Implicitly Defined Functions · 隐式定义函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| implicitly defined/ɪmˈplɪsɪtli dɪˈfaɪnd/ | 隐式定义 | yǐn shì dìng yì |
| conic section/ˈkɒnɪk ˈsekʃn/ | 圆锥曲线 | yuán zhuī qū xiàn |
When y refuses to stand alone
- Most functions we meet are explicit: $y$ sits alone, equal to some formula in $x$.
- But an equation like $x^2 + y^2 = 25$ tangles $x$ and $y$ together.
- You cannot cleanly write "$y = \dots$" — yet the equation still describes a perfect curve.
- Such a relationship is defined implicitly, and it opens up a whole family of shapes.
当 y 拒绝单独站立
- 我们遇到的大多数函数是显式的:$y$ 单独站着,等于某个关于 $x$ 的式子。
- 但像 $x^2 + y^2 = 25$ 这样的方程,把 $x$ 和 $y$ 缠在一起。
- 你无法干净地写出"$y = \dots$"——可这个方程仍描述着一条完美的曲线。
- 这样的关系是隐式定义的,它开启了一整个形状家族。
Explicit versus implicit
- An explicit function isolates the output: $y = x^2 - 3$.
- An implicitly defined 隐式定义 relation leaves $x$ and $y$ mixed: $x^2 + y^2 = 25$.
- The implicit form states a condition that points on the curve must satisfy.
- It is often the most natural way to describe a shape.
显式与隐式
- 一个显式函数把输出分离出来:$y = x^2 - 3$。
- 一个隐式定义(implicitly defined)的关系让 $x$ 和 $y$ 混着:$x^2 + y^2 = 25$。
- 隐式形式陈述了曲线上的点必须满足的一个条件。
- 它常常是描述一个形状最自然的方式。
An implicitly defined relationship is one where… · 隐式定义的关系是指……
An implicitly defined curve, like $x^2 + y^2 = 25$, mixes $x$ and $y$ without isolating $y$. · 一条隐式定义的曲线，如$x^2 + y^2 = 25$，混合了$x$和$y$而没有分离出$y$。
A circle, implicitly
- The equation $x^2 + y^2 = 25$ says "the distance from the origin is $5$".
- Every point obeying it lies on a circle of radius $5$ — a classic conic section 圆锥曲线.
- No rearranging is needed to see the shape; the equation already captures it.
- Ellipses, parabolas, and hyperbolas are all defined this same implicit way.
圆的隐式形式
- 方程 $x^2 + y^2 = 25$ 说的是"离原点的距离是 $5$"。
- 每一个满足它的点都落在一个半径为 $5$ 的圆上——一条经典的圆锥曲线(conic section)。
- 不需要移项就能看出形状;方程已经把它抓住了。
- 椭圆、抛物线和双曲线,都用这同样的隐式方式定义。
The equation · 方程 $x^2 + y^2 = 25$ describes… · 方程$x^2 + y^2 = 25$描述了……
All points at distance $5$ from the origin satisfy $x^2 + y^2 = 25$ — a circle of radius $5$. · 所有到原点距离为$5$的点都满足$x^2 + y^2 = 25$——这是一个半径为$5$的圆。
An implicit relation like $x^2 + y^2 = 25$ need not be a function of $x$. · 像$x^2 + y^2 = 25$这样的隐式关系不一定是$x$的函数。
Most $x$-values give two $y$-values (top and bottom of the circle), so it fails the vertical-line test. · 大多数$x$-值对应两个$y$-值（圆的顶部和底部），因此它通过了竖直线测试失败。
Select all · 所有 true statements about implicitly defined relations. · 选择关于隐式关系的所有正确陈述。
Many implicit relations cannot be solved neatly for $y$ at all. The other three are correct. · 许多隐式关系根本无法整齐地解出$y$。其他三项是正确的。
Not a function of x
- For most $x$-values, an implicit curve gives two $y$-values (top and bottom).
- So it usually fails the vertical-line test — it is a relation, not a function.
- $x^2 + y^2 = 25$ solved for $y$ is $y = \pm\sqrt{25 - x^2}$: two semicircle branches.
- Each branch on its own is a function; together they form the whole curve.
不是 x 的函数
- 对大多数 $x$ 值,隐式曲线给出两个 $y$ 值(上和下)。
- 所以它通常通不过垂直线检验——它是一个关系,而不是函数。
- $x^2 + y^2 = 25$ 解出 $y$ 是 $y = \pm\sqrt{25 - x^2}$:两条半圆分支。
- 每条分支单独是一个函数;合起来它们构成整条曲线。
Explicit or implicit? · 显式还是隐式？
An implicit equation relates x and y without solving for y. Sort each. · 隐式方程在不解出y的情况下关联x和y。对每一项进行分类。
Solving $x^2 + y^2 = 25$ for $y$ gives $y = \pm\sqrt{25 - x^2}$ — that is ____ separate branches. · 解出$x^2 + y^2 = 25$关于$y$得到$y = \pm\sqrt{25 - x^2}$——那是____个独立分支。
The $\pm$ splits the circle into an upper and a lower semicircle — two explicit functions. · $\pm$将圆分为上半圆和下半圆——两个显式函数。
Working with implicit curves
- To find points, substitute one coordinate and solve for the other.
- To sketch, recognise the shape (circle, ellipse, …) from the equation's form.
- Some implicit relations cannot be solved for $y$ at all — you work with them as they are.
- In calculus, implicit differentiation finds slopes without ever isolating $y$.
处理隐式曲线
- 要找点,代入一个坐标,解出另一个。
- 要作图,从方程的形式认出形状(圆、椭圆……)。
- 有些隐式关系根本无法解出 $y$——你就按原样处理它们。
- 在微积分里,隐函数求导能在从不分离 $y$ 的情况下求出斜率。
An implicit equation is a relation, not automatically a function. Do not assume you can write $y = f(x)$ — for a circle you get $y = \pm\sqrt{25 - x^2}$, and forgetting the $\pm$ silently drops half the curve.
一个隐式方程是一个关系,并不自动是一个函数。不要假设你能写出 $y = f(x)$——对一个圆你得到 $y = \pm\sqrt{25 - x^2}$,忘了那个 $\pm$ 就悄悄丢掉了一半曲线。
Find the points on $x^2 + y^2 = 25$ where $x = 3$.
- Substitute: $9 + y^2 = 25$, so $y^2 = 16$.
- Then $y = \pm 4$ — two points, $(3, 4)$ and $(3, -4)$.
- One $x$-value, two $y$-values: the mark of a non-function relation.
求 $x^2 + y^2 = 25$ 上 $x = 3$ 处的点。
- 代入:$9 + y^2 = 25$,所以 $y^2 = 16$。
- 于是 $y = \pm 4$——两个点,$(3, 4)$ 和 $(3, -4)$。
- 一个 $x$ 值,两个 $y$ 值:非函数关系的标志。
An implicitly defined relation mixes $x$ and $y$ in one equation, like $x^2 + y^2 = 25$, rather than isolating $y$. It can describe circles and other conic sections, usually is not a function of $x$ (solving gives $\pm$ branches), and is often the most natural way to state a curve.
一个隐式定义的关系把 $x$ 和 $y$ 混在一个方程里,比如 $x^2 + y^2 = 25$,而不是把 $y$ 分离出来。它能描述圆和其他圆锥曲线,通常不是 $x$ 的函数(解出来有 $\pm$ 分支),而且常常是陈述一条曲线最自然的方式。