knowledge Point 3: Why $\frac{d}{dx}x^2=2x$
- Tangent Line：图像上这是 $y=x^2$ 的切线斜率。
- 微积分BC第13次正课-无穷级数判敛 · Knowledge Point A: Polar Slope Means $dy/dx$
slope of the tangent line
- 微积分BC第13次正课-无穷级数判敛 · 9. Exam Language / 关键句中英对照
slope of tangent line -> 瞬时斜率 -> derivative
- 微积分BC第13次正课-无穷级数判敛 · 8. Deep Phrases / 深句对应表
| Slope of the tangent line means $dy/dx$. | 切线斜率就是 $dy/dx$，极坐标也一样。 |
- 02 - The Paradox of the Derivative · 1. What This Episode Is About / 这一集在讲什么
这一集不是只告诉你“导数 = 瞬时变化率”，而是解释这句话为什么表面上有矛盾，以及微积分如何用 limit 和 tangent line 合理地解决这个矛盾。
- 02 - The Paradox of the Derivative · 2. Link to Previous Episode / 和上一集的连接
derivative = slope of tangent line