Exploring Behaviors of Implicit Relations · 探索隐式关系的行为
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| implicit relations/ɪmˈplɪsɪt rɪˈleɪʃnz/ | 隐关系 | yǐn guān xì |
| implicit differentiation/ɪmˈplɪsɪt ˌdɪfəˌrenʃɪˈeɪʃn/ | 隐函数求导 | yǐn hán shù qiú dǎo |
Analyzing curves that aren't functions
- Circles, ellipses, and other implicit relations 隐关系 fail the vertical-line test — they aren't $y=f(x)$.
- But the calculus tools still work through implicit differentiation 隐函数求导.
- You can find their tangent slopes, spot horizontal and vertical tangents, and even their concavity.
- This lesson applies Unit 3's implicit method to Unit 5's analysis questions.
分析不是函数的曲线
- 圆、椭圆及其他隐关系通不过竖直线检验——它们不是 $y=f(x)$。
- 但微积分工具仍能通过隐函数求导发挥作用。
- 你能求它们的切线斜率,找出水平和竖直切线,甚至它们的凹凸性。
- 这一课把第 3 单元的隐式方法用到第 5 单元的分析问题上。
Horizontal tangents: numerator zero
- From implicit differentiation you get $\tfrac{dy}{dx}$ as a fraction.
- A horizontal tangent occurs where $\tfrac{dy}{dx}=0$ — set the numerator to zero (with denominator $\neq0$).
- For $x^2+y^2=25$: $\tfrac{dy}{dx}=-\tfrac{x}{y}=0$ when $x=0$, i.e. at $(0,5)$ and $(0,-5)$ (top and bottom).
- These are the highest and lowest points of the curve.
水平切线:分子为零
- 由隐函数求导,你得到 $\tfrac{dy}{dx}$ 作为一个分数。
- 水平切线出现在 $\tfrac{dy}{dx}=0$ 之处——令分子为零(且分母 $\neq0$)。
- 对 $x^2+y^2=25$:$\tfrac{dy}{dx}=-\tfrac{x}{y}=0$ 当 $x=0$,即在 $(0,5)$ 和 $(0,-5)$(顶和底)。
- 这些是曲线的最高和最低点。
Tangents on a circle · 圆上的切线
y = √(25 − x²) (upper half) · y = √(25 − x²)（上半部分）
On $x^2+y^2=25$ the top and bottom have horizontal tangents; the left and right have vertical tangents. · 在$x^2+y^2=25$上，顶部和底部有水平切线；左侧和右侧有垂直切线。
A horizontal tangent on an implicit curve occurs where $\tfrac{dy}{dx}$... · 隐式曲线上的水平切线出现在$\tfrac{dy}{dx}$...
Horizontal ⇔ slope $0$ ⇔ numerator of $\tfrac{dy}{dx}$ is zero. · 水平 ⇔ 斜率$0$ ⇔ $\tfrac{dy}{dx}$的分子为零。
For · 支持 $x^2+y^2=25$ ($\tfrac{dy}{dx}=-\tfrac{x}{y}$), select all · 所有 points with a horizontal tangent. · 对于 $x^2+y^2=25$ ($\tfrac{dy}{dx}=-\tfrac{x}{y}$)，选择所有具有水平切线的点。
$\tfrac{dy}{dx}=0$ when $x=0$: the top and bottom points. · 当 $\tfrac{dy}{dx}=0$ 时 $x=0$：最高点和最低点。
Vertical tangents: denominator zero
- A vertical tangent occurs where $\tfrac{dy}{dx}$ is undefined — set the denominator to zero (with numerator $\neq0$).
- For $x^2+y^2=25$: $-\tfrac{x}{y}$ is undefined when $y=0$, i.e. at $(5,0)$ and $(-5,0)$ (left and right).
- The tangent line is vertical there — the curve turns straight up and down.
- Numerator zero → horizontal; denominator zero → vertical.
竖直切线:分母为零
- 竖直切线出现在 $\tfrac{dy}{dx}$ 无定义之处——令分母为零(且分子 $\neq0$)。
- 对 $x^2+y^2=25$:$-\tfrac{x}{y}$ 在 $y=0$ 时无定义,即在 $(5,0)$ 和 $(-5,0)$(左和右)。
- 那里切线竖直——曲线笔直地上下转。
- 分子为零 → 水平;分母为零 → 竖直。
A vertical tangent occurs where the ____ of $\tfrac{dy}{dx}$ is zero. · 垂直切线出现在$\tfrac{dy}{dx}$的____为零的地方。
Denominator zero ⇒ slope undefined ⇒ vertical tangent. · 分母为零 ⇒ 斜率未定义 ⇒ 垂直切线。
Setting the denominator of $\tfrac{dy}{dx}$ to zero finds tangents that are... · 令 $\tfrac{dy}{dx}$ 的分母为零可找到与...相切的切线
Denominator zero ⇒ vertical tangent (numerator zero ⇒ horizontal). · 分母为零 ⇒ 垂直切线（分子为零 ⇒ 水平切线）。
Concavity: the implicit second derivative
- To get concavity, differentiate $\tfrac{dy}{dx}$ again, implicitly, to find $\tfrac{d^2y}{dx^2}$.
- Remember every $y$ still depends on $x$, so the chain rule reappears, and you substitute the known $\tfrac{dy}{dx}$.
- The sign of $\tfrac{d^2y}{dx^2}$ gives concave up/down at a point, just as before.
- It's more algebra, but the same ideas: $f''>0$ up, $f''<0$ down.
凹凸性:隐式二阶导数
- 要得到凹凸性,对 $\tfrac{dy}{dx}$ 再次隐式求导,求出 $\tfrac{d^2y}{dx^2}$。
- 记住每个 $y$ 仍依赖 $x$,所以链式法则再次出现,并代入已知的 $\tfrac{dy}{dx}$。
- $\tfrac{d^2y}{dx^2}$ 的符号给出某点的上凹/下凹,和之前一样。
- 代数更多,但思想相同:$f''>0$ 上凹,$f''<0$ 下凹。
A candidate point must satisfy the original equation (lie on the curve) to count. · 候选点必须满足原始方程（位于曲线上）才算数。
Check the point is actually on the curve, not just the slope condition. · 检查该点是否确实在曲线上，而不仅仅是斜率条件。
To find concavity on an implicit curve you... · 要在隐函数曲线上找到凹凸性，你...
The implicit second derivative $\tfrac{d^2y}{dx^2}$ gives concavity by its sign. · 隐式二阶导数 $\tfrac{d^2y}{dx^2}$ 通过其符号给出凹凸性。
For horizontal tangents set the numerator of $\tfrac{dy}{dx}$ to zero; for vertical tangents set the denominator to zero — don't swap them. And a point only qualifies if it actually lies on the curve: check it satisfies the original equation, not just the derivative condition.
求水平切线时令 $\tfrac{dy}{dx}$ 的分子为零;求竖直切线时令分母为零——别弄反。而且一个点只有当它确实在曲线上才算数:检查它满足原方程,而不只是导数条件。
Find the horizontal and vertical tangents of $x^2+y^2=25$.
- $2x+2y\tfrac{dy}{dx}=0\Rightarrow\tfrac{dy}{dx}=-\tfrac{x}{y}$.
- Horizontal ($\tfrac{dy}{dx}=0$): $x=0\Rightarrow(0,5),(0,-5)$.
- Vertical ($\tfrac{dy}{dx}$ undefined): $y=0\Rightarrow(5,0),(-5,0)$.
求 $x^2+y^2=25$ 的水平和竖直切线。
- $2x+2y\tfrac{dy}{dx}=0\Rightarrow\tfrac{dy}{dx}=-\tfrac{x}{y}$。
- 水平($\tfrac{dy}{dx}=0$):$x=0\Rightarrow(0,5),(0,-5)$。
- 竖直($\tfrac{dy}{dx}$ 无定义):$y=0\Rightarrow(5,0),(-5,0)$。
Analyze an implicit relation with implicit differentiation: $\tfrac{dy}{dx}=0$ (numerator zero) gives horizontal tangents; $\tfrac{dy}{dx}$ undefined (denominator zero) gives vertical tangents. Differentiate again implicitly for $\tfrac{d^2y}{dx^2}$ and read concavity from its sign. Always confirm points lie on the curve.
用隐函数求导分析隐关系:$\tfrac{dy}{dx}=0$(分子为零)给水平切线;$\tfrac{dy}{dx}$ 无定义(分母为零)给竖直切线。再次隐式求导得 $\tfrac{d^2y}{dx^2}$,从其符号读凹凸性。永远确认点在曲线上。