Second Derivative

Core Sentence / 核心句

The second derivative measures how the first derivative, or slope, changes with the input.

中文理解：一阶导数看斜率，二阶导数看斜率怎样变化。

Formula / 公式或规则

$$f^{\prime \prime }(x)=\frac{d}{dx}{f}^{\prime }(x)=\frac{d^{2}f}{d{x}^2}$$

Graph Meaning / 图像含义

$$f^{\prime \prime }>0\Rightarrow \text{slope increasing},f^{\prime \prime }<0\Rightarrow \text{slope decreasing}$$

Motion Meaning / 运动含义

$$a(t)=s^{\prime \prime }(t)$$

Watch Out / 易错点

$f^{\prime \prime }>0$ 不代表 $f$ increasing；它只代表 $f^{\prime }$ increasing。$f^{\prime \prime }(c)=0$ 也不能单独证明 inflection point。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第6次正课 · First Versus Second Derivative
  • second-derivative signs control concavity, inflection analysis, and a local-extremum test
• Chapter 10 · Changing Slope
• Chapter 10 · Acceleration

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Inflection Points
• First Derivative Test
• 10 - Higher Order Derivatives
• Higher-Order Derivatives
• Derivatives
• Concavity
• Acceleration
• Taylor Series